Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence
We generalize Aganagic-Okounkov's theory of elliptic stable envelopes, and its physical realization in Dedushenko-Nekrasov's and Bullimore-Zhang's works, to certain varieties without holomorphic symplectic structure or polarization. These classes of varieties include, in particular, c...
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| Cite this: | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence. Nafiz Ishtiaque, Seyed Faroogh Moosavian and Yehao Zhou. SIGMA 20 (2024), 099, 95 pages |
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| citation_txt | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence. Nafiz Ishtiaque, Seyed Faroogh Moosavian and Yehao Zhou. SIGMA 20 (2024), 099, 95 pages |
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| description | We generalize Aganagic-Okounkov's theory of elliptic stable envelopes, and its physical realization in Dedushenko-Nekrasov's and Bullimore-Zhang's works, to certain varieties without holomorphic symplectic structure or polarization. These classes of varieties include, in particular, classical Higgs branches of 3d = 2 quiver gauge theories. The Bethe/gauge correspondence relates such a gauge theory to an isotropic/elliptic superspin chain, and the stable envelopes compute the -matrix that solves the dynamical Yang-Baxter equation (dYBE) for this spin chain. As an illustrative example, we solve the dYBE for the elliptic (1|1) spin chain with fundamental representations using the corresponding 3d = 2 SQCD whose classical Higgs branch is the Lascoux resolution of a determinantal variety. Certain Janus partition functions of this theory on × for an interval and an elliptic curve compute the elliptic stable envelopes, and in turn the geometric elliptic -matrix, of the anisotropic (1|1) spin chain. Furthermore, we consider the 2d and 1d reductions of elliptic stable envelopes and the -matrix. The reduction to 2d gives the K-theoretic stable envelopes, and the trigonometric -matrix, and a further reduction to 1d produces the cohomological stable envelopes and the rational -matrix. The latter recovers Rimányi-Rozansky's results that appeared recently in the mathematical literature.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 20 (2024), 099, 95 pages
Elliptic Stable Envelopes for Certain Non-Symplectic
Varieties and Dynamical R-Matrices for Superspin
Chains from the Bethe/Gauge Correspondence
Nafiz ISHTIAQUE a, Seyed Faroogh MOOSAVIAN b and Yehao ZHOU c
a) Institut des Hautes Études Scientifiques, 35 Rte de Chartres, 91440 Bures-sur-Yvette, France
E-mail: ishtiaque@ihes.fr
b) Department of Physics, McGill University, Ernest Rutherford Physics Building,
3600 Rue University, Montréal, QC H3A 2T8, Canada
E-mail: sfmoosavian@gmail.com
c) Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo,
Kashiwa, Chiba 277-0882, Japan
E-mail: yehao.zhou@ipmu.jp
Received February 07, 2024, in final form October 11, 2024; Published online October 31, 2024
https://doi.org/10.3842/SIGMA.2024.099
Abstract. We generalize Aganagic–Okounkov’s theory of elliptic stable envelopes, and
its physical realization in Dedushenko–Nekrasov’s and Bullimore–Zhang’s works, to certain
varieties without holomorphic symplectic structure or polarization. These classes of varieties
include, in particular, classical Higgs branches of 3d N = 2 quiver gauge theories. The
Bethe/gauge correspondence relates such a gauge theory to an anisotropic/elliptic superspin
chain, and the stable envelopes compute the R-matrix that solves the dynamical Yang–
Baxter equation (dYBE) for this spin chain. As an illustrative example, we solve the dYBE
for the elliptic sl(1|1) spin chain with fundamental representations using the corresponding
3d N = 2 SQCD whose classical Higgs branch is the Lascoux resolution of a determinantal
variety. Certain Janus partition functions of this theory on I × E for an interval I and
an elliptic curve E compute the elliptic stable envelopes, and in turn the geometric elliptic
R-matrix, of the anisotropic sl(1|1) spin chain. Furthermore, we consider the 2d and 1d
reductions of elliptic stable envelopes and the R-matrix. The reduction to 2d gives the
K-theoretic stable envelopes and the trigonometric R-matrix, and a further reduction to 1d
produces the cohomological stable envelopes and the rational R-matrix. The latter recovers
Rimányi–Rozansky’s results that appeared recently in the mathematical literature.
Key words: equivariant elliptic cohomology; elliptic stable envelope; 3d N = 2 theory; Janus
interfaces; elliptic genus
2020 Mathematics Subject Classification: 81R12; 81T60; 55N34
ishtiaque@ihes.fr
mailto:sfmoosavian@gmail.com
yehao.zhou@ipmu.jp
https://doi.org/10.3842/SIGMA.2024.099
2 N. Ishtiaque, S.F. Moosavian and Y. Zhou
Contents
1 Introduction 3
1.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Glossary of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
I Stable envelopes: beyond symplectic varieties 15
2 Classical Higgs branches of 3d N = 2 gauge theories 15
2.1 Example: GIT quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Example: quiver varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Elliptic stable envelopes for partially-polarized varieties 20
3.1 Chambers and attracting sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Attractive line bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Universal line bundle, Kähler torus and resonant locus . . . . . . . . . . . . . . . 25
3.4 Elliptic stable envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.5 Triangle lemma and duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6 Abelianization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.7 R-matrices and dynamical Yang–Baxter equations . . . . . . . . . . . . . . . . . 31
3.8 R-matrices for quiver varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.9 K-theory and cohomology limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
II The solution to dYBE for sl(1|1) from 3d N = 2 SQCD 37
4 Stable envelopes and R-matrix from gauge theory: the setup 37
4.1 The Bethe/Gauge correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 3d N = 2 SQCD and its parameters . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 Branes and the Bethe/Gauge correspondence . . . . . . . . . . . . . . . . . . . . 42
4.4 Gauge-theoretic definition of elliptic stable envelope . . . . . . . . . . . . . . . . 44
5 Boundaries and interfaces in 3d N = 2 SQCD 46
5.1 Vacua and BPS equations in 3d N = 2 Gauge theories . . . . . . . . . . . . . . . 46
5.1.1 Vacuum equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.1.2 BPS equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Classical Higgs branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2.1 Classical Higgs branches of 3d N = 2 SQCDs . . . . . . . . . . . . . . . . 49
5.2.2 Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.3 Attracting sets and Morse flow . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3 Boundaries and interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.3.1 Thimble (exceptional Dirichlet), DC(p) . . . . . . . . . . . . . . . . . . . . 55
5.3.2 Enriched Neumann, NC(p) . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.3.3 Massless boundary condition, BL(p)(p) . . . . . . . . . . . . . . . . . . . . 58
5.3.4 Janus interface, J (mC, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Elliptic Stable Envelopes and Dynamical R-Matrices 3
6 Stable envelopes and R-matrix from gauge theory: the computation 60
6.1 Elliptic stable envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2 R-matrix for elliptic sl(1|1) spin chains . . . . . . . . . . . . . . . . . . . . . . . . 63
7 2d and 1d avatars of elliptic stable envelopes and the R-matrix 65
7.1 The K-theory limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.2 The cohomology limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A Equivariant elliptic cohomology 69
A.1 Chern class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
A.2 Gysin map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.3 Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.4 Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.5 Degree of a line bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
B Notations and conventions for supersymmetry 72
B.1 Conventions for spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
B.2 3d N = 2 supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
B.3 Localizing supercharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
C sl(1|1) rational R-matrix from the A-model localization 75
C.1 Construction of geometric rational R-matrix . . . . . . . . . . . . . . . . . . . . . 75
C.2 Further details on two-point functions of 2d GLSM observables . . . . . . . . . . 84
C.2.1 Brief recap of Closset–Cremonesi–Park localization formula . . . . . . . . 84
C.2.2 Explicit computation of two-point functions . . . . . . . . . . . . . . . . . 87
References 89
1 Introduction
1.1 Background and motivation
In this paper, we study some mathematical and physical aspects of one-dimensional quantum
integrable spin chains with anisotropy based on Lie superalgebras g of type A. Given a Lie
algebra/superalgebra g, there is a family of integrable spin chains with varying spectral curves
and spectrum-generating algebras:
Spin chain Spectral curve Spectrum generating algebra
XXX/Rational C Yangian, Yℏ(g)
XXZ/Trigonometric C× Quantum affine, Uℏ(ĝ)
XYZ/Elliptic Eτ Elliptic dynamical quantum group, Eτ,ℏ(g)
Table 1. One-dimensional integrable quantum spin chains based on the Lie algebra g. Here ℏ is the
usual quantization parameter, τ denotes the complex structure of the elliptic curve Eτ , and ĝ is the affine
Lie algebra of g.
Given one of these choices, to complete the description of a specific spin chain, we need to
choose the number of sites L in the chain, along with the g-module Vi and the spectral curve
valued inhomogeneity vi at the ith site for i = 1, . . . , L. As an example, the XYZ sl2 spin chain
4 N. Ishtiaque, S.F. Moosavian and Y. Zhou
with Vi = C2 being the fundamental representation for all i can be defined by the following
Hamiltonian [7]:
HXYZ- 1
2
= −1
2
L∑
i=1
3∑
j=1
Jjσ
j
(i)σ
j
(i+1), (1.1)
where σ1, σ2, σ3 are the standard Pauli matrices, σ(i) acts on Vi, and J1, J2, J3 are three mutually
distinct coupling constants. We consider periodic spin chains by identifying the (L+1)st site with
the first one. Starting from the Hamiltonian, one can construct a large commutative algebra of
operators containing the Hamiltonian, called the Bethe subalgebra of the spectrum-generating
algebra. These operators provide the large number of conserved charges for the spin chain,
which is the source of its integrability. The wave function and energy spectrum of such spin
chains are obtained using its equivalence to the 2d statistical eight-vertex model and the Bethe
ansatz [7, 50, 79, 129, 131, 132, 136, 138, 149, 150].
It became apparent from the pioneering works [5, 6, 90, 128, 149, 150] that the more general
and fundamental approach to defining integrable spin chains is utilizing the R-matrix and the
Yang–Baxter equation (YBE). For any two g-modules U and V with inhomogeneities u and v,
the R-matrix is an operator R(u− v) ∈ End(U ⊗ V ) depending meromorphically on the differ-
ence u− v.1 It is sometimes convenient to use a diagrammatic notation for the R-matrix, as in
Figure 1.
U(u) V (v)
V (v) U(u)
R
Figure 1. Diagrammatic representation of the R-matrix R(u− v) : U ⊗V ! U ⊗V . The notation U(u)
means that at the site with the module U the inhomogeneity is u. For rational spin chains, given a g-
module U , U(u) becomes the evaluation module for the Yangian Yℏ(g) with the evaluation parameter u.
The defining property of the R-matrix is that it satisfies the YBE. For g-modules U , V andW
with inhomogeneities u, v and w, the YBE can be written as
RUV (u− v)RUW (u− w)RVW (v − w) = RVW (v − w)RUW (u− w)RUV (u− v). (1.2)
It is an equality of operators in U ⊗ V ⊗W . Here RUV is the R-matrix in End(U ⊗ V ) and its
action is extended to U ⊗ V ⊗W by identity, similarly for RVW and RUW . Figure 2 shows the
diagrammatic YBE.
The R-matrix is used to construct the commuting charges of the spin chain as follows. Given
the spin chain Hilbert spaceH =
⊗L
i=1 Vi, we add an auxiliary site U to it with inhomogeneity u.2
Then we define the transfer matrix
t(u) := trU [RUV1(u− v1)RUV2(u− v2) · · ·RUVL(u− vL)] ∈ End(H),
1The difference is taken in the appropriate abelian group, C, C× or Eτ .
2Inhomogeneity of the auxiliary site is also referred to as the spectral parameter in literature. We use the
terms inhomogeneity of a site and its spectral parameter interchangeably.
Elliptic Stable Envelopes and Dynamical R-Matrices 5
U(u) V (v) W (w)
W (w) V (v) U(u)
RUW
RUV
RVW
=
U(u) V (v) W (w)
W (w) V (v) U(u)
RUW
RUV
RVW
Figure 2. Diagrammatic Yang–Baxter equation. A line joining two R-matrices is a contraction of indices
for the vector space labeling the line. In particular, the order of matrix multiplication in the YBE (1.2)
is read off from the diagrams by reading from bottom to top.
which can also be depicted in a diagram as
t(u) =
RUV1 RUV2
RUVL· · ·
· · ·
· · ·
· · ·V1(v1) V2(v2) VL(vL)
U(u)
.
As a consequence of the YBE, these transfer matrices commute with each other for arbitrary in-
homogeneities [t(u), t(v)] = 0. Therefore, if we expand the transfer matrix as t(u) =
∑∞
n=0 u
nhn,
then the coefficients hn ∈ End(H) form a large commutative algebra of operators: [hm, hn] = 0.
This becomes the Bethe subalgebra of the spin chain. Any element of this subalgebra can be
taken to be the Hamiltonian of the quantum mechanical system.
The R-matrix, as a solution of the YBE, has been computed in all cases from Table 1 for
bosonic Lie [8, 9, 11, 12, 13, 14, 81, 82, 116, 130, 135] and affine Lie algebra [71, 116]. For Lie
superalgebras, the first examples of solutions to YBE appeared in [82]. Certain rational solutions
were also obtained in [80]. The classical trigonometric solutions to YBE are constructed in [85]
while the explicit trigonometric quantum R-matrices first appeared in [10]. Furthermore, the
rational solution for g = sl(1|1) is constructed using the notion of stable envelopes in [117]. In this
paper, we propose a general construction of elliptic R-matrices for Lie superalgebras of type A
and we explicitly compute the elliptic R-matrix for g = sl(1|1). Reduction of the elliptic R-
matrix automatically gives the trigonometric R-matrix. Further reduction of the trigonometric
R-matrix gives the rational R-matrix. The reduction of our elliptic sl(1|1) R-matrix down to
the rational case matches with the previously computed rational sl(1|1) R-matrix from [117].
The tool that we use to construct the elliptic R-matrices is a conjectural relationship between
quantum integrable systems and supersymmetric gauge theories, called the Bethe–Gauge corre-
spondence, proposed by Nekrasov and Shatashvili [102, 103, 104]. Part of their conjecture is an
action of the spectrum-generating algebra of a quantum integrable system, Yangian for example,
on the cohomology (or generalized cohomologies) of the Higgs branch of the corresponding gauge
theory. The correspondence and its string-theory origin, when g is a bosonic Lie algebra, have
been extensively studied [23, 27, 33, 54, 57, 74, 77, 97, 98, 102, 103, 104, 105, 110, 111, 113],
generalized to higher dimensions [44, 45, 69, 70, 84, 101, 102, 105], and the mathematical for-
mulation has been established [1, 2, 25, 88, 106, 107]. For g being a Lie superalgebra, the
correspondence and its string-theory origin were studied recently [20, 66, 99, 112] but despite
progress [26, 117], a full mathematical treatment was lacking. Providing such a mathematical
6 N. Ishtiaque, S.F. Moosavian and Y. Zhou
formulation for g being a Lie superalgebra of type A has been one of the central motivations of
this work.
Notably, for a 3d gauge theory with N = 4 supersymmetry, the Higgs branch is a complex
symplectic variety. A particular example is the 3d N = 4 quiver gauge theory, whose Higgs
branch is the Nakajima quiver variety [94], and in this case, the works of Nakajima [95] and
Varagnolo [141] establish the quantum affine algebra Uℏ(ĝQ) (resp. Yangian Yℏ(gQ)) actions
on equivariant cohomology (resp. equivariant K-theory) of the Nakajima quiver variety MQ,
where gQ is the Kac–Moody algebra associated to the quiver Q. Their construction relies on
the generators and relations of the Yangian Yℏ(gQ) or quantum affine algebra Uℏ(ĝQ).
From the quantum integrability point of view, the spectrum-generating algebras are better
explained in the framework of R-matrices and Yang–Baxter equations (1.2), instead of generators
and relations. In the context of the Bethe–Gauge correspondence, the vector spaces on which the
R-matrices act are generalized cohomologies of the Higgs branches of the corresponding gauge
theories.
Providing the mathematical formulation of the Bethe–Gauge correspondence has been one of
the main motivations of Maulik and Okounkov in their fundamental work [88]. To construct R-
matrices acting on the cohomologies, Maulik and Okounkov [88] came up with a map, called the
stable envelope, from the equivariant cohomology of the torus fixed point set to the equivariant
cohomology of the whole Higgs branch
Stab: HT
(
XA
)
! HT(X). (1.3)
Here X is assumed to be a smooth complex symplectic variety, with a torus T action such that
the subtorus A ⊂ T preserves the symplectic structure. Their construction depends on a choice
of chamber C which is a certain subset of the real Lie algebra Lie(A)R, and the R-matrix is
defined as
RC2�C1 := Stab−1
C2
◦ StabC1 : HT
(
XA
)
! HT
(
XA
)
. (1.4)
Then it follows from the definition that R-matrices satisfy the braiding relation
RC1�Cn ◦RCn�Cn−1 ◦ · · · ◦RC2�C1 = 1, (1.5)
which implies the Yang–Baxter equation. The choice of chamber C induces a partial order ⪯
on the set of connected components of XA such that StabC is triangular with respect to ⪯
and R−C�C = Stab−1
−C ◦ StabC gives a Gauss decomposition of R−C�C. A similar kind of Gauss
decomposition of R-matrices has been discussed by Khoroshkin and Tolstoy in the algebraic
setting [76, 115]. Since the paper of Maulik and Okounkov came out, the construction of
stable envelopes has been extended to K-theory [106] and elliptic cohomology [2], giving rise to
trigonometric and elliptic solutions to the Yang–Baxter equations respectively.
From the physics point of view, a strategy for the construction of cohomological stable en-
velopes was suggested by Nekrasov [97, 98], and was successfully implemented for gauge theories
associated with sl(2) spin chains in [23]. The latter also provides a recipe for the computation
of the R-matrix by an A-model localization computation. The systematic gauge-theoretic con-
struction of elliptic stable envelopes was developed in [24, 40]. In these works the elliptic stable
envelope was identified as certain Janus partition functions of 3d N = 4 theories on I × Eτ
where I is some interval and Eτ is the elliptic curve. The Janus interface interpolates between
certain boundary conditions corresponding to the vacua of massless and massive 3d theories.
These boundary conditions correspond to equivariant elliptic cohomology classes of the Higgs
branch of the 3d theory and the (flavor) torus fixed points of the Higgs branch respectively [39].
The elliptic stable envelope is then seen as a map between these cohomologies (the elliptic version
of (1.3)). Dedushenko and Nekrasov [40] construct Janus interfaces for more general 3d N = 2
Elliptic Stable Envelopes and Dynamical R-Matrices 7
theories, though they provide concrete boundary conditions corresponding to vacua of 3d N = 4
theories and they compute the stable envelopes for complex symplectic varieties corresponding
to 3d N = 4 Higgs branches. In this paper we build on the techniques of [40] and generalize
the boundary conditions to vacua of 3d N = 2 theories and use them to compute the elliptic
stable envelopes of non-symplectic varieties corresponding to 3d N = 2 Higgs branches. Some
related topics such as vertex functions for Higgs branches of 3d N = 4 (and N = 2) theories
and the role of elliptic stable envelopes in the transformation of vertex functions under mirror
symmetry/symplectic duality have been discussed in [35, 41].
A key ingredient in the mathematical construction of elliptic stable envelope [2] is a polar-
ization, i.e., a decomposition of the tangent bundle in the K-theory
TX = T
1/2
X + ℏ−1
(
T
1/2
X
)∨ ∈ KT(X). (1.6)
This kind of decomposition shows up naturally on the Higgs branches of 3d N = 4 gauge
theories, for example, hypertoric varieties and Nakajima quiver varieties. Computations of
stable envelopes have been done for many examples of these types of varieties [46, 47, 53, 121,
122, 123, 125]. However, for more general 3d N = 2 theories, their Higgs branches do not
have complex symplectic structure or polarization, even worse they can be singular. There have
been efforts to construct stable envelopes for varieties without complex symplectic structure or
polarization [117], the authors constructed cohomological stable envelopes using the super weight
functions, which generalized their previous works on weight functions and stable envelopes on
cotangent bundles of flag varieties [121, 142]. They found the R-matrix of Yℏ(sl(1|1)) as the
geometric R-matrix of the total space of the bundle O(−1)⊕2 ! P1. The super Lie algebra sl(1|1)
justifies the terminology “super” weight functions.
Another approach to generalizing the construction of stable envelopes was proposed by Ok-
ounkov in [109], with a focus on the elliptic version. An observation in [109] is that polarization
is not crucial for the existence of elliptic stable envelopes. A technical alternative to polarization
is a notion called an attractive line bundle SX on the T-equivariant elliptic cohomology EllT(X)
(see Appendix A for a review on equivariant elliptic cohomology). Roughly speaking, SX pro-
vides a “square root” of the elliptic Thom line bundle Θ(TX), in parallel to a polarization being
“half” of the tangent bundle. For a precise definition, see Section 3.2. It is shown in [109] that
if X admits a polarization (1.6), then the elliptic Thom line bundle Θ
(
T
1/2
X
)
is an attractive line
bundle, which makes the construction of elliptic stable envelopes in [2] a special case of the more
general one in [109].
Part of the objectives in this paper is to use the idea developed in [109] to extend the con-
struction of elliptic stable envelopes in [2] to the classical Higgs branches of a certain class
of 3d N = 2 gauge theories which do not have N = 4 supersymmetry. In mathematical ter-
minology, we construct the elliptic stable envelopes for certain smooth varieties which do not
necessarily admit complex symplectic structure or polarization. The replacement for the po-
larization in our construction is a certain structure called the partial polarization, which means
a decomposition of the tangent bundle in the K-theory TX = PolX + ℏ−1Pol∨X + E ∈ KT(X),
where E ∈ KT(X) is a K-theory class that is required to have the following properties:
� E = E ∨ in KT(X), and that
� the elliptic Thom bundle Θ(E ) admits a square root.
Then it turns out that Θ(PolX)⊗
√
Θ(E ) is an attractive line bundle, thus the machinery
developed in [109] is applicable to our situation.
We will give a sufficient condition for the existence of partial polarization in Proposition 3.1.
A class of examples that satisfy the criterion in Proposition 3.1 are quiver varieties. They are
similar to Nakajima quiver varieties, namely our quiver variety is built from the cotangent bundle
8 N. Ishtiaque, S.F. Moosavian and Y. Zhou
to the representation of a quiver by imposing some relations and stability conditions and then
taking the quotient by the gauge group; but the difference is that we allow some of the nodes
to be free from imposing complex moment map equations, these nodes will be called odd, and
the ordinary nodes where complex moment map equations are imposed will be called even. We
use the following notation for the odd/even nodes
evenodd
.
The following is an example of a type A quiver (framed and doubled)
· · ·
. (1.7)
Our notation might remind the reader of Kac–Dynkin diagrams for Lie superalgebras [55], for
example, the Kac–Dynkin diagram corresponding to (1.7) is the following
· · · . (1.8)
Indeed our choice of notation is on purpose, due to the following conjecture we make.
Conjecture 1.1. Let Q be a finite or affine type A quiver with nodes decorated as above,
and let gQ be the Lie superalgebra associated with the corresponding Kac–Dynkin diagram. For
a generic stability parameter ζ, and a pair of dimension vectors v,w ∈ NQ0, denote by Mζ(v,w)
the quiver variety of the given data. There exists an action of elliptic dynamical quantum super
group Eℏ,τ (gQ) on the extended equivariant elliptic cohomology scheme
ET
(
Mζ(w)
)
=
⊔
v
ET
(
Mζ(v,w)
)
.
Similarly, there exists an action of quantum affine super algebra Uℏ(ĝQ) on the equivariant
K-theory
KT
(
Mζ(w)
)
=
⊕
v
KT
(
Mζ(v,w)
)
,
and an action of super Yangian Yℏ(gQ) on the equivariant cohomology
HT
(
Mζ(w)
)
=
⊕
v
HT
(
Mζ(v,w)
)
.
Moreover, all the actions factor through the corresponding Maulik–Okounkov quantum groups
constructed via the stable envelopes for Mζ(w).
In anisotropic spin chains, as in (1.1), when all coupling constants are distinct, the R-matrix
depends non-trivially on them. This dependence is manifested as a dependence of the R-matrix
on an additional spectral curve valued parameter called the Dynamical parameter. Thus the
elliptic R-matrix, and the corresponding YBE which involves nontrivial shifts of the spectral
parameters, are also called the dynamical R-matrix and the dynamical Yang–Baxter equation
(dYBE) [51, 52]. Our Conjecture 1.1 is supported by the evidence of the weight structure
occurring in the shifts in the dYBE. In fact, we will show (see (3.17)) that the R-matrices
constructed from the elliptic stable envelopes for quiver varieties satisfy the following dynamical
Yang–Baxter equation
R12(z)R13
(
z − ℏµ(2)
)
R23(z) = R23
(
z − ℏµ(1)
)
R13(z)R12
(
z − ℏµ(3)
)
, (1.9)
where z is the dynamical parameter and we have suppressed the spectral parameter dependence
of R-matrices. Here the dynamical shifts µ are weights in a certain highest-weight module
of ŝl(m|n) or sl(m|n).
Elliptic Stable Envelopes and Dynamical R-Matrices 9
1.2 Summary of the results
Let us now briefly summarize the main contributions of this work.
Elliptic stable envelopes for partially polarized varieties.
� We define the notion of partial polarization on a smooth variety X with a torus T action
(see Definition 3.1), and give a sufficient condition for the existence of partial polarization
(see Proposition 3.1). It turns out that a large class of GIT quotients admits partial
polarization (see Example 3.1), this includes all abelian gauge group cases, and all the
quiver varieties considered in Section 2.2, and the total spaces of vector bundles on partial
flag varieties (see Example 3.3).
� We show that a partial polarization induces an attractive line bundle on the equivariant
elliptic cohomology EllT(X), a notion defined by Okounkov in [109]. Then the existence
and uniqueness of elliptic stable envelope for partially polarized varieties follow from the
general result in [109], see Theorem 3.1. In particular, elliptic stable envelopes exist for all
cases considered in Examples 3.1, 3.2, 3.3 and 3.4. The last example was also considered
in the recent work of Rimányi and Rozansky [117, Section 2.3.3], and our result implies
the existence of stable envelopes in this example.
� We compute the elliptic stable envelope for the quiver with a single node which is odd,
i.e., the total space of direct sum of L copies of tautological bundles on Gr(N,L), see
equation (3.6).
� The triangle lemma and the duality for the elliptic stable envelopes on partially polarized
varieties are obtained verbatim to that in [2], see Section 3.5.
� We extend the abelianization procedure described by Aganagic and Okounkov [2] to the
partially polarized case in Section 3.6. This is potentially helpful in the computation of
stable envelopes for more general quivers.
� In Section 3.7, the R-matrices are defined in the same way as [2, 88], and dynamical
Yang–Baxter equations (3.15) follows from the braiding relations. The R-matrices and
dynamical Yang–Baxter equations for the quiver varieties are discussed in Section 3.8. For
general quivers of finite or affine type A, we observe that the dynamical shift µ is closely
related to a highest-weight module of the corresponding Lie superalgebra.
� In Section 3.9, we show that the analog of the limit procedure in [2] also works for the par-
tially polarized varieties and produces K-theoretic and cohomological stable envelopes, see
Corollary 3.2. We present the explicit K-theoretic and cohomological stable envelopes for
the quiver with a single node which is odd, see (3.26) and (3.27). The cohomological stable
envelope (3.27) recovers the recent result of Rimányi and Rozansky [117, Theorem 7.3].
A couple of distinctions between 3d N = 2 and 3d N = 4 cases. For the purpose
of illustration, we solve the dYBE for the Lie superalgebra g = sl(1|1) from the corresponding
gauge theory, which is a 3d N = 2 SQCD. While rather similar in computation, there are a few
important conceptual differences from the computation of sl(2) elliptic stable envelopes from
3d N = 4 SQCDs.
� Higgs branch: Non-hyperkähler and quantum correction. Higgs branches of 3d
N = 2 theories without N = 4 supersymmetry are Kähler manifolds, unlike 3d N = 4
Higgs branches which are hyperkähler. For example, in the case of 3d N = 2 SQCD, the
classical Higgs branch MH(N,L) is given by the direct sum of L copies of the tautological
bundle over Gr(N,L), the so-called resolved determinantal variety [83]. Also, unlike in case
of the 3d N = 4 Higgs branches, 3d N = 2 Higgs branches receive quantum corrections
[3, 36]. In this paper, unless explicitly mentioned otherwise, all mentions of 3d N = 2
Higgs branches will refer to the classical Higgs branches.
10 N. Ishtiaque, S.F. Moosavian and Y. Zhou
� Non-isolated fixed points. Another departure from the 3d N = 4 SQCD case is that
there can still be continuous Higgs branch moduli in 3d N = 2 SQCDs in the presence
of generic flavor twisted masses (even at the quantum level [3, 36]). As a result, we can
not have a general correspondence between fully massive vacua and Bethe eigenstates
of integrable superspin chains, as is more familiar in the bosonic Bethe/Gauge correspon-
dence [102, 103]. More specifically, in the context of interval partition functions computing
stable envelopes or R-matrices for bosonic spin chains, boundary conditions are chosen rep-
resenting these fully massive vacua [23, 40]. On the Higgs branch, these fully massive vacua
correspond to isolated fixed points of the (flavor) torus action. In contrast, the fixed point
set of the 3d N = 2 Higgs branches decomposes into connected components of nonzero
dimensions (5.26). So we introduce brane-type boundary conditions preserving N = (0, 2)
supersymmetry mimicking these connected components (see Section 5.3.1), rather than
isolated points.
The solution to dYBE for sl(1|1) from 3d N = 2 SQCD. Let us now present the
formulas.
� Elliptic stable envelopes and the elliptic R-matrix for sl(1|1). We compute the
stable envelopes as Janus partition functions in 3d N = 2 SQCD on I × Eτ . The theory
has U(N) gauge group and SU(L) × U(1)ℏ flavor symmetry. There is also a topolog-
ical U(1)top symmetry acting only on monopoles. A Janus interface is created by vary-
ing U(1)L−1 twisted massesm1, . . . ,mL (satisfyingm1+· · ·+mL = 0) along I. The masses
interpolate between nonzero generic values satisfying m1 < · · · < mL and zero. A choice
of such ordering creates a chamber C in RL−1. The formula for the stable envelope is (see
Section 6.1 for details)
StabC(p) := (−1)
∑
a #(i<p(a))
× SymSN
[(∏
a
fC,p(a)(sa, x, ℏ, z)
)( ∏
p(a)>p(b)
1
ϑ
(
sas
−1
b
))], (1.10)
where ϑ is the elliptic theta function (defined in (5.43)), SymSN
denotes the symmetrization
with respect to the gauge holonomies sa, and
fC,n(s,x, ℏ, z) :=
(∏
i<n
ϑ(sxi)
)
ϑ
(
sxnzℏn−L
)
ϑ
(
zℏn−L
) (∏
i>n
ϑ(sxiℏ)
)
.
The elliptic equivariant parameters s, x, ℏ and z correspond to background holonomies
for the U(N), SU(L), U(1)ℏ, and U(1)top connections respectively.
The corresponding R-matrix, which satisfies dYBE, is given by (see Section 6.2 for details)
R−C C(u, z) =
1 0 0 0
0 ϑ(u)ϑ(z)ϑ(zℏ−2)
ϑ(u−1ℏ)ϑ(zℏ−1)2
ϑ(ℏ)ϑ(uzℏ−1)
ϑ(u−1ℏ)ϑ(zℏ−1)
0
0
ϑ(ℏ)ϑ(u−1zℏ−1)
ϑ(u−1ℏ)ϑ(zℏ−1)
ϑ(u)
ϑ(u−1ℏ) 0
0 0 0 ϑ(uℏ)
ϑ(u−1ℏ)
, (1.11)
where −C denotes the opposite chamber, and the diagonal blocks from top to bottom cor-
respond to zero, one, and two-magnon sectors, respectively. To the best of our knowledge,
Elliptic Stable Envelopes and Dynamical R-Matrices 11
the elliptic stable envelopes and the elliptic R-matrix for sl(1|1) have not been constructed
in the literature before. Furthermore, this is also the first time that a new solution to
dYBE has been constructed directly from the gauge theory using the Bethe/Gauge corre-
spondence that we could find.
� K-theoretic stable envelopes and the trigonometric R-matrix for sl(1|1). The
3d ! 2d reduction of elliptic stable envelopes gives K-theoretic stable envelopes and the
trigonometric R-matrix. The precise way to get the K-theoretic stable envelope is de-
scribed in Section 3.9. First, we define the slope parameter s associated to the Kähler
parameter z by
s := lim
ln z!∞
ln q!∞
Re
(
− ln z
ln q
)
∈ R\Z,
where q := e2πiτ . Then, the K-theoretic stable envelope, as the limit of (1.10), is given by
(see Example 3.10 and Section 7.1 for details)
StabC,s(p) = (−1)
∑
a #(i<p(a))ℏ
1
2
∑
a #(i>p(a))
×SymSN
( N∏
a=1
fC,s,p(a)(sa,x, ℏ)
)
·
(∏
a>b
1(
sas
−1
b
) 1
2 −
(
sas
−1
b
)− 1
2
), (1.12)
where
fC,s,n(s,x, ℏ) ≡ (sxn)
⌊s⌋
∏
i<n
(
1− (sxi)
−1
)∏
i>n
(
1− (sxiℏ)−1
)
.
The corresponding trigonometric R-matrix, as the reduction of (1.11), is G(x) := x
1
2 − x−
1
2
R−C C(u) =
1 0 0 0
0 G(u)
G
(
u−1ℏ
) u⌊s⌋+
1
2
G(ℏ)
G
(
u−1ℏ
) 0
0 u−⌊s⌋− 1
2
G(ℏ)
G
(
u−1ℏ
) G(u)
G
(
u−1ℏ
) 0
0 0 0 G(uℏ)
G
(
u−1ℏ
)
. (1.13)
Note that all holonomies are still multiplicative but are now C×-valued.
� Cohomological stable envelopes and the rational R-matrix for sl(1|1). A fur-
ther 2d ! 1d reduction of (1.12) gives cohomological stable envelopes (see Section 3.9 for
the precise mathematical procedure)
StabC(p) = (−1)
∑
a #(i<p(a))SymSN
[(
N∏
a=1
fC,m(s,x, ℏ)
)
·
(∏
a>b
1
sa − sb
)]
,
where fC,n(s,x, ℏ) :=
∏
i<n(s+ xi)
∏
i>n(s+ xi + ℏ), while the reduction of (1.13) pro-
duces the rational R-matrix
R−C C(u) =
1 0 0 0
0 u
ℏ−u
ℏ
ℏ−u 0
0 ℏ
ℏ−u
u
ℏ−u 0
0 0 0 ℏ+u
ℏ−u
.
12 N. Ishtiaque, S.F. Moosavian and Y. Zhou
All holonomies are now additive and C-valued. These reproduce the results of Rimányi
and Rozansky [117] (see Example 3.10 and Section 7.2 for details).
Dynamical R-matrices for the fundamental representation of sl(n|1). We write
down explicit elliptic dynamicalR-matrices for the fundamental representation of sl(n|1). Details
are in Example 3.9. Let us label the basis for Cn+1 by vα where α ∈ {0, 1, . . . , n}. Then there
exists a dynamical R-matrix for the tensor product Cn+1 ⊗ Cn+1 which reads as follows (see
Example 3.9 for details)
R(u, z1, . . . , zn)(vα ⊗ vβ)
=
vα ⊗ vβ, α = β < n,
D(u)vα ⊗ vβ, α = β = n,
C(u)vα ⊗ vβ +B
(
u, ℏ−δβ,n
β∏
i=α+1
zi
)
vβ ⊗ vα, α < β,
A
u, ℏ−δα,n
α∏
i=β+1
zi
vα ⊗ vβ +B
u, ℏδα,n
α∏
i=β+1
z−1
i
vβ ⊗ vα, β < α.
(1.14)
Here we use the shorthand notations for some special functions
A(u, z) =
ϑ(zℏ)ϑ
(
zℏ−1
)
ϑ(u)
ϑ(z)2ϑ
(
uℏ−1
) , B(u, z) =
ϑ(ℏ)ϑ(uz)
ϑ(z)ϑ
(
u−1ℏ
) , C(u) =
ϑ(u)
ϑ
(
uℏ−1
) ,
D(u) =
ϑ(uℏ)
ϑ
(
u−1ℏ
) .
Then R satisfies the dynamical Yang–Baxter equation (1.9) for µ = (µ1, . . . , µn) given by
weights for the fundamental representation of sl(n|1) (see (3.21) for details on the weights).
This suggests that we should actually treat Cn+1 as Cn|1, with basis elements v0, v1, . . . , vn−1
being even and vn being odd.
The rational limit of (1.2) is
R(u) = P
(
u
u− ℏ
Π− ℏ
u− ℏ
1
)
.
Here P is the usual swapping-tensor operator, i.e., P(vα ⊗ vβ) = vβ ⊗ vα, and Π is the super
swapping-tensor operator, i.e., Π(vα ⊗ vβ) = (−1)|vα|·|vβ |vβ ⊗ vα, where |v| is the parity of the
vector v which equals to 0 if v is even and 1 if v is odd. Note that we can rewrite
(u− ℏ)ΠPR(u) = u1− ℏΠ,
the right hand side is the more familiar rational R-matrix for the fundamental representation
of sl(n|1) in the literature [114].
1.3 Future directions
Let us mention some directions related to the study of this paper that we find interesting:
Stable envelopes from 4d Chern–Simons theory. A unifying theme for (1 + 1)d in-
tegrable systems has emerged in recent years. 4d Chern–Simons theory describes quantum
integrable spin-chain models and a large class of classical 2d integrable field theories [29, 30, 31,
32, 34]. In the particular case of integrable spin-chain models, the R-matrix can be constructed
by a relatively straightforward Feynman diagram calculation. However, the mathematical for-
mulation of the BGC shows that more basic objects to study are stable envelopes through which
Elliptic Stable Envelopes and Dynamical R-Matrices 13
R-matrix can be constructed easily. It would be very interesting to provide a 4d Chern–Simons
perspective of the mathematical works [2, 88, 106, 107] and specially the construction of various
stable envelopes. The string-theory realization of the Bethe–Gauge correspondence in which
4d Chern–Simons theory plays a crucial role should be a very useful guide in providing such
a perspective (see [33, 38, 66] for some related works in this direction).
3d N = 2 mirror symmetry for stable envelopes. It is believed that the stable en-
velopes for two mirror-dual pairs
(
X,X∨) must match after certain identification of parameters.
This conjecture has been proven in many special cases related to 3d N = 4 mirror symmetry:
cotangent bundle of Grassmannian [120], complete flag variety of type-An [119] and arbitrary
type [124], hypertoric varieties [127, 134], and many examples of Bow varieties [118]. We expect
that a similar conclusion holds for the case of 3dN = 2 mirror symmetry. Therefore, the first step
in proving this conjecture would be to construct the elliptic stable envelope for the Higgs branch
of the mirror-dual of the gauge theory we considered here. It is expected that the physical setup
of [40] which we employed here in the presence of a Janus interface in the space of FI parameters
(instead of a mass-Janus interface) produces the elliptic stable envelopes for the mirror-dual of
Higgs branches considered in this paper. This would provide a solid ground for progress towards
the mathematical formulation of 3d N = 2 mirror symmetry (or symplectic duality).
1.4 Organization of the paper
The mathematical and physical contents of the paper are mainly organized in Part I and Part II
respectively. In Section 2, we characterize the varieties we are interested in, including the
classical Higgs branches of 3d N = 2 theories, and in Section 3, we lay down the mathematical
foundation of elliptic stable envelopes for these spaces. In Section 4, we give motivations, in
the context of the Bethe/Gauge correspondence, for computing the stable envelopes as certain
mass Janus interfaces. In particular, we argue that certain mass Janus partition functions
for 3d N = 2 SQCDs on I × Eτ should give us the sl(1|1) elliptic stable envelopes. To compute
the Janus partition functions for the 3d theories on intervals we need specific half-BPS boundary
conditions corresponding to the vacua of the theory – we describe these in Section 5. In Section 6,
we do the explicit computation of the Janus partition functions and find the formulas for the
elliptic sl(1|1) stable envelopes and the R-matrix satisfying the dYBE. In the last main section,
Section 7, we take certain limits of the elliptic stable envelopes and recover the K-theoretic
and the cohomological stable envelopes and their respective R-matrices for the sl(1|1) superspin
chains.
We include a brief review of equivariant elliptic cohomology in Appendix A. Our conventions
regarding supersymmetry are summarized in Appendix B. Lastly, in Appendix C, we provide an
alternative computation of the rational R-matrix for sl(1|1) spin chains based on the observations
in [23, 97, 98] using the A-model computation.
Note added. After the appearance of this work, a related work appeared in the context of
stable envelopes and 3d N = 2 gauge theories in [139]. This work does not have any overlap
with the present paper.
1.5 Glossary of notations
Some often-used symbols:
Latin
A Maximal torus of FA.
a Lie algebra of A.
C
A chamber for the A-twisted masses,
for FA = SU(L) it corresponds to a choice of ordering for the mis.
14 N. Ishtiaque, S.F. Moosavian and Y. Zhou
Eτ An elliptic curve with complex moduli τ , part of the 3d space-time.
EllT(X) T-equivariant elliptic cohomology of X.
ET(X) Extended T-equivariant elliptic cohomology of X.
ET T-equivariant elliptic cohomology of a point.
EK K-equivariant elliptic cohomology of a point.
F
Total flavor symmetry,
F = FA × C×
ℏ in Part I and F = FA ×U(1)ℏ in Part II.
FA
Part of the flavor symmetry, the framing in quivers,
complex in Part I and real in Part II. In SQCDs FA = SU(L).
Fp
Connected component of the A-fixed point set in MH labeled by p,
MA
H =
⊔
pFp. In SQCDs p is a choice of N integers out of {1, . . . , L}.
F̂p
Lift of Fp to the space M of chirals,
projection: π : Ms ↠ MH , π
(
F̂p
)
= Fp.
f Lie algebra of F .
fA Lie algebra of FA.
G
Gauge group, complex in Part I and real in Part II,
in SQCDs G = SU(N).
g Lie algebra of G.
H Maximal torus of G
h Lie algebra of H.
ℏ Equivariant elliptic parameter/fugacity for U(1)ℏ.
I A finite interval [y−, y+], part of the 3d space-time.
K Kähler torus.
M Space of chiral multiplets. M = T ∗Hom
(
CL,CN
)
for SQCDs.
Ms Stable locus of M.
M0 Space of massless chirals in the vacuum p.
MC;±(p) Space of +/−-vely massive chirals in the vacuum p and chamber C.
MH Classical Higgs branch.
mi, m
C
i A-twisted (in 3d) and AC-twisted (in 2d) masses.
mI
i Triplet of A-twisted masses in 1d, I = 1, 2, 3.
PolX Partial polarization of X.
PolopX Opposite partial polarization of X.
Q Quiver/Kac–Dynkin diagram (e.g., (1.8)).
Q Framed and doubled quiver (e.g., (1.7)).
Q̃ Tripled/gauge theory quiver (e.g., (4.1)).
Q, Q̃ Chiral multiplets in SQCDs valued in Hom
(
CL,CN
)
and Hom
(
CN ,CL
)
.
Q, Q, Q Supercharges in 3d, 2d, and 1d.
R, R̃ Space of the Qs and Q̃s in SQCD.
R0(p), R̃(p) Spaces of massless Qs and Q̃s in the vacuum p.
RC;±(p), R̃C;±(p) +/−-ve mass subspaces of R, R̃ in the vacuum p and chamber C.
R, Rs, R Elliptic, K-theoretic, and rational R-matrices.
Rℏ, Rtop Lie algebras of U(1)ℏ, and U(1)top.
s Slope in the K-theoretic stable envelope.
sa Equivariant elliptic parameter/fugacity for H. In SQCDs a = 1, . . . , N .
S1
A, S
1
B
Non-contractible cycles of Eτ = S1
A × S1
B,
we go from 3d ! 2d ! 1d by reducing S1
B first and then S1
A.
Stab, Stabs, Stab Elliptic, K-theoretic, and cohomological stable envelopes.
T Maximal torus of F .
t Lie algebra of T.
Elliptic Stable Envelopes and Dynamical R-Matrices 15
TX Tangent bundle of X.
U Universal line bundle on elliptic cohomology scheme.
U(1)ℏ, U(1)top Parts of the flavor symmetry. U(1)top acts only on monopoles in 3d.
X Smooth quasi-projective complex variety with T action.
xi Equivariant elliptic parameter/fugacity for A. In SQCDs i = 1, . . . , L.
y Real coordinate along I.
z Equivariant elliptic parameter/fugacity for U(1)top.
Greek
ζ, ζC Real (in 3d) and complex in 2d FI parameters.
ζI Triplet of real FI parameters in 1d, I = 1, 2, 3.
µ Complex moment map for the gauge group G action.
µev Complex moment map for the even part Gev of the gauge group action.
µg Real moment map for the gauge group G action.
σ Real adjoint scalar in 3d N = 2 vector multiplet.
σC Complex adjoint scalar in 2d N = (2, 2) vector multiplet.
Part I
Stable envelopes: beyond symplectic
varieties
2 Classical Higgs branches of 3d N = 2 gauge theories
The essential data extracted from a 3d N = 2 gauge theory is a complex algebraic group G,
a complex G-representation M, with a G-invariant algebraic function W : M! C, and a char-
acter ζ : G! C×. The Higgs branch of the 3d N = 2 gauge theory associated to (G,M,W, ζ)
is then defined as the GIT quotient
MH(G,M,W, ζ) := Crit(W)ζ−ss/G, (2.1)
where Crit(W)ζ−ss is the ζ-semistable locus of the critical locus Crit(W) which is the solution
to the equation dW = 0.
Assumption. We assume that the semistable locus Crit(W)ζ−ss is smooth and the action
of G on it is free so that we get a smooth Higgs branch MH(G,M,W, ζ). As we will see later,
this assumption is satisfied in many examples that we consider. This setting is not the most
general one for generic 3d N = 2 gauge theories, but this is enough for our purpose.
2.1 Example: GIT quotients
A typical example of MH(G,M,W, ζ) is as follows. Let G = Gev×Godd that acts on a represen-
tationR and then we takeM := R⊕R∨⊕gev. We choose a complex moment map µ : R⊕R∨! g
for theG action, and define µev : R⊕R∨! g∨ev to be the composition prev◦µ, where prev : g∨! g∨ev
is the projection to the even part. Note that µev is the moment map for the action of Gev on M.
We take W = ⟨X,µev⟩ where X is the coordinate on gev and ⟨·, ·⟩ is the pairing between gev
and g∨ev. Then we choose a character ζ : G ! C×. In this case, the Higgs branch is then iso-
morphic to MH(G,M,W, ζ) ∼= µ−1
ev (0)
ζ−ss//G, where µ−1
ev (0)
ζ−ss is the ζ-semistable locus of
16 N. Ishtiaque, S.F. Moosavian and Y. Zhou
the µ−1
ev (0). In other words, MH(G,M,W, ζ) is a further quotient of a Hamiltonian reduction
by a group action
MH(G,M,W, ζ) ∼=
((
R⊕R∨)ζ−ss///Gev
)
//Godd. (2.2)
Notice that we have a G-equivariant closed embedding µ−1(0) ↪! µ−1
(
g⊥ev
)
= µ−1
ev (0), which
leads to a closed embedding
µ−1(0)ζ−ss//G ↪! µ−1
ev (0)
ζ−ss//G.
In other words, if we consider the moment map µ for the whole gauge group G, and perform the
Hamiltonian reduction for the G-action, then the resulting variety is a smooth and closed (but
could be empty) subvariety of MH(G,M,W, ζ).
Example 2.1. When G is abelian, G acts trivially on its Lie algebra g, and we have a commu-
tative diagram
µ−1(0)ζ−ss//G µ−1
ev (0)
ζ−ss//G
(
R⊕R∨)ζ−ss//G
{0} g⊥ev g∨
such that every square is Cartesian. The whole diagram is equivariant under the action of
the flavour torus A = (C×)rkR/G. The central fiber µ−1(0)ζ−ss//G is a hypertoric variety and
the GIT quotient
(
R⊕R∨)ζ−ss//G is known as the Lawrence toric variety [64]. For a generic
choice of ζ, we have
(
R⊕R∨)ζ−ss = (R⊕R∨)ζ−s and the latter is smooth over the base g∨,
therefore
(
R⊕R∨)ζ−ss//G is flat over the base g∨ and µ−1
ev (0)
ζ−ss//G is flat over g⊥ev. In the
case when the charge matrix A : ZrkR ! Char(G) is surjective and unimodular,3 the action
of G on
(
R⊕R∨)ζ−ss is free,4 so
(
R⊕R∨)ζ−ss//G is smooth over g∨, thus µ−1
ev (0)
ζ−ss//G is
a smooth deformation of the hypertoric variety µ−1(0)ζ−ss//G over the base g⊥ev.
In the abelian gauge group example, we have the implication
µ−1
ev (0)
ζ−ss ̸= ∅ ⇒ µ−1(0)ζ−ss ̸= ∅.
In fact, if µ−1
ev (0)
ζ−ss is nonempty, then µ−1
ev (0)
ζ−ss//G is projective over the affine scheme
µ−1
ev (0)//G. The projection is equivariant under the C×
r which acts on R ⊕ R∨ with weight r
and acts on g⊥ev and g∨ with weight r2. C×
r contracts µ−1
ev (0)//G to a unique point, which is the
image of 0 ∈ R⊕R∨, therefore the image of µ−1
ev (0)
ζ−ss//G in µ−1
ev (0)//G, which is a C×
ℏ -invariant
closed subset, must contain the image of 0. In particular, µ−1(0)ζ−ss//G is nonempty.
To summarize, in the case when the gauge group is abelian and under the unimodular as-
sumption, we always get a smooth deformation of the corresponding hypertoric variety over the
base g⊥ev, and the deformation is equivariant under the action of A×C×
r , where A is the flavour
torus (C×)rkR/G. The smooth deformation gives rise to an isomorphism between cohomol-
ogy groups H∗(µ−1(0)ζ−ss//G
) ∼= H∗(µ−1
ev (0)
ζ−ss//G
)
, and we will see that the stable envelopes
for µ−1(0)ζ−ss//G and µ−1
ev (0)
ζ−ss//G are the same (see Example 3.6).
However, when G is nonabelian, the cohomologies of µ−1(0)ζ−ss//G and µ−1
ev (0)
ζ−ss//G are not
isomorphic in general; more drastically, it is possible that µ−1
ev (0)
ζ−ss is nonempty but µ−1(0)ζ−ss
is empty. See Example 2.4 below.
3The action of G on R gives a homomorphism G ! (C×)rkR, the charge matrix A is the induced map on
characters. A is called unimodular if and only if every rkG× rkG submatrix has determinant ∈ {0,±1}.
4In fact, G acts on
(
R⊕R∨)ζ−ss
freely ⇐⇒ G acts on µ−1(0)ζ−ss freely ⇐⇒ charge matrix is surjective and
unimodular [64].
Elliptic Stable Envelopes and Dynamical R-Matrices 17
Remark 2.1 (tautological generation). We say that the K-theory K
(
µ−1
ev (0)
ζ−ss//G
)
is gen-
erated by tautological classes if the natural map KG(pt) ! K
(
µ−1
ev (0)
ζ−ss//G
)
is surjective.
This property is also known as Kirwan surjectivity. For hyperkähler quotients (i.e., Godd = 1),
it is known that both hypertoric varieties (i.e., G is torus) and the Nakajima quiver varieties
have the Kirwan surjectivity property, see [63, 89]. Let ζb be the restriction of ζ to Gev, then
we have a natural open embedding
(
R⊕R∨)ζ−ss ⊂ (R⊕R∨)ζb−ss, which induces open embed-
ding: (
R⊕R∨)ζ−ss///Gev ↪!
(
R⊕R∨)///ζbGev,
where the right-hand-side is the usual hyperkähler reduction. Assume that KGodd
((
R ⊕ R∨)
///ζbGev
)
is generated by KGev×Godd
(pt) = KG(pt), then KGodd
((
R⊕R∨)ζ−ss///Gev
)
is also
generated by KG(pt), whence (2.2) implies that K
(
µ−1
ev (0)
ζ−ss//G
)
is generated by KG(pt). In
other words,
� Godd-equivariant Kirwan surjectivity of
(
R⊕R∨)///ζbGev implies the Kirwan surjectivity
of µ−1
ev (0)
ζ−ss//G.
Since for the hypertoric varieties and the Nakajima quiver varieties, the Kirwan surjectivity
holds for flavour group equivariant K-theory, we conclude that: if either G is a torus or (R, G) is
constructed from a quiver representation (see Section 2.2), then µ−1
ev (0)
ζ−ss//G has the Kirwan
surjectivity.
2.2 Example: quiver varieties
A large class of GIT quotients comes from 3d N = 2 quiver gauge theories, and we recall the
construction of the Higgs branches of such theories here.
Let Q = (Q0, Q1) be a quiver, i.e., Q0 and Q1 are finite sets (known as the set of nodes
and edges respectively) together with two maps h, t : Q1 ! Q0 sending an edge to its head and
tail respectively. We add one more structure to the quiver Q, namely, we separate Q0 into two
parts Q0 = Qev
0 ⊔Qodd
0 , which will be called even and odd respectively. Our notation for the
even and odd nodes is as follows
evenodd
.
Let w,v ∈ NQ0 be Q0-tuples of natural numbers, then the gauge group and the chiral multiplets
are built from the above data as follows.
The gauge group G = Gev ×Godd is such that
Gev =
∏
i∈Qev
0
GL(vi), Godd =
∏
i∈Qodd
0
GL(vi).
And the space of chiral multiplets M = R⊕R∨ ⊕ gev is such that
R =
⊕
i∈Q0
Hom(Vi,Wi)⊕
⊕
a∈Q1
Hom(Vt(a), Vh(a)),
where Wi is the wi-dimensional complex vector space and Vi is the vi-dimensional complex
vector space. The action of G on R is the obvious one, namely GL(vi) acts on Vi as the funda-
mental representation. We give notations to the field contents of R by letting αi ∈ Hom(Vi,Wi)
for i ∈ Q0 and xa ∈ Hom(Vt(a), Vh(a)) for a ∈ Q1; for the dual representation R∨, we use the
notation α̃i ∈ Hom(Wi, Vi) and x̃a ∈ Hom(Vh(a), Vt(a)) for a ∈ Q1.
18 N. Ishtiaque, S.F. Moosavian and Y. Zhou
The holomorphic symplectic form on R⊕R∨ is
ω =
∑
a∈Q1
dxa ∧ dx̃a +
∑
i∈Q0
dαi ∧ dα̃i,
so there is a moment map µ : R⊕R∨ ! g∨ such that
µ
(
xa, x̃a, αi, α̃i
)
=
∑
a∈Q1
[
xa, x̃a
]
+
∑
i∈Q0
α̃iαi.
We define µev : R⊕R∨ ! g∨ev to be the composition of µ with the projection g∨ ! g∨ev.
We introduce the flavour symmetry group F = GW × C×
ℏ , such that GW =
∏
i∈Q0
GL(wi),
GL(wi) acts on Wi by fundamental representation, and C×
ℏ acts on M by scaling R∨ with
weight ℏ−1 and scaling gev with weight ℏ and fixing R. Note that the action of F on M
commutes with the action of G. We also fix a maximal torus T of F , namely T = TW ×C×
ℏ such
that TW =
∏
i∈Q0
(C×)wi is a maximal torus of GW .
Finally, we choose a character ζ : G! C×, ζ can be written as ζ(g) =
∏
i∈Q0
det(gi)
ζi . Then
the Higgs branch of the 3d N = 2 quiver gauge theory associated to the quiver Q with gauge
rank v and flavour rank w is isomorphic to the GIT quotient
Mζ(v,w) = µ−1
ev (0)
ζ−ss//G, (2.3)
and we write
Mζ(w) :=
⊔
v∈NQ0
Mζ(v,w),
for a fixed w. In other words, we impose complex moment map equations only for even nodes.
If ζ equals to ζ+ := (1, . . . , 1) or ζ− := (−1, . . . ,−1), then according to the King’s criterion
for the stability [78], a quiver representation
(
V, xa, x̃a, αi, α̃i
)
∈ R ⊕R∨ is ζ-semistable if and
only if the following condition is satisfied respectively:
(ζ+) If Si ⊂ Vi are subspaces such that S is preserved under the maps
(
xa, x̃a
)
, and that
Si ⊃ Im(α̃i) for all i ∈ Q0, then S = V .
(ζ−) If Ti ⊂ Vi are subspaces such that T is preserved under the maps
(
xa, x̃a
)
, and that
Ti ⊂ Ker(αi) for all i ∈ Q0, then T = 0.
In the following discussion, we always assume that
� ζ is generic, i.e.,
(
R⊕R∨)ζ−s = (R⊕R∨)ζ−ss.
For example, ζ± are generic.
It is well known that if ζ is generic, then µ :
(
R⊕R∨)ζ−ss ! g∨ is smooth, and the action
of G on
(
R⊕R∨)ζ−ss is free. Since the projection prev : g
∨ ! g∨ev is smooth, the composi-
tion µev = prev ◦ µ is smooth. Therefore, we have the following lemma.
Lemma 2.1. Assume that ζ is generic then, Mζ(v,w) is a smooth variety and the quotient
map µ−1
ev (0)
ζ−ss !Mζ(v,w) is a principal G-bundle.
Example 2.2. If there is no odd node, i.e., Qodd
0 is empty, then µev=µ and in this caseMζ(v,w)
is a Nakajima quiver variety [93, 94].
Elliptic Stable Envelopes and Dynamical R-Matrices 19
Example 2.3. Let Q be an An quiver with Qev
0 = {1, . . . , n − 1} and Qodd
0 = {n}, take
framing dimension w = (r, 0, . . . , 0) and gauge dimension v = (v1, . . . , vn). Below is the doubled
quiver Q,
v1 v2
· · ·
vn−1 vn
r
Mζ+(v,w) is nonempty if and only if r ≥ v1 ≥ · · · ≥ vn, and if nonempty then Mζ+(v,w)
is isomorphic to GLr×Pm, where P is the parabolic subgroup of GLr that stabilizes a fixed
flag F• = F1 ⊂ F2 ⊂ · · · ⊂ Fn ⊂ Fn+1 = Cr such that dimFn+1/Fi = vi, and m is the Lie algebra
of the subgroup M ⊂ P that acts on Fi+1/Fi as identity for all i except for i = n. Mζ+(v,w) is
a vector bundle over the flag variety Flv = GLr/P . Note that M contains the unipotent radical
of P , therefore Mζ+(v,w) contains T ∗Flv as a closed subvariety.
Example 2.4. Let Q be an An quiver with Qev
0 = {2, . . . , n} and Qodd
0 = {1}, take fram-
ing dimension w = (r, 0, . . . , 0) and gauge dimension v = (v1, . . . , vn). Below is the doubled
quiver Q,
v1 v2
· · ·
vn−1 vn
r
Mζ+(v,w) is nonempty if and only if vn ≤ r and vi+1 ≤ vi ≤ vi+1 + r. Mζ+(v,w) can be
described as follows. Let di = vi − vi+1 for 1 ≤ i ≤ n − 1 and dn = vn, then Mζ+(v,w) is
a vector bundle over the convolution of affine Grassmannian orbits
G̃r
λ⃗
GLr
= Gr
ωdn
GLr
×̃Gr
ωdn−1
GLr
×̃ · · · ×̃Gr
ωd1
GLr
,
where λ⃗ = (ωdn , . . . , ωd1) and ωi is the i-th fundamental coweight of GLr. The convolution of the
affine Grassmannian gives a map m : G̃r
λ⃗
GLr
! Gr
v1ω1
GLr
, and the latter represents sub-lattices L
in C[z]⊕r of codimension v1, so it has a v1-dimensional vector bundle V which is the universal
quotient C[z]⊕r/Luniv. Mζ+(v,w) is the total space of the vector bundle Hom(V,O⊕r) on
the G̃r
λ⃗
GLr
.
In the last example, if we replace the odd node with a even one, then Mζ+(v,w) is nonempty
if and only if r ≥ v1 ≥ v2 ≥ · · · ≥ vn. In particular, if n ≥ 2 then there exists v such
that µ−1(0)ζ+−ss is empty but µ−1
ev (0)
ζ+−ss is nonempty, for instance v = (nr, (n− 1)r, . . . , r).
Let A ⊂ TW be a subtorus, such that W decomposes as eigenspaces
W =
⊕
λ∈Char(A)
W λ,
and we write w =
∑
λw
λ for the dimension vector, then it is well known that the A-fixed points
set Mζ(w)A is isomorphic to the product
Mζ(w)A ∼=
∏
λ∈Cochar(A)
M
(
wλ
)
.
20 N. Ishtiaque, S.F. Moosavian and Y. Zhou
3 Elliptic stable envelopes for partially-polarized varieties
Aganagic–Okounkov defined the elliptic stable envelope and showed their existence and unique-
ness for hypertoric varieties and Nakajima quiver varieties [2], later Okounkov extended the
result on the existence and uniqueness of elliptic stable envelope to algebraic stacks with a po-
larization [108] and to varieties with an attractive line bundle on its equivariant elliptic coho-
mology [109].
A polarization, namely a decomposition of tangent bundle TX = T
1/2
X +
(
T
1/2
X
)∨
in the K-
theory class of a variety or stack X, is manifested for the Higgs branches of 3d N = 4 gauge
theories, since the moduli space is holomorphic symplectic by construction.
However, in the 3d N = 2 setting, we do not have a polarization in general. In fact, the
Higgs branch can be odd-dimensional, for example, the resolved conifold in the case of SQED
with two flavours (see equation (5.15)). In this section, we show that the condition of having
a polarization can be weakened to having a partial polarization (defined below). We basically
follow the line of arguments in [109], the main point is that we can always construct an attractive
line bundle from a partial polarization.
Throughout this section, we denote by X a smooth quasi-projective complex variety with
a torus T action, we fix a nontrivial group homomorphism T ! C×
ℏ and a subtorus A ⊂
ker
(
T! C×
ℏ
)
.
Definition 3.1. A partial polarization on X is the following data:
� a decomposition of the tangent bundle
TX = Pol+X + ℏ−1
(
Pol+X
)∨
+ Pol−X + ℏ(Pol−X )
∨ + E ∈ KT(X),
for some T-equivariant K-theory classes PolX := Pol+X + Pol−X and E ,
such that
(1) E = E ∨ in KT(X), and that
(2) the elliptic Thom line bundle Θ(E ) admits a square root.5
We call such X a partially-polarized variety, with partial polarization PolX. And we define the
opposite partial polarization to be PolopX = Polop,+X + Polop,−X = ℏ−1
(
Pol+X
)∨
+ ℏ(Pol−X )
∨.
Remark 3.1. In particular, the existence of partial polarization implies that TX equals to T∨
X
in KA(X).
We give a criterion when a variety X has a partial polarization in the following.
Proposition 3.1. Let P±
X ∈ KT(X), and moreover assume that there exist
(1) a finite set of pairs (Gi, ni)i∈I , where ni ∈ Z̸=0 and Gi is a reductive group whose derived
subgroup [Gi, Gi] is simply connected and every simple constituent is of type A, C, D, E6,
or G2,
(2) for every i ∈ I, a T-equivariant principal Gi bundle Pi on X,
such that the tangent bundle TX decomposes in the T-equivariant K-theory into
TX = P+
X + ℏ−1
(
P+
X
)∨
+ P−
X + ℏ(P−
X )
∨ +
∑
i∈I
niadj(Pi) ∈ KT(X),
where adj(Pi) := Pi×Gi gi is the adjoint bundle associated to Pi. Then PX = P+
X+P−
X is a partial
polarization on X.
5For a review of elliptic cohomology and Theta bundles, see Appendix A.
Elliptic Stable Envelopes and Dynamical R-Matrices 21
The above proposition is the direct consequence of the following lemma.
Lemma 3.1. Let G be a reductive group whose derived subgroup is simply connected and every
simple constituent is of type A, C, D, E6, or G2. Let P be a T-equivariant principal G-bundle
on X, then the elliptic Thom line bundle Θ(adj(P)) has a square root in Pic(EllT(X)).
Proof. It suffices to prove the existence of a square root in the universal case, i.e., Θ(g)
on E⊗Z X∗/W has a square root, where X∗ is the cocharacter lattice of G and W is the Weyl
group and for any G-representation V , Θ(V ) is the pullback of the Θ-bundle along the induced
map E⊗Z X∗/W ! SrkV E between the moduli of semistable bundles on E.
We first assume that G is simple, then the cocharacter lattice X∗ agrees with the coroot
lattice Q∨. According to the result of Looijenga [86], E⊗ZQ
∨/W is isomorphic to the weighted
projective space P(1, g1, . . . , gr) such that
θ∨ =
r∑
i=1
giα
∨
i ,
is the decomposition of the dual of the highest root in terms of simple coroots. It is well known
that the Picard group of P(1, g1, . . . , gr) is freely generated by O(m), where m = lcm(g1, . . . , gr)
[15], and Looijenga also shows that Θ(g) is isomorphic to O
(
2h∨
)
where h∨ = 1+
∑r
i=1 gi is the
dual Coxeter number [86], therefore Θ(g) has a square root if and only if h∨ is divisible by m,
which is true if and only if G is of type A, C, D, E6 or G2.
In the general case, G is the central quotient of G̃ := H × G′ by a finite abelian group Γ,
where H is a torus and G′ is the derived subgroup of G. Then E ⊗Z X∗/W is the quotient
of E⊗Z X̃∗/W by H1(E,Γ), where X̃∗ is the cocharacter lattice of G̃, so there is a fiber sequence
H1(E,Γ)∨ −! Pic(E⊗Z X∗/W ) −! Pic(E⊗Z X̃∗/W )H
1(E,Γ), (3.1)
where H1(E,Γ)∨ is the Pontryagin dual of H1(E,Γ). Note that we can identify E ⊗Z X̃∗/W =
EH ×
(
E ⊗Z Q
∨/W
)
, where Q∨ is the coroot lattice of G′. We also note that there is an étale
locally trivial fibration E⊗ZX∗/W ! EH/Γ with fibers isomorphic to E⊗ZQ
∨/W , such that the
trivial fibration EH ×
(
E ⊗Z Q
∨/W
)
is the pullback along the quotient map EH ! EH/Γ. The
fiber sequence (3.1) is compatible with its counterpart coming from the quotient map between
the bases EH ! EH/Γ. In other words, we have the commutative diagram of fiber sequences
H1(E,Γ)∨ Pic(EH/Γ) Pic(EH)
H1(E,Γ)
H1(E,Γ)∨ Pic(E⊗Z X∗/W ) Pic
(
E⊗Z X̃∗/W
)H1(E,Γ)
.
Now the image of Θ(g) in Pic
(
E⊗Z X̃∗/W
)H1(E,Γ)
is the OEH
⊠ Θ(g′), which has a square
root OEH
⊠Θ(g′)⊗
1
2 by the previous step. The square root is invariant under the action H1(E,Γ)
since H1(E,Γ) acts trivially on Pic
(
E ⊗Z Q
∨/W
)
. We lift such square root to L ∈ Pic(E ⊗Z
X∗/W ) so that Θ(g) ∼= L ⊗2 ⊗ K for some K ∈ H1(E,Γ)∨. Since H1(E,Γ)∨ ∈ Pic0(EH/Γ), so
we can find K1 ∈ Pic0(EH/Γ) such that K ∼= K ⊗2
1 , hence we have Θ(g) ∼= (L ⊗ K1)
⊗2. ■
Remark 3.2. When G = GLn, there is an alternative proof of the above lemma. In this
case adj(P) ∼= End(V) where V is a T-equivariant vector bundle. Notice that Θ
(
VV∨) is iso-
morphic to pullback ∆∗Θ
(
V1V∨
2
)
along the diagonal morphism EllT(X) ↪! EllT(X) × EllT(X),
where Vi = pr∗iV are pullback of V along projections to the first and the second component.
Moreover, (12)∗Θ
(
V1V∨
2
) ∼= Θ
(
V∨
1 V2
)
which is isomorphic to Θ
(
V1V∨
2
)
, where (12) is the permu-
tation of factors, thus we can apply [100, Lemma 6.1] to conclude that Θ
(
VV∨) admits a square
root.
22 N. Ishtiaque, S.F. Moosavian and Y. Zhou
Example 3.1. Suppose that we are in the situation of Section 2.1, namely, G = Gev × Godd
acts on a representation R and µev : R⊕R∨ ! gev is the moment map for Gev action. Assume
moreover that Godd is the reductive group whose derived subgroup is simply connected and
every simple constituent is of type A, C, D, E6, or G2. Then X := µ−1
ev (0)
ζ−ss//G is a partially-
polarized variety with PolX = Pol+X = R − adj(Pb), where R is the bundle associated to the
representation R and Pb×Pf is the principal Gev×Godd = G bundle µ−1
ev (0)
ζ−ss ! X. We have
the decomposition TX = PolX + ℏ−1Pol∨X − adj(Pf ) ∈ KT(X).
Example 3.2. In the quiver case, adjoint bundles can be written in terms of tautological
bundles. Let Q be a quiver with decomposition of nodes Q0 = Qev
0 ⊔ Qodd
0 into even and
odd parts. In this case, T = A × C×
ℏ , where A = TW is the maximal torus of the flavour
group. Our choice of C×
ℏ -weight convention is as follows. We split the set of arrows into two
parts Q1 = Q+
1 ⊔Q−
1 , such that
(⋆) All arrows connected to an even node are from the same group, i.e., either all of them
are in Q+
1 or all of them are in Q−
1 . Denote by Qev,±
0 the subset of Qev
0 consisting of even
nodes such that arrows connected to them are all in Q±
1 .
Then we have Qev
0 = Qev,+
0 ⊔ Qev,−
0 by our assumption. We define the action of C×
ℏ on the
doubled quiver Q by assigning the C×
ℏ -weights as follows:
xa x̃a, a ∈ Q±
1 αi α̃i, i ∈ Qodd
0 α̃i, i ∈ Qev,±
0
C×
ℏ -weight 0 ∓1 0 −1 ∓1
Let ζ be generic stability parameter, then Mζ(v,w) is smooth. Denote by Vi the tautological
bundle on Mζ(v,w) corresponding to the i-th gauge node, and denote by Wi the tautological
bundle on Mζ(v,w) corresponding to the i-th framing, then PolM = Pol+M + Pol−M such that
Pol+M :=
∑
i∈Q0\Qev,−
0
WiV∨
i +
∑
a∈Q+
1
Vh(a)V∨
t(a) −
∑
j∈Qev,+
0
VjV∨
j ,
Pol−M :=
∑
i∈Qev,−
0
WiV∨
i +
∑
a∈Q−
1
Vh(a)V∨
t(a) −
∑
j∈Qev,−
0
VjV∨
j (3.2)
is a partial polarization on M(v,w). In fact,
TMζ(v,w) = Pol+M + ℏ−1
(
Pol+M
)∨
+ Pol−M + ℏ(Pol−M)∨ −
∑
i∈Qodd
0
ViV∨
i ,
in KT
(
Mζ(v,w)
)
.
Example 3.3. Let G be a semisimple and simply connected algebraic group, P be a parabolic
subgroup, and U ⊂ P be the unipotent radical. Let M be a connected normal subgroup of P
such that M contains U , and m be its Lie algebra. Denote by K = P/M , which is a quotient of
the Levi L ∼= P/U . Take X = G×Pm, and T = A× C×
ℏ acts on X such that A is maximal torus
of G and C×
ℏ only acts on the fiber m with weight ℏ−1. Then tangent bundle of X decomposes
into
TX = TG/P + ℏ−1T∨
G/P + L − K ∈ KT(X),
where TG/P is the pullback of the tangent bundle of G/P via the canonical projection G×Pm!
G/P , and L (resp. K ) is the associated adjoint bundle of the principal L-bundle (resp. K-
bundle) induced from the principal P -bundle G×m! X. Assume that every simple constituent
of L is of type A, C, D, E6, or G2, then the same holds for K. Therefore, TG/P is a partial
polarization of X under this assumption.
Elliptic Stable Envelopes and Dynamical R-Matrices 23
Example 3.4. A special case of the previous example is when G = SLn and P is the subgroup
fixing a flag F• = (0 = F0 ⊂ F1 ⊂ · · · ⊂ Fk−1 ⊂ Fk = Cn), we choose the Lie algebra ms
according to a marking number s ∈ {±1}k such that
ms(Fi) ⊂
{
Fi−1, si = +1,
Fi, si = −1,
then X = G×Pms has a partial polarization for any choice of marking s. These varieties show
up in the work of Rimányi and Rozansky [117, Section 2.3.3] as the moduli space of certain
quiver-like diagram construction.
3.1 Chambers and attracting sets
Let Cochar(A) be the cocharacter lattice of A, and we denote aR := Cochar(A)⊗ZR ⊂ a = Lie(A).
We define the roots of the pair (X,A) to be the set of weights {α} appearing in the normal bundle
to XA. A chamber is defined to be a connected component of the complement of hyperplanes
cut out by roots, i.e.,
aR
∖ ⋃
α∈roots
α⊥ =
⊔
i
Ci. (3.3)
For every σ ∈ Ci ∩ Cochar(A), we have Xσ = XA.
Let C be a chamber, then we say that a root α is attracting (resp. repelling) if α is positive
(resp. negative) on C. For a connected component F ⊂ XA, we define the attracting part N+
X/F
of normal bundle to be the span of attracting root space in NX/F ; similarly, the repelling
part N−
X/F is defined to be the span of repelling root space in NX/F . Then we define attracting
submanifold AttrC(F) to be the subset
{x ∈ X | lim
t!0
σ(t) · x ∈ F}, (3.4)
for some σ ∈ C∩Cochar(A). AttrC(F) does not depend on the choice of σ, in fact, it is the expo-
nential of attracting part of the normal bundle N+
X/F . Define the union AttrC :=
∐
F AttrC(F),
then AttrC admits an immersion AttrC ↪! X×XA, x 7! (x, limt!0 σ(t) ·x). In general, AttrC is not
closed in X×XA, since limt!∞ σ(t) · x, if exists, is not in AttrC. We define AttrfC to be the set of
pairs (x, y) that belongs to a chain of closures of attracting A-orbits. AttrfC is closed in X× XA.
A partial order ⪯ on the set of connected components of XA is generated by letting
Fj ⊂ AttrC(Fi) ⇒ Fj ⪯ Fi.
We define closed subvarieties Attr<C ⊂ Attr≤C ⊂ X× XA by
Attr<C :=
⋃
Fj≺Fi
AttrC(Fj)×Fi, Attr≤C :=
⋃
Fj⪯Fi
AttrC(Fj)×Fi.
Note that AttrfC is a closed subvariety of Attr≤C and AttrfC ∩
(
Attr≤C \ Attr<C
)
= AttrC.
Example 3.5. In the case when X is constructed from quiver representation, there is a nice
description of attracting sets, due to Andrei Neguţ [96]. The setting in loc. cit. is for a Naka-
jima quiver variety (i.e., Godd = 1), nevertheless the arguments are applicable to the situation
when Godd ̸= 1. We record the result here. Consider X = Mζ+(v,w) for a quiver Q with
dimension vectors (v,w) and the stability condition ζ+, let A = C×
a that acts on framing
24 N. Ishtiaque, S.F. Moosavian and Y. Zhou
vector space as W = aW (1) ⊕W (2), so that the framing dimension vector decomposes accord-
ingly mathbfw = w(1) +w(2), then
XA =
⊔
v(1)+v(2)=v
Mζ
(
v(1),w(1)
)
×Mζ
(
v(2),w(2)
)
.
We choose the chamber C such that a > 0, then the same argument in [96, Lemma 3.10] can be
applied, and AttrC consists of quiver representations (V,W ),
(
V (1),W (1)
)
,
(
V (2),W (2)
)
in X×XA
that fits into exact sequences 0! V (1) ↪! V ↠ V (2) ! 0, that commutes with
(
xa, x̃a
)
and
also commutes with (αi, α̃i) through the splitting exact sequence 0!W (1) ↪!W ↠W (2) ! 0.
Moreover, the same argument in [96, Lemma 3.11] can be applied (since the moment map
equation is not essential in the proof), and AttrfC consists of quiver representations (V,W ),(
V (1),W (1)
)
,
(
V (2),W (2)
)
in X× XA that fits into the complex V (1) f
−! V
g
−! V (2), such that
� the composition g ◦ f = 0, and
� f and g commutes with (xa, x̃a) and also commutes with (αi, α̃i) through the splitting
exact sequence 0!W (1) ↪!W ↠W (2) ! 0.
� there exist filtrations of quiver representations
V (1) = E0 ↠ E1 ↠ · · · ↠ Ek−1 ↠ Ek = Im(f),
Ker(g) = F k ↠ F k−1 ↠ · · · ↠ F 1 ↠ F 0 = V (2),
such that the kernels of El ↠ El+1 and F l+1 ↠ F l are isomorphic quiver representations.
In particular,
Mζ
(
v(1),w(1)
)
×Mζ
(
v(2),w(2)
)
⪯ Mζ
(
v(1) + u,w(1)
)
×Mζ
(
v(2) − u,w(2)
)
,
for all u ∈ NQ0 .
The restriction of partial polarization PolX to XA decomposes according to the chamber C as
PolX|XA = PolX|XA,>0 + PolX|XA,fixed + PolX|XA,<0.
Definition 3.2. We define the index bundle ind = Pol+X |XA,>0 − Pol−X |XA,>0 ∈ KT
(
XA
)
, the al-
ternating sum of attracting part of partial polarization.
Lemma 3.2. PolX|XA,fixed is a partial polarization on XA.
Proof. Let TX = Pol+X + ℏ−1
(
Pol+X
)∨
+Pol−X + ℏ(Pol−X )
∨+E , then there exists a decomposition
of E into eigenspaces of A-actions
E |XA =
∑
λ∈Cochar(A)
E λ.
By the duality E ∼= E ∨, we have E −λ ∼=
(
E λ
)∨
, therefore we have
Θ
(
E 0
)
= Θ(E |XA)⊗
⊗
λ(C)>0
Θ
(
E λ
)⊗2
.
In particular, we see that Θ
(
E 0
)
admits a square root. Moreover, we have the decomposition of
the tangent bundle of XA in KT
(
XA
)
TXA = Pol+X |XA,fixed + ℏ−1
(
Pol+X |XA,fixed
)∨
+ Pol−X |XA,fixed + ℏ
(
Pol−X |XA,fixed
)∨
+ E 0,
which implies that PolX|XA,fixed is a partial polarization on XA. ■
We denote by PolXA the restriction of partial polarization PolX|XA,fixed on XA.
Elliptic Stable Envelopes and Dynamical R-Matrices 25
3.2 Attractive line bundle
This is the most crucial part of our construction of elliptic stable envelopes for partially-polarized
varieties. An important ingredient that we need from [109] is the following.
Definition 3.3. A line bundle L on EllT(X) is called attractive for a given chamber C if
degA L = degAΘ(N−
X/XA), where degA L is the degree of the restriction of L to the fiber along
the projection EllT
(
XA
)
! EllT/A
(
XA
)
.6 The degA takes value in H0
(
XA, S2Char(A)
)
.
Definition 3.4. From now on, we fix a square root for Θ(E ) and define SX := Θ(PolX) ⊗
Θ(E )⊗
1
2 .
Proposition 3.2. SX is an attractive line bundle for every chamber C of Cochar(A).
Proof. Since S2Char(A) is a torsion-free abelian group, it suffices to show that degA
(
S ⊗2
X
)
=
degA
(
Θ
(
N−
X/XA +N−∨
X/XA
))
. Since N−∨
X/XA is A-equivariantly isomorphic to N+
X/XA , we have
degA
(
Θ
(
N−
X/XA +N−∨
X/XA
))
= degA
(
Θ(NX/XA)
)
= degA(Θ(TX)).
On the other hand, S ⊗2
X
∼= Θ
(
PolX + Pol∨X + E
)
, and its A-degree equals to that of
Θ
(
Pol+X + ℏ−1
(
Pol+X
)∨
+ Pol−X + ℏ(Pol−X )
∨ + E
)
= Θ(TX),
this proves the lemma. ■
Remark 3.3. For a line bundle L on EllT(X), define L ▽ := L ∨ ⊗ Θ(TX). Then there is an
isomorphism S ▽
X
∼= Θ
(
PolopX
)
⊗Θ(E )⊗
1
2 , therefore S ▽
X is the attractive line bundle associated
to the opposite partial polarization.
The attractive line bundle associated to a polarization naturally restricts to A-fixed points. In
fact we can define SX,A := i∗SX ⊗Θ(−N−
X/XA) where i : EllT
(
XA
)
! EllT(X) is the map induced
by the inclusion XA ↪! X. On the other hand, we also have the line bundle SXA defined using the
restriction of the partial polarization PolXA ∈ Pic
(
EllT
(
XA
))
, i.e., SXA = Θ(PolXA)⊗Θ
(
E 0
)⊗ 1
2 .
SX,A is not isomorphic to SXA in general, and they are related as follows
SX,A ⊗ U ∼= SXA ⊗Θ(ℏ)− rk ind ⊗ τ(−ℏdet ind)∗U . (3.5)
U is the universal line bundle and τ(−ℏdet ind) is the translation [2], whose definitions are
recalled below.
3.3 Universal line bundle, Kähler torus and resonant locus
Let A be an abelian variety and A∨ its dual abelian variety, then there is a universal line
bundle UPoincaré on A∨×A. The sections of UPoincaré on the universal cover of A∨
s ×Az has the
same factor of automorphy as the function ϑ(sz)
ϑ(s)ϑ(z) .
Assumption. We assume that Pic(X) is finitely generated as an abelian group, and we fix
a set of generators {L ◦
i }ri=1, which induces a group homomorphism K = (C×)r ↠ Pic(X)⊗ZC×.
K is called the Kähler torus.
We choose an equivariant lift Li ∈ PicT(X) for each L ◦
i , then the elliptic Chern class gives
a map ci : EllT(X)! E. We consider ci × 1: EllT(X)× Ezi ! E× Ezi and define
U (Li, zi) := (ci × 1)∗UPoincaré,
6The degree of a line bundle L on an abelian variety A is defined as the image of L in the Néron–Severi
group NS(A) = Pic(A)/Pic0(A), see Appendix A for details.
26 N. Ishtiaque, S.F. Moosavian and Y. Zhou
where we identify E ∼= E ∨
zi . Moreover, we define the extended equivariant elliptic cohomol-
ogy ET(X) := EllT(X)× EK which is a scheme over BT,X := ET × EK. Using the identifica-
tion EK
∼=
∏r
i=1 Ezi , the line bundle U (Li, zi) is naturally defined on the extended equivariant
elliptic cohomology ET(X), and we set U :=
⊗r
i=1 U (Li, zi).
Let Fj ⪯ Fi be a pair of connected components of XA and consider the commutative diagram
ET(Fj ×Fi) ET/A(Fj ×Fi)
BT,X BT/A,X,
ϕ
ψ
p p
ϕ
where BT/A,X := ET/A × EK.
Definition 3.5. Let SX,F be the restriction of SX,A to a connected component F of XA. The
resonant locus ∆ is defined as the union of
p
(
suppRϕ∗
(
SX,Fj
⊗ U ⊠ (SX,Fi
⊗ U )∨
))
⊂ BT/A,X,
over all pairs Fj ⪯ Fi of connected components of XA. We also use the same notation ∆ for its
preimages.
Verbatim to that of [109, Proposition 2.6], we have the following.
Lemma 3.3. The complement of ∆ in ET(X) is open and dense.
For a homomorphism between abelian varieties g : ET ! EK, we associate an automor-
phism τ(g) : BT,X
∼= BT,X by (t, z) 7! (t, z + g(t)), where t is the coordinate on ET. We also use
the same notation τ(g) for its pullback to ET(X). Such transformation does not affect EllT(X),
only the Kähler parameters are shifted. An example is as follows: for a pair
µ ∈ Char(T) = Hom(ET,E), λ ∈ Cochar(K) = Hom(E,EK),
λµ is an abelian variety homomorphism, so we have a shift map τ(λµ).
The line bundle τ(g)∗U ⊗ U −1 depends linearly on g, i.e.,
τ(g1)
∗U
U
⊗ τ(g2)
∗U
U
∼=
τ(g1 · g2)∗U
U
.
In the case when µ ∈ Char(T) and λ ∈ Cochar(K), there is a meromorphic section ϑ(λ·µ)
ϑ(λ)ϑ(µ) ,
of the line bundle τ(λµ)∗U ⊗ U −1, here λ · µ is the coordinate multiplication [2, Lemma 2.4].
We will write U (λ, µ) for the line bundle τ(λµ)∗U ⊗ U −1.
3.4 Elliptic stable envelope
Now we have all the ingredients, and the main result of this section: the existence and uniqueness
of the elliptic stable envelope for partially-polarized varieties, directly follows from the general
construction in [109].
Theorem 3.1. If X is partially polarized, then there exists a unique section
StabC,SX
∈ Γ
(
ET
(
X× XA
)
\∆,SX ⊗ U ⊠ (SX,A ⊗ U )▽
)
,
such that it is supported on AttrfC and that its restriction to the complement of Attr<C is given
by (−1)rk ind[AttrC].
7
7For the definition of supports, see Appendix A.
Elliptic Stable Envelopes and Dynamical R-Matrices 27
Proof. This is a special case of [109, Theorem 1] applied to the attractive line bundle SX. ■
Equivalently, StabC,SX
gives rise to a map between sheaves SX,A ⊗ U ! SX ⊗ U over the
non-resonant locus BT,X\∆, which is uniquely characterized by
(1) The support of StabC,SX
is triangular with respect to C, i.e., StabC,SX
([Fi])|Fj = 0 for
a pair of connected components of XA such that Fj ⪯̸ Fi.
(2) The diagonal StabC,SX
([Fi])|Fi = (−1)rk indϑ(N−
X/Fi
), where ϑ(N−
X/Fi
) is the canonical sec-
tion of the elliptic Thom line bundle Θ(N−
X/Fi
).
We will use the above point of view on the stable envelope in the following discussions.
According to (3.5), the elliptic stable envelope is a map between sheaves
StabC,SX
: SXA ⊗Θ(ℏ)− rk ind ⊗ τ(−ℏdet ind)∗U ! SX ⊗ U .
This turns out to be more practical since it is usually easier to compute PolXA than SX,A.
Change of partial polarization. If we change the partial polarization by
Pol′X = PolX − (F+ + F−) + ℏ−1F∨
+ + ℏF∨
− ,
for some F+,F− ∈ KT(X), then the associated attractive line bundle changes by
S ′
X ⊗ U ∼= SX ⊗ τ(ℏdet(F+ − F−))
∗U ⊗Θ(ℏ)rkF ,
and the elliptic stable envelope changes accordingly
StabC,S ′
X
(z) = StabC,SX
(z + ℏdet(F+ − F−)) · (−1)rkFmoving ,
where z is the Kähler parameter (coordinate on EK), and Fmoving is the A-moving part of F |XA .
In particular, if we choose the opposite partial polarization PolopX = ℏ−1
(
Pol+X
)∨
+ ℏ(Pol−X )
∨,
then
StabC,S ▽
X
(z) = StabC,SX
(
z + ℏdet
(
Pol+X − Pol−X
))
· (−1)rkPolX,moving .
Example 3.6 (abelian Gauge groups). Let us consider the Higgs branch of an Abelian N = 2
gauge theory, i.e., Example 2.1. In this case X := µ−1
ev (0)
ζ−ss//G is a T-equivariant smooth defor-
mation of X0 := µ−1(0)ζ−ss//G over the base g⊥ev, where T = A×C×
ℏ =
((
C×)rkR/G)×C×
ℏ . C
×
ℏ
acts on R⊕R∨ as R⊕ℏ−1R∨ and acts on g as ℏ−1g. Then every A-fixed point of the hypertoric
variety X0 extends to an étale cover of the base g⊥ev, which must be trivial cover since every
point in this cover is repelled by the C×
ℏ action from the fixed point. Let Fp be the component
of XA that contains fixed point p ∈ XA
0 , then for any chamber C of Lie(A)R, Fp ⪯ Fq in X if and
only if p ⪯ q in X0. In other words, the chamber C induces exactly the same partial order on XA
and XA
0 . Moreover, the inclusion X0 ↪! X induces isomorphism on equivariant elliptic coho-
mology EllT(X0) ∼= EllT(X). It is straightforward to see that SX
∼= Θ(R) ∼= SX0 , i.e., X and X0
have the same attractive line bundle. The normal bundles are isomorphic N±
X/Fp
|p = N±
X0/p
|p,
so the index bundles are also isomorphic. To summarize, X and X0 have the same elliptic stable
envelope StabC,SX
([Fp]) = StabC,SX0
([p]). The explicit formula of the elliptic stable envelope
for hypertoric varieties can be found in [2], see also [133].
Example 3.7 (resolved determinantal varieties). Consider the quiver Q with one odd node,
i.e., Q0 = Qodd
0 = {1} and v = N,w = L, and ζ = ζ−, then Mζ(N,L) is nonempty if and
only if N ≤ L, and if nonempty then it is isomorphic to the total space of the vector bundle
Tot
(
V⊕L ! Gr(N,L)
)
, where V is the tautological bundle of rank N on the Grassmannian. It
is a resolution of the determinantal variety which parameterizes L× L matrices of rank ≤ N .
28 N. Ishtiaque, S.F. Moosavian and Y. Zhou
In this case A = TW = (C×)L and T = A × C×
ℏ . A acts on both the base and the fiber of
the bundle Tot
(
V⊕L ! Gr(N,L)
)
, and C×
ℏ scales the fiber with weight ℏ−1. Let the equivariant
parameters of the gauge group GLN be s = {sa}Na=1 and the equivariant parameters of A
be x = {xi}Li=1. Connected components of A-fixed points are labelled by order-preserving
embedding p : {1, . . . , N} ↪! {1, . . . , L}. Denote by {m1, . . . ,mL} the coordinates on Lie(A),
then we choose the chamber C such that m1 < · · · < mL. We choose the partial polarization as
in (3.2). Then we claim that the stable envelope is the following:8
StabC,SX
([Fp]) = SymSN
[(
N∏
a=1
fp(a)(sa,x, ℏ, z)
)
·
(∏
a>b
1
ϑ
(
sas
−1
b
))] , (3.6)
where SymSN
means summation over permutations {sa} 7! {sσ(a)}σ∈SN
, and fm(s,x, ℏ, z) is
the following function:
fm(s,x, ℏ, z) :=
ϑ
(
sxmℏm−Lz
)
ϑ
(
ℏm−Lz
) ∏
i<m
ϑ(sxi)
∏
j>m
ϑ(sxjℏ),
where z is the Kähler parameter corresponding to the ample line bundle (detV)−1.
Let us show that (3.6) is indeed the stable envelope. The first thing to check is that (3.6) is
a map between line bundles SXA ⊗ Θ(ℏ)− rk ind ⊗ τ(−ℏdet ind)∗U ! SX ⊗ U . Note that SXA
is a trivial line bundle, and ind has equivariant weights xi/xp(a) such that i > p(a). Every
summand in the symmetrization (3.6) can be rewritten as(∏
i,a
ϑ(saxi)
)(∏
a
ϑ
(
saxp(a)ℏp(a)−Lz
)
ϑ(saxp(a))ϑ
(
ℏp(a)−Lz
))( ∏
i>p(a)
ϑ(saxiℏ)
ϑ(saxi)ϑ(ℏ)
)
×
(∏
a>b
1
ϑ
(
sas
−1
b
))ϑ(ℏ)#(i>p(a)),
which is a meromorphic section of the line bundle
Θ(PolX)⊗ U ⊗ (τ(−ℏdet ind)∗U |Fp)
−1 ⊗Θ
(
VV∨)⊗−1
2 ⊗Θ(ℏ)rk ind,
which is isomorphic to SX ⊗ U ⊗ (τ(−ℏdet ind)∗U )−1 ⊗ Θ(ℏ)rk ind, this shows that (3.6) lives
in the correct line bundle.
Next, we verify that (3.6) is triangular with respect to C. In fact, if Fp′ ⪯̸ Fp is another
connected component of fixed points, in other words, p′ : {1, . . . , N} ↪! {1, . . . , L} is an order-
preserving embedding and there exists 1 ≤ a ≤ N such that p′(a) < p(a), then for every σ ∈ SN
there exists b such that p′(σ(b)) < p(b), so every summand in the symmetrization (3.6) vanishes.
Finally, the restriction of (3.6) to diagonal is
(−1)#(i>p(a))
N∏
a=1
∏
i<p(a)
i/∈Im(p)
ϑ
(
xi
xp(a)
) ∏
i>p(a)
ϑ
(
xp(a)
xiℏ
)
,
which equals to (−1)rk indϑ(N−
X/Fp
). This verifies that (3.6) is indeed the stable envelope for
Mζ(N,L) with respect to the chamber C.
8For the gauge-theoretic derivation of (3.6), see Section 6.1. Note that the gauge-theoretic computation uses
the normalization of stable envelope such that StabC,SX([F ])|F = ϑ(N−
X/F ).
Elliptic Stable Envelopes and Dynamical R-Matrices 29
3.5 Triangle lemma and duality
Let C′ ⊂ C be a face, and let A′ ⊂ A be the subtorus associated to the span of C′ in Lie(A).
Proposition 3.3 (triangle lemma). We have StabC,SX
= StabC′,SX
◦ StabC/C′,SX,A′
. Moreover,
StabC/C′,SX,A′
(z) = StabC/C′,S
XA
′ (z − ℏdet indXA′ ), (3.7)
where indXA′ = Pol+X |XA′ ,>0 − Pol−X |XA′ ,>0 is the alternating sum of the attracting part of partial
polarization with respect to the chamber C′.
Proof. This is verbatim to that of [2, Proposition 3.3], replacing polarization with partial
polarization. ■
Assume that XT is proper, then we have the equivariant pushforward f⊛ : Θ(TX)! OET,localized
for the map f : X ! pt, and accordingly we define Stab∗
C,SX
(z) := Stab−C,S ▽
X
(−z)∨, where
z ∈ EPicT(X) is the Kähler parameter and the dual means transpose with respect to the duality
SX ⊗ S ▽
X
product
−! Θ(TX)
f⊛
! OET,localized.
Proposition 3.4 (duality). We have
Stab∗
C,SX
◦ StabC,SX
= 1. (3.8)
Proof. This is verbatim to that of [2, Proposition 3.4], replacing polarization by partial polar-
ization. ■
Combine the duality (3.8) with the formula of the stable envelope for opposite partial polar-
ization, and we obtain the following corollary.
Corollary 3.1 (inverse stable envelope for the opposite chamber). We have
Stab−1
−C,SX
(z) = (−1)rkPolX,movingStabC,SX
(
−z + ℏdet
(
Pol+X − Pol−X
))∨
.
where z is the Kähler parameter (coordinate on EK).
From now on, unless specified, we will drop the attractive line bundle in the notation and
write StabC for StabC,SX
and StabC/C′ for StabC/C′,S
XA
′ , etc.
3.6 Abelianization
A general strategy to compute elliptic stable envelope for GIT quotients by nonabelian gauge
group is abelianization. This is described in detail for the symplectic reduction in [2, Section 4.3],
and in this subsection, we adapt their technique to the more general setting in Section 2.1. In
the following, we point out the necessary modification to [2, Section 4.3].
Let Sb (resp. Sf ) be a maximal torus of Gev (resp. Godd) and let Bb ⊃ Sb (resp. Bf ⊃ Sf )
be a Borel subgroup with Lie algebra bb (resp. bf ). Denote by S := Sb × Sf and B := Bb ×Bf ,
then we have a diagram similar to that of [2, equation (59)]
Fl µ−1
(
b⊥b
)ζ−ss
//S XS
X.
j+
π
j−
(3.9)
Fl = µ−1
(
g⊥ev
)ζ−ss
//B ∼= (µ× µR)
−1
(
g⊥ev × {ζ}
)
/U ∩ S, where µR : R⊕R∨ ! Lie(U)∨ is the real
moment map with respect to the action of a maximal compact subgroup U ⊂ G and a U -invariant
30 N. Ishtiaque, S.F. Moosavian and Y. Zhou
Hermitian metric on R ⊕ R∨, and XS = µ−1
(
s⊥b
)
//ζS is the GIT quotient of µ−1
(
s⊥b
)
by the
torus S. Note that j+ is a non-holomorphic embedding and j− is a holomorphic immersion. π is
a G/B-bundle, so every statement in [2, Section 4.3.2] still holds for π.
We have decompositions
π∗Gev = Nb + N ∨
b + Lie(Sb)⊗ OFl, π∗Godd = Nf + N ∨
f + Lie(Sf )⊗ OFl,
where Nb (resp. Nb) is the bundle on XS associated to the adjoint action of S on nb = [bb, bb]
(resp. nf = [bf , bf ]). The relative tangent bundle of π is Tπ = N ∨ = N ∨
b + N ∨
f , and the
normal bundle of j+ is Nj+ = N + ℏ−1Nb, and the normal bundle of j− is Nj− = ℏ−1N ∨
b .
For simplicity, let us choose the partial polarizations
PolX = Pol+X = R − Gev, PolXS
= Pol+XS
= R + ℏ−1N ∨
b − Nb − Lie(Sb)⊗ OXS
.
Note that the partial polarization on XS differs from the choice in Example 3.1 by ℏ−1N ∨
b −Nb,
which affects the stable envelope by
StabC(z) 7! StabC(z + ℏdetNb) · (−1)rkNb,moving ,
thus (3.24) still holds for such modified choice of partial polarization.
Now we have the modified version of [2, equation (63)]
j∗
(
PolXS
− ℏ−1N ∨
b + N ∨
f
)
= π∗PolX + Tπ,
where j = j− ◦ j+. Also notice that Θ
(
N ∨
f
) ∼= Θ(Nf ) ∼= π∗Θ(Godd)
⊗ 1
2 . Therefore, we have the
modified version of [2, equation (64)]
SX π∗SX ⊗Θ(Tπ) j∗−SXS
⊗Θ
(
−ℏ−1N ∨
b
)
SXS
.
π⊛ j∗+ j−⊛
(3.10)
The map j∗+ in (3.10) is surjective since any tautological class extends to XS .
Next, we choose a connected component F ⊂ XA, and let F ′ be the unique connected com-
ponent of π−1(F)A such that normal weights of F ′ in π−1(F) are repelling with respect to the
chamber C, i.e., (Tπ|F ′)>0 = 0. This implies that
(
N ∨|F ′
)
>0
= 0 by the same argument with [2,
Section 4.3.8]. Consider the following analog of (3.9)
F ′ FS ∩ µ−1
(
b⊥b
)ζ−ss
//S FS
F,
j′+
π′
j′−
where FS is the connected component of XA
S that contains F ′. Then by the same argument
with [2, Section 4.3.9], j′∗+ is surjective, and π∗(indF) = j′∗(indFS), where j′ = j′− ◦ j′+.
Now we have all ingredients to state the abelianization result. We claim that the following
analog of [2, equation (72)] is well defined and the diagram is commutative
SF ⊗ U ′ SFS
⊗ U ′
SX ⊗ U ⊗Θ(ℏ)rk ind SXS
⊗ U ⊗Θ(ℏ)rk ind,
j′−⊛(j′∗+)
−1
(π′
⊛)−1
Stab StabS
π⊛j∗+(j−⊛)−1
(3.11)
where U ′ = τ(−ℏdet ind)∗U . In fact, the argument in [2, Section 4.3.11] works verbatim,
so the inverse of the top horizontal arrow is well defined; the normal weights to b⊥b ⊂ g∨
Elliptic Stable Envelopes and Dynamical R-Matrices 31
are non-attracting then it follows that Attrf
(
µ−1
(
b⊥b
)A) ⊂ µ−1
(
b⊥b
)
, thus StabS ◦ j′−⊛ factors
through j−⊛, so the inverse of the bottom horizontal arrow is also well defined. To show that
the diagram commutes, we only need to show that Stab agrees with the composition of the
rest of the arrows in the diagram when restricted to the diagonal. The point is to compare
the difference between π∗N−
X/F and N−
XS/F ′ , and from our previous discussion we learn that
the new repelling directions are (Tπ|F ′)moving and (Nj− |F ′)moving. (Tπ|F ′)moving is cancelled out
by π⊛, and (Nj− |F ′)moving is cancelled with (j−⊛)
−1 ◦ StabS ◦ j′−⊛. This concludes the proof of
abelianization (3.11).
3.7 R-matrices and dynamical Yang–Baxter equations
Definition 3.6. Let C1, C2 be two chambers in Lie(A), then we define the R-matrix
RC2�C1 := Stab−1
C2
◦ StabC1 . (3.12)
This is a map from SXA ⊗ τ(−ℏdet ind1)∗U to SXA ⊗ τ(−ℏdet ind2)∗U , where ind1 and ind2
are index bundles for the chambers C1 and C2 respectively.
Obviously it follows from definition that
RC3�C2RC2�C1 = RC3�C1 . (3.13)
Moreover, if C1 and C2 share a face C′, then (3.7) implies that
RC2�C1(z) = RC2/C′�C1/C′(z − ℏdet indXA′ ,C′), (3.14)
where A′ is the subtorus corresponding to the chamber C′ and indXA′ ,C′ is the index bundle
on XA′
for the chamber C′. If C′ is of codimension one in both C1 and C2, then we say C′
is a wall and define Rwall := RC2/C′�C1/C′ . The dynamical Yang–Baxter equation is a direct
consequence of (3.13) and (3.14) for A = (C×)3. To see this, let m1, m2, m3 be coordinates
on Lie(A), then the two paths connecting the chamber C123 = {m1 < m2 < m3} to its oppo-
site C321 = {m3 < m2 < m1}, namely C123 ! C132 ! C312 ! C321 and C123 ! C213 ! C231 !
C321, gives rise to equation between two sequence of composing R-matrices
R12(z − ℏdet ind3(12))R13(z − ℏdet ind(13)2)R23(z − ℏdet ind1(23))
= R23(z − ℏ det ind(23)1)R13(z − ℏdet ind2(13))R12(z − ℏ det ind(12)3), (3.15)
where Rij is the R-matrix for the ij-wall, and the subscript of ind means the wall and chamber
for which the index bundle is taken, for example, 3(12) means the chamber {m3 < m1 = m2}.
The particular case C3 = C1 in (3.13) implies the unitarity R21(−u, ℏ, z)R(u, ℏ, z) = 1,
where R(u, ℏ, z) is the wall R-matrix and subscript 21 means swapping the two tensor compo-
nents.
3.8 R-matrices for quiver varieties
Let us write down the dynamical Yang–Baxter equation for the quiver variety Mζ−(w) for
a quiver Q with decomposition of nodes Q0 = Qev
0 ⊔ Qodd
0 into even and odd parts and de-
composition of arrows Q1 = Q+
1 ⊔ Q−
1 such that the condition (⋆) in Example 3.2 is satisfied.
We take A = (C×)3 ⊂ TW such that the framing vector space decomposes into eigenspaces as
W = a1W
(1) + a2W
(2) + a3W
(3). Choosing the partial polarization (3.2), we can write down
the index bundle for the chamber corresponding to the weight decomposition W = aW ′ + bW ′′
such that b/a is attracting
ind =
∑
i,j∈Q0
DijHom
(
V ′
i,W ′′
j
)
+ (Qij − Pij)Hom
(
V ′
i,V ′′
j
)
,
32 N. Ishtiaque, S.F. Moosavian and Y. Zhou
where Qij is the weighted adjacency matrix of Q defined by
Qij = #
(
a ∈ Q+
1 : h(a) = j, t(a) = i
)
−#(a ∈ Q−
1 : h(a) = j, t(a) = i),
and Dij is the diagonal matrix such that
Dii =
{
1, i /∈ Qev,−
0 ,
−1, i ∈ Qev,−
0 ,
and Pij is the diagonal matrix such that
Pii =
1, i ∈ Qev,+
0 ,
−1, i ∈ Qev,−
0 ,
0, i ∈ Qodd
0 .
Define the quiver R-matrix RQ(z) := Rwall
(
z + ℏ
(
P − Qt
)
v
)
, where z := {zi}i∈Q0 are the
equivariant parameters corresponding to the ample line bundles (detVi)−1. Let µ := Dw− Cv,
where
C = 2P− Q− Qt, (3.16)
then (3.15) is specialized to
RQ
12(z)R
Q
13
(
z − ℏµ(2)
)
RQ
23(z) = RQ
23
(
z − ℏµ(1)
)
RQ
13(z)R
Q
12
(
z − ℏµ(3)
)
. (3.17)
This kind of equation is called the dynamical Yang–Baxter equations and was first studied by
Felder [51, 52].
Example 3.8 (sl(1|1)). Consider the resolved determinantal variety case (see Example 3.7),
namely a quiver Q with one odd node, i.e., Q0 = Qodd
0 = {1}. We choose stability condi-
tion ζ = ζ− and partial polarization (3.2). We choose the framing dimensions w(1) = w(2) =
w(3) = 1, then Mζ
(
w(1)
)
is disjoint union of two points, and
Mζ
(
w(1) +w(2)
)
= pt ⊔ Tot
(
O(−1)⊕2 ! P1
)
⊔ C4.
Let us label the basis for the elliptic cohomology ET
(
Mζ
(
w(1)
))
, which is a free module of rank
two over Ex × Eℏ × Ez, by v0 =
[
Mζ(0, 1)
]
and v1 =
[
Mζ(1, 1)
]
, then we can write down the
quiver R-matrix in the basis {v0 ⊗ v0, v1 ⊗ v0, v0 ⊗ v1, v1 ⊗ v1}
RQ(u, z) =
1 0 0 0
0 ϑ(z)ϑ(zℏ−2)ϑ(u)
ϑ
(
zℏ−1
)2
ϑ
(
uℏ−1
) ϑ(ℏ)ϑ(uzℏ−1)
ϑ
(
zℏ−1
)
ϑ
(
u−1ℏ
) 0
0 ϑ(ℏ)ϑ(zu−1ℏ−1)
ϑ
(
zℏ−1
)
ϑ
(
u−1ℏ
) ϑ(u)
ϑ
(
uℏ−1
) 0
0 0 0 ϑ(uℏ)
ϑ
(
u−1ℏ
)
, (3.18)
where u = x2/x1 is the ratio between equivariant weights in the decomposition W = x1W
′ +
x2W
′′. In this case C = 0, which is the Cartan matrix for sl(1|1), and µ(v0) = µ(v1) = 1, so the
dynamical Yang–Baxter equation (3.17) reads
RQ
12(u, z)R
Q
13(uw, z/ℏ)R
Q
23(w, z) = RQ
23(w, z/ℏ)R
Q
13(uw, z)R
Q
12(u, z/ℏ), (3.19)
which is the dynamical Yang–Baxter equation for sl(1|1) in the fundamental representation.
Elliptic Stable Envelopes and Dynamical R-Matrices 33
Example 3.9 (sl(n|1)). Next, we consider a generalization to the above example, namely the
quiver in Example 2.3. Let Q be an An quiver with Qev
0 = {1, . . . , n − 1} and Qodd
0 = {n},
and the orientation is such that h(a) < t(a) for all arrows a ∈ Q1. We take Q1 = Q+
1 . We
choose stability condition ζ = ζ− and the partial polarization (3.2). We choose framing dimen-
sionsw(1) = w(2) = w(3) = (1, 0, . . . , 0), thenMζ
(
w(1)
)
is the disjoint union of n+1 points which
is labeled by integers α ∈ {0, 1, . . . , n} so that the gauge dimension vector of the corresponding
quiver variety is
vα = (
n︷ ︸︸ ︷
1, . . . , 1︸ ︷︷ ︸
α
, 0, . . . , 0).
Using the above notation, we have
Mζ
(
vα + vβ,w
(1) +w(2)
)
=
pt, α = β < n,
T ∗P1, α < β < n or β < α < n,
Tot
(
O(−1)⊕2 ! P1
)
, α < β = n or β < α = n,
C4, α = β = n.
(3.20)
Let us label the basis for the elliptic cohomology ET
(
Mζ
(
w(1)
))
, which is a free module of
rank n+ 1 over Ex × Eℏ × Ez, by vα =
[
Mζ
(
vα,w
(1)
)]
. Here the Kähler parameter z = (zi)
n
i=1
corresponds to the ample line bundles
(
detV∨
i
)n
i=1
, where Vi is the tautological bundle corre-
sponding to the i-th node.
For the quiver Q, we have the Cartan matrix
Ci,j = 2δi,j(1− δi,n)− δi+1,j − δi−1,j , 1 ≤ i, j ≤ n,
which is the symmetric Cartan matrix of sl(n|1) [55]. The weights of vα are
µ(vα) = w(1) − Cvα =
(1, 0, . . . , 0), α = 0,
(0, . . . , 0︸ ︷︷ ︸
α−1
,−1, 1, 0, . . . , 0), 0 < α < n,
(0, . . . , 0, 1), α = n,
(3.21)
which is aligned with the weights for the fundamental representation of sl(n|1). Moreover, the
Kähler parameter shifts between the wall R-matrix and the quiver R-matrix is
(
P− Qt
)
vα =
(0, . . . , 0︸ ︷︷ ︸
α−1
, 1, 0, . . . , 0), 0 < α < n,
(0, . . . , 0), α = 0 or α = n.
Using the known results on the R-matrices for all four varieties in (3.20), see (3.18) and [2,
equation (85)], we can write down the quiver R-matrix in this case. We introduce shorthand
notations for some special functions
A(u, z) =
ϑ(zℏ)ϑ
(
zℏ−1
)
ϑ(u)
ϑ(z)2ϑ
(
uℏ−1
) , B(u, z) =
ϑ(ℏ)ϑ(uz)
ϑ(z)ϑ
(
u−1ℏ
) , C(u) =
ϑ(u)
ϑ
(
uℏ−1
) ,
D(u) =
ϑ(uℏ)
ϑ
(
u−1ℏ
) ,
where we have suppressed the ℏ-dependence. Then the quiver R-matrix in this case is the
following:
RQ(u, z)(vα ⊗ vβ)
34 N. Ishtiaque, S.F. Moosavian and Y. Zhou
=
vα ⊗ vβ, α = β < n,
D(u)vα ⊗ vβ, α = β = n,
C(u)vα ⊗ vβ +B
(
u, ℏ−δβ,n
∏β
i=α+1 zi
)
vβ ⊗ vα, α < β,
A
(
u, ℏ−δα,n
∏α
i=β+1 zi
)
vα ⊗ vβ +B
(
u, ℏδα,n
∏α
i=β+1 z
−1
i
)
vβ ⊗ vα, β < α.
RQ satisfies the dynamical Yang–Baxter equation (3.17) for µ given by weights for the funda-
mental representation of sl(n|1) (3.21).
For quivers of finite or affine type A, we make the following observation. In addition to the
condition (⋆) in Example 3.2, we also require that
(⋆⋆) if the valency at an odd node is two then the arrows connected to it are from different
groups, i.e., exactly one of the arrows is from Q+
1 and the other is from Q−
1 .
Here is an example of an allowed quiver (after adding framing and doubling)
− − − + + −
.
For simplicity, we consider an Âm+n−1 quiver Q and label the nodes by Q0 = Z/(m+ n)Z. We
choose the orientation Q1 = {i− 1! i : i ∈ Z/(m+n)Z}, and choose a splitting Q1 = Q+
1 ⊔Q−
1
such that |Q+
1 | = m and |Q−
1 | = n. Define
ci =
{
+1, if (i− 1! i) ∈ Q+
1 ,
−1, if (i− 1! i) ∈ Q−
1 .
Then the condition (⋆) is equivalent to ci = ci+1 for all i ∈ Qev
0 , and the condition (⋆⋆) is
equivalent to ci = −ci+1 for all i ∈ Qodd
0 . The Cartan matrix (3.16) reads
Ci,j = (ci + ci+1)δi,j − ciδi,j+1 − cjδi+1,j .
Ci,j is exactly the symmetric Cartan matrix of the Kac–Dynkin diagram for ŝl(m|n) associ-
ated to the quiver Q. Moreover, the quiver R-matrix RQ satisfies the dynamical Yang–Baxter
equation (3.17) for µ given by weights in a certain highest-weight module of ŝl(m|n).
3.9 K-theory and cohomology limit
In the q ! 0 limit, the elliptic curve E = C×/qZ degenerates to a nodal compactification of C×,
and the elliptic cohomology EllT(X) degenerates to the K-theory KT(X)⊗C, such that sections
of line bundles on EllT(X) becomes sections of line bundles on KT(X) ⊗ C. Strictly speaking,
because of half-periodicity property of ϑ-function: ϑ
(
e2πix
)
= −ϑ(x), their q ! 0 limit is defined
on the double cover of C×.
We expect that the elliptic stable envelope degenerates to the K-theoretic stable envelope in
the q ! 0 limit, such that the Kähler parameter z degenerates to the slope parameter in the
K-theory side. Suppose that
lim
ln z!∞
ln q!∞
Re
(
− ln z
ln q
)
= s ∈ PicT(X)⊗Z R, (3.22)
Elliptic Stable Envelopes and Dynamical R-Matrices 35
such that the slope s is generic, and then define
StabsC := lim
q!0
[
(det PolX)
− 1
2 ◦ StabC ◦ (det PolXA)
1
2
]
∈ KT
(
X× XA
)
. (3.23)
For every connected component F ⊂ XA,
StabsC([F ])|F = (−1)rk ind
(
detN−
X/F
detPolX|F ,moving
) 1
2 ∧
(N−
X/F )
∨
=
(
−
√
ℏ
)rk ind
(det PolX|F ,>0)
−1(detGodd|F ,<0)
− 1
2
∧
(N−
X/F )
∨.
Moreover, StabsC is supported on AttrfC, i.e., if F2 ⪯̸ F1 then StabsC([F1])|F2 = 0. We expect
that StabsC is the K-theoretic stable envelope with slope s for the chamber C. In other words,
StabsC also satisfies the degree condition, which is checked in the following.
For a connected component F of XA, define
∆F := Convex hull (SuppAStab
s
C([F ])|F ) ,
where SuppA of a KT(F) class is the set of A-weights that appears in that class.
Proposition 3.5. Let X be a quotient of Hamiltonian reduction (2.1), assume moreover that
either
� the gauge group is abelian, i.e., the situation in Example 2.1, or
� X is constructed from the quiver representation, i.e., the situation in Section 2.2,
then for every pair of connected components F2 ⪯ F1 in XA with respect to the chamber C,
SuppAStab
s
C([F1])|F2 ⊂ ∆F2 + A-weight of s|F2 − A-weight of s|F1 . (3.24)
Proof. Suppose that X = µ−1
ev (0)
ζ−ss//G, where G = Gev × Godd acts on the representa-
tion R with moment map µ : R ⊕ R∨ ! g∨ and µev is the composition of µ with projec-
tion prev : g
∨ ! g∨ev, and ζ is a generic stability such that µ−1
ev (0)
ζ−ss = µ−1
ev (0)
ζ−s. As we have
assumed in the setup of Section 2.1, µ−1
ev (0)
ζ−ss is smooth with free G action.
Step 1. Assume that G is abelian, then according to our previous discussions in Exam-
ple 3.6, the equivariant elliptic cohomology of X is isomorphic to that of the central fiber
X0 = µ−1(0)ζ−ss//G: EllT(X) ∼= EllT(X0), and elliptic stable envelopes of X and X0 are identified
via this isomorphism. X0 is a hypertoric variety, thus the result follows from [2, Proposition 4.2].
Step 2. If G is nonabelian, then we shall utilize the abelianization (3.11). It suffices to prove
in the special case when A = C×, by the same argument of [2, Proposition 4.2]. By step 1, we
have
SuppAStab
s
S
([
F ′
1
])
|F ′
2
⊂ ∆F ′
2
+ A-weight of s|F ′
2
− A-weight of s|F ′
1
.
The abelianization gives
Stabs([F1])|F2 =
(
detN ∨
f |F ′
2
)− 1
2StabsS
([
F ′
1
])
|F ′
2∧
((Nj− |F ′
2
)moving)∨ ·
∧
((Tπ|F ′
2
)moving)∨
(
detN ∨
f |F ′
1,fixed
) 1
2 ,
Stabs([F2])|F2 =
(detN ∨
f |F ′
2,moving)
− 1
2StabsS
([
F ′
2
])
|F ′
2∧
((Nj− |F ′
2
)moving)∨ ·
∧
((Tπ|F ′
2
)moving)∨
.
36 N. Ishtiaque, S.F. Moosavian and Y. Zhou
The first line in the above equation implies that
SuppAStab
s([F1])|F2 ·
∧
((Nj− |F ′
2
)moving)
∨ ·
∧
((Tπ|F ′
2
)moving)
∨
⊂ ∆F ′
2
− 1
2
A-weight of detN ∨
f |F ′
2
+ A-weight of s|F ′
2
− A-weight of s|F ′
1
,
and the second line implies that
∆F ′
2
= ∆F2 ·
∧
((Nj− |F ′
2
)moving)
∨ ·
∧
((Tπ|F ′
2
)moving)
∨ +
1
2
A-weight of detN ∨
f |F ′
2
,
thus we have
SuppAStab
s([F1])|F2 ⊂ ∆F2 + A-weight of s|F2 − A-weight of s|F1 .
This finishes the proof. ■
A further reduction from the K-theory to cohomology can be defined by
StabC := lowest cohomological degree term in ch
(
StabsC
)
, (3.25)
where ch: KT
(
X× XA
)
! HT
(
X× XA
)
is the T-equivariant Chern character map. The compo-
sition of reductions StabC ! StabsC ! StabC amounts to replacing ϑ(x) by x.
For every connected component F ⊂ XA, StabC([F ])|F = (−1)rk inde(N−
X/F ), where e(·) is
the T-equivariant Euler class. Moreover, StabC is supported on AttrfC, i.e., if F2 ⪯̸ F1 then
StabC([F1])|F2 = 0. We expect that StabC is the cohomological stable envelope for the cham-
ber C. In other words, StabC also satisfies the degree condition, which is checked in the following.
Proposition 3.6. Under the assumption of Proposition 3.5, then for every pair of connected
components F2 ⪯ F1 in XA with respect to the chamber C, degA StabC([F1])|F2 < dimN−
X/F2
,
where degA of an element in HT
(
XA
)
is the degree in C[a] under the non-canonical splitting
HT
(
XA
) ∼= HT/A
(
XA
)
⊗ C[a].9
Proof. The idea is essentially the same as Proposition 3.5. Namely, the abelian case is the
same as hypertoric varieties, which is shown by using the explicit formula [2, equation (56)]. For
the nonabelian case, we use the abelianization and get
Stab([F1])|F2 =
StabS
([
F ′
1
])
|F ′
2
e((Nj− |F ′
2
)moving) · e((Tπ|F ′
2
)moving)
.
Since the abelian stable envelope satisfies degA
(
StabS
([
F ′
1
])
|F ′
2
)
< dimN−
XS/F2,S
, we conclude
that
degA Stab([F1])|F2 = degA(StabS
([
F ′
1
])
|F ′
2
)− dim(Nj− |F ′
2
)moving − dim(Tπ|F ′
2
)moving
< dimN−
XS/F2,S
− dim(Nj− |F ′
2
)moving − dim(Tπ|F ′
2
)moving
= dimN−
X/F2
.
This finishes the proof. ■
Propositions 3.5 and 3.6 are summarized in the following
Corollary 3.2. Under the assumption of Proposition 3.5, StabsC is the K-theoretic stable envelope
with slope s for the chamber C, and StabC is the cohomological stable envelope for the chamber C.
9Different choice of splitting does not change the degree in C[a].
Elliptic Stable Envelopes and Dynamical R-Matrices 37
Example 3.10 (Resolved Determinantal Varieties). We apply (3.23) to (3.6) and obtain the
K-theoretic stable envelope for the resolved determinantal variety10
StabsC([Fp]) = ℏ
#(i>p(a))
2 SymSN
[(
N∏
a=1
fp(a)(sa,x, ℏ, s)
)
·
(∏
a>b
â
(
sas
−1
b
))]
, (3.26)
where fm(s,x, ℏ, s) is the following function
fm(s,x, ℏ, s) := (sxm)
⌊s⌋
∏
i<m
(
1− s−1x−1
i
) ∏
j>m
(
1− ℏ−1s−1x−1
j
)
,
and â is a version of Â-genus â(w) = 1
w
1
2−w− 1
2
. A further reduction to cohomology using (3.25)
gives the cohomological stable envelope
StabC([Fp]) = SymSN
[(
N∏
a=1
fp(a)(sa,x, ℏ)
)
·
(∏
a>b
1
sa − sb
)]
, (3.27)
where fm(s,x, ℏ) is the following function fm(s,x, ℏ) :=
∏
i<m(s+ xi)
∏
j>m(s+ xj + ℏ). Note
that (3.27) equals to (−1)#(i>p(a))W
(10)
w0,I
(−s,x,−ℏ), where W (10)
w0,I
is the super weight function
in [117] for the longest element w0 ∈ SN and I = w0(image(p)). The sign (−1)#(i>p(a)), which
equals to (−1)rk ind, reflects the choice of normalization of the stable envelope in the loc. cit.
being StabC([F ])|F = e(N−
X/F ) as opposed to (−1)rk inde(N−
X/F ), see Axiom A1 in Definition 4.1
of loc. cit.
Part II
The solution to dYBE for sl(1|1)
from 3d N = 2 SQCD
4 Stable envelopes and R-matrix from gauge theory: the setup
In the previous sections, we have defined the elliptic stable envelope for certain non symplectic
varieties. Examples of such varieties include Higgs branches of 3d N = 2 theories. We have
characterized the stable envelopes by their formal properties and using these stable envelopes,
we defined the R-matrix satisfying the dynamical Yang–Baxter equations for superspin chains.
In practice, it is rather difficult to construct stable envelopes from their definitions alone. To
write down explicit formulas for these elliptic functions, we turn to the 3d N = 2 theories. In
this section, we explain the motivation behind computing certain partition functions of these
theories that we argue should give us stable envelopes. Finally, in Section 6 when we actually
compute these partition functions, we shall give an a posteriori justification for the computation
by showing that the partition functions satisfy the criteria for elliptic stable envelopes set up
earlier and that the R-matrix satisfies the dynamical Yang–Baxter equations. In fact, results
from these computations were already used to define the functional forms of the elliptic sl(1|1)
stable envelopes (3.6) and the R-matrix (3.18) and we checked that they indeed fulfill all the
necessary criteria. Let us now go through the details of how we actually computed them.
10Here we use the identities limq!0 ϑ(w) = w
1
2 − w− 1
2 and limq!0
ϑ(wz)
ϑ(z)
= w⌊s⌋+ 1
2 .
38 N. Ishtiaque, S.F. Moosavian and Y. Zhou
4.1 The Bethe/Gauge correspondence
The correspondence relates integrable spin chains to supersymmetric quiver gauge theories, for
both purely bosonic [102, 103] and superspin [99] chains. One way to introduce this is from
a correspondence between Kac–Dynkin diagrams and quivers.
An integrable spin chain is based on a Lie algebra, say sl(m|n), which can be characterized
by its Kac–Dynkin diagram. For example, the following is a Kac–Dynkin diagram for sl(1|3)
.
A hollow circle represents an even simple root and a crossed circle represents an odd simple root.
In order to fully characterize a spin chain, in addition to the symmetry, we also need to specify
the representations or the spins that appear in the chain and the anisotropy. For simplicity, let
us only consider spin chains with highest-weight representations, with the highest-weight being
the fundamental weight, then we just need to specify the number of sites in the spin chain. For
the anisotropy, we have three choices (see Table 1), fully isotropic (XXX/rational), partially
anisotropic (XXZ/trigonometric/K-theoretic), and fully anisotropic (XYZ/elliptic).
The Bethe/Gauge correspondence relates this data to a quiver such as the following:11
N1 N2 N3
L
(4.1)
Each circular/gauge node corresponds to a node in the Kac–Dynkin diagram – even nodes are
distinguished from the odd ones by the presence of a self-loop. The presence of a single framing
node attached to the left-most gauge node reflects our choice regarding the spin chain only
having the fundamental highest weights. The value L of the framing node is the size of the spin
chain. The gauge dimensions N1, N2 and N3 are not part of the data defining the spin chain.
Rather, they correspond to a sector of the spin chain with definite magnon numbers. To find
the gauge theory dual of the full spin chain we need to consider multiple quivers, varying the
gauge dimensions over all possible magnon numbers. The choice of anisotropy is not part of the
quiver itself, but it will be reflected in the choice of gauge theory that we shall assign to the
quiver. The quiver (4.1) is a special case of Example 2.4 with n = 3.
To quivers like (4.1), we can assign supersymmetric gauge theories with 4 supercharges in
various dimensions. In the version of the Bethe/Gauge correspondence in which we are interested
in this paper, the relation between anisotropy and gauge theory is as follows:
Rational 1d N = 4 (quantum mechanics),
Trigonometric 2d N = (2, 2),
Elliptic 3d N = 2.
(4.2)
In the absence of any odd node, the amount of supersymmetry is doubled in all cases. All these
theories possess the same classical Higgs branch MH , which is the quiver variety Mζ(v,w) (2.3)
with v = (N1, N2, N3) and w = (L, 0, 0). The stability condition ζ corresponds to the real FI
11In the hierarchy of quivers, the first one is just a Kac–Dynkin diagram (e.g., (1.8)), then there’s the framed
and doubled quiver (e.g., (1.7)), and finally the one in (4.1) is a tripled quiver that is traditionally used in physics
to define gauge theories with 4 supercharges.
Elliptic Stable Envelopes and Dynamical R-Matrices 39
parameter in gauge theories. Supersymmetric ground states of these theories correspond to dif-
ferent cohomologies of the Higgs branch. For quantum mechanics, Witten’s Morse theory argu-
ments [144] describe the space of ground states as the ordinary equivariant cohomologyHA(MH)
where A is the maximal torus of the SU(L) flavor symmetry of the quiver. In the 2d and 3d cases
the cohomology is replaced by K-theory KA(MH) and elliptic cohomology EllA(MH) respec-
tively [145, 146]. On the spin chain side, for the rational, trigonometric, and elliptic cases, we
have the Yangian Yℏ(g), the quantum affine algebra Uℏ(ĝ), and the elliptic dynamical quantum
group Eτ,ℏ(g) as the spectrum generating algebras. The Bethe/Gauge correspondence identifies
the Bethe eigenstates of the spin chain with the space of supersymmetric vacua of the corre-
sponding gauge theories. Consequently, it conjectures an action of the spectrum generating
algebra on the corresponding cohomology [53, 88, 106].12
There are more choices of gauge theories that we can assign to quivers such as (4.1) and find
dual gauge theoretic interpretations of spin chain quantities. For example, in the original formu-
lation of the Bethe/Gauge correspondence [99, 102, 103], for both bosonic and supersymmetric
spin chains, the gauge dual of the rational spin chain was presented as a 2d N = (4, 4) (bosonic
spin chain) or an N = (2, 2) (superspin chain) theory. The quantum mechanics from (4.2) is
simply the dimensional reduction of these 2d theories. The supersymmetric vacua of these 2d
theories correspond to the quantum deformation of the equivariant cohomology. In principle,
there should be 3d and 4d theories with 4 supercharges that quantize K-theory [72, 73, 75, 106]
and elliptic cohomology, and whose dimensional reductions correspond to the 2d and 3d theories
in (4.2). In this paper we shall not be concerned with quantum cohomology, regardless, it is of
some interest to look at the 2d dual of the rational spin chains. We consider the example of
a sl(1|1) spin chain in the following.
(a) Kac–Dynkin diagram.
N L
Q
Q̃
(b) Gauge theory quiver.
Figure 3. sl(1|1) Kac–Dynkin diagram and the corresponding quiver.
Example 4.1 (sl(1|1) spin chains and 2d N = (2, 2) Gauge theories). Consider a length L spin
chain whose sites are labeled by i = 1, . . . , L. At the ith site we assign the sl(1|1) representation
with the highest-weight
(
λ
(1)
i , λ
(2)
i
)
. We are using the gl(1|1) notation to write down the weights,
i.e., λ
(1)
i + λ
(1)
i is the sl(1|1) weight of the ith site. We also consider complex inhomogeneities vi.
Excitations containing N magnons with rapidities σCa for a = 1, . . . , N correspond to Bethe
eigenstates if the rapidities satisfy the Bethe ansatz equations (BAE) [114] (also see [143])
L∏
i=1
+σCa + i
2ϵ− vi + iϵλ
(1)
i
−σCa − i
2ϵ+ vi + iϵλ
(2)
i
= eiφ, a = 1, . . . , N, (4.3)
The inhomogeneities and the phase on the right-hand side are arbitrary auxiliary parameters
of the spin chain. ϵ is the quantization parameter of the spin chain.13 Note that the value
of λ
(1)
i − λ
(2)
i is somewhat ambiguous since it can be changed by shifting the (arbitrary) inho-
mogeneity vi. The sl(1|1) charge λ(1)i + λ
(2)
i is invariant under such shifts.
12Since a single gauge theory corresponds to a specific magnon sector, we need to take direct sum of the
cohomologies over gauge dimensions to get the full spin chain Hilbert space and the algebra action (cf. the sums
over v in Conjecture 1.1).
13Usually the quantization parameter in a quantum mechanical system is written as ℏ, however in this paper,
we are using ℏ for the elliptic deformation parameter which is roughly related to ϵ by ϵ ∼ ln ℏ.
40 N. Ishtiaque, S.F. Moosavian and Y. Zhou
The BAE is the condition of extremizing a potential function called the Yang–Yang function
Y
(
σC) = N∑
a=1
L∑
i=1
(
σCa −mC
i
)(
ln
(
σCa −mC
i
)
− 1
)
+
N∑
a=1
L∑
i=1
(
−σCa − m̃C
i
)(
ln
(
−σCa − m̃C
i
)
− 1
)
− iφ
N∑
a=1
σCa , (4.4)
where we have defined the new parameters
mC
i := vi − iϵλ
(1)
i − i
2
ϵ, m̃C
i := −vi − iϵλ
(2)
i +
i
2
ϵ. (4.5)
In terms of this function, the BAE becomes: exp
(
2πi∂Y (σC)
∂σC
a
)
=1, a=1, . . . , N . The Bethe/Gauge
correspondence identifies the Yang–Yang function (4.3) as the effective twisted superpotential of
the 2d N = (2, 2) theory defined by the quiver Figure 3b in the presence of twisted masses [99].
We identify the rapidities σCa with the adjoint scalars in the Cartan part of the N = (2, 2)
U(N) vector multiplet. Then we recognize the 1-loop contributions of the fundamental and the
anti-fundamental chirals Qi
a, Q̃a
i in the first and the second line of (4.4) [65, 147]. Masses of
these chirals are precisely the parameters defined in (4.5). We also notice the microscopic linear
twisted superpotential as the last term in the second line.
For positive real FI parameter, the 2d theory flows in the infrared to a phase where it is
described by a sigma model into its Higgs branch [147]. The Higgs branch is acted on by SU(L)
and the fixed points of the U(1)L−1 action correspond to the vacua of the theory. When put on
an interval, such as I × S1 for some interval I, the theory has brane-type boundary conditions
supported on the attracting and repelling sets of these fixed points. All this Higgs branch
structure can just as well be studied by looking at the dimensionally reduced N = 4 quantum
mechanics (from (4.2)) on I using Morse theory [144].
4.2 3d N = 2 SQCD and its parameters
The 3d theory assigned to the quiver Figure 3b is the N = 2 SQCD. The theory has a gauge
group G := U(N) with Lie algebra g := u(N). The flavor symmetry visible in the quiver
is FA := SU(L) with Lie algebra fA := su(L). There is an additional U(1)ℏ symmetry with Lie
algebra Rℏ. The total flavor symmetry we consider is F := SU(L) × U(1)ℏ, whose Lie algebra
we denote by f.
Let trA be the component of the gauge field valued in the center of u(N). From this abelian
gauge field, we can form a current ⋆ trF in 3d which is conserved by Bianchi identity. This
gives rise to an abelian topological symmetry U(1)top. It is topological in the sense that only
the monopoles and none of the fields in the Lagrangian are coupled to this current. The Lie
algebra of U(1)top is written as Rtop. This global symmetry can be weakly gauged by introducing
a background vector multiplet for it. Let Atop, σtop and Dtop be the gauge field, the scalar, and
the auxiliary scalar in this vector multiplet. The purely bosonic part of the Lagrangian coupling
this multiplet to the topological current is
Atop ∧ trF + σtop trD+ Dtop trσ. (4.6)
The theory has a g-valued vector multiplet σ and chiral multiplets14(
Q, Q̃
)
∈ R⊕R∨, where R = Hom
(
CL,CN
)
, R∨ = Hom
(
CN ,CL
)
. (4.7)
14We refer to a multiplet by its scalar component. Their full field content and our conventions about super-
symmetry are presented in Section B.2.
Elliptic Stable Envelopes and Dynamical R-Matrices 41
We also define M := R ⊕ R∨ to refer to the space of all the chirals. The chiral multiplets
are accompanied by their conjugate anti-chiral multiplets
(
Q, Q̃
)
∈ R∨ ⊕R. Charges of these
multiplets under the groups defined above are as follows
U(N) SU(L) U(1)ℏ U(1)top
σ ad 1 0 0
Q N L∨ 1
2 0
Q̃ N∨ L 1
2 0
Table 4. Charges of 3d N = 2 multiplets in the quiver gauge theory of Figure 3b. N and L refer to the
fundamental representations of U(N) and SU(L) respectively.
There is also a U(1)R R-symmetry but charges of various multiplets under this symmetry are
a bit ambiguous since the symmetry can be mixed with any other U(1) global symmetry. Also,
we do not turn on any background for the R-symmetry.
We put the 3d theory on a Euclidean space-time manifold [y−, y+] × Eτ . Here Eτ is the
elliptic curve with complex structure τ , defined as the quotient Eτ = C×/qZ, q := e2πiτ . For
the global symmetries SU(L)× U(1)ℏ × U(1)top we turn on background vector multiplets. The
multiplets contain, among other fields, flat connections.15 These flat connections are parame-
terized by their holonomies around the non-contractible cycles, S1
A and S1
B, of Eτ . For example,
if Aa is the background connection for the FA = SU(L) flavor symmetry, then we can compute
the aC = su(L)⊗ C valued periods aa :=
∮
S1
B
Aa − τ
∮
S1
A
Aa. We can always take the background
fields to be Cartan valued
aa = diag((aa)1, . . . , (aa)L), (aa)1 + · · ·+ (aa)L = 0.
Once we quotient by equivalences due to gauge transformations, these periods are really valued
in the torus aa ∈ aC
Λ∨
a ⊕τΛ∨
a
, where Λ∨
a is the coroot lattice of a. Consequently, the holonomies
of this connection around the cycles of the elliptic curve, also called the elliptic equivariant
parameters, are valued in the elliptic curve as well
(xi, . . . , xL) :=
(
e2πiτ(aa)1 , . . . , e2πiτ(aa)L
)
∈ ELτ , i = 1, . . . , L.
The tracelessness condition translates to x1 · · ·xL = 1 for the equivariant parameters. Another
common term for these parameters is fugacity, which we shall also use.
In a completely analogous fashion, we turn on elliptic fugacities for the flavor symme-
tries U(1)ℏ, U(1)top, as well as for the gauge symmetry U(N)
Symmetry Fugacities
U(1)ℏ ℏ ∈ Eτ
U(1)top z ∈ Eτ
U(N) (s1, . . . , sN ) ∈ ENτ
.
The values of the flavor fugacities are constant along the interval [y−, y+] due to BPS equa-
tions (5.6a). The values of the gauge fugacities along the interval are unconstrained and only
their values at the boundaries with fixed holonomy are parameters of the theory. These fugacities
are the same equivariant parameters as in Example 3.7.
15The connections need to be flat to preserve supersymmetry (cf. (5.5a)). The full constraints on the background
fields follow from the BPS equations of Section 5.1.2.
42 N. Ishtiaque, S.F. Moosavian and Y. Zhou
In addition to the holonomies, we also turn on SU(L) twisted masses, by giving nonzero VEV
to the adjoint scalar in the background SU(L) vector multiplet ⟨σa⟩ = m = diag(m1, . . . ,mL),
as well as U(1)top twisted mass ⟨σtop⟩ = ζ. Note that, due to the coupling (4.6) between the
topological and the dynamical gauge multiplets, the ζ is nothing but the real FI parameter.
The absolute values of these mass parameters do not play any role in this paper. However, the
relative ordering of their values matter. We fix the sign of the FI parameter once and for all
localization computations ζ > 0. For the SU(L)-masses, we shall refer to different chambers
at different times. A chamber simply refers to a particular ordering of the masses. For any
permutation ς : {1, . . . , L} ↪! {1, . . . , L} we have a chamber
Cς = {(m1, . . . ,mL) | mς(1) < mς(2) < · · · < mς(L)} ⊆ a. (4.8)
These are the same chambers as in (3.3). Two arbitrary but specific chambers defined by i 7! i
and i 7! L + 1 − i, reintroduced in (5.41), will be used for explicit computations of stable
envelopes and the R-matrix in Section 6.
Reduction to 2d. In Section 7, we shall consider dimensional reduction of the 3d partition
functions down to 2d and 1d. The 3d ! 2d reduction is done by compactifying the S1
B cycle of
the elliptic curve. Then the period of the connections along the S1
B cycle becomes a real adjoint
scalar field in the 2d theory. Combined with the real adjoint scalar already in the 3d vector
multiplet, this becomes the complex adjoint scalar of the 2d N = (2, 2) vector multiplet. For
example, for the FA vector multiplet we define the following complex scalar
σCa := σa + i
∮
S1
B
Aa. (4.9)
We use (σg)
C, (σℏ)
C, and ζC for the similarly defined complex adjoint scalars in the dynamical
vector multiplet, in the U(1)ℏ vector multiplet, and the complex FI parameter in 2d respectively.
A further reduction on S1
A gives an N = 4 quantum mechanics.
4.3 Branes and the Bethe/Gauge correspondence
In this section, we are going to look at the brane construction of the 3d quiver theory and
relations between certain string dualities and the Bethe/Gauge correspondence. The goal is to
find hints regarding which gauge theoretic quantities may correspond to stable envelopes and the
R-matrix. The brane constructions will serve merely as a motivation to set up the purely gauge
theoretic computations of the future sections, we will be rather schematic in the present section.
Gauge theories with 4 supercharges have standard brane constructions in terms of D-branes
suspended between rotated NS5s [37, 48, 49]. We consider a 10d space-time Ry × R1 × Eτ ×
R2 × R3 × R2
+ϵ × R2
−ϵ. Here Ry is a real direction parameterized by the coordinate y and R1,
R2, R3 are three real directions parameterized by some unspecified coordinates. R2
±ϵ are two
Ω-deformed planes, ϵ being the deformation parameter. The 3d N = 2 SQCD can be seen as
the world-volume theory of a stack of N D3 branes in this background. The complete brane
configuration is in Table 5, and we draw a cartoon of the branes in Figure 4.
T-dualizing the elliptic curve and the R1 direction takes us to a type IIA configuration
with N D2 branes suspended between rotated NS5 branes, with L D4 branes about. This is
precisely the configuration studied in [66]. In this duality frame, the world-volume theory of
the D2 branes is the 2d N = (2, 2) theory from Example 4.1. If we do not T-dualize R1, then we
find D1 branes suspended between NS5s and additional D3 branes. The D1 branes are described
the quantum mechanics from (4.2) for which the D3 branes provide flavor symmetry. The main
result of [66] was to relate this setup to 4d Chern–Simons theory.16 By a further S-duality we can
16Also see [33] for relations between Ω-deformed D5s and bosonic 4d Chern–Simons theory.
Elliptic Stable Envelopes and Dynamical R-Matrices 43
No. Ry R1 Eτ R2 R3 R2
+ϵ R2
−ϵ
1 NS5 × × × ×
1 NS5′ × × × ×
N D3 × × ×
L D5 × × × ×
Table 5. Brane configuration for 3d N = 2 SQCD with U(N) gauge group and SU(L) flavor (see
Figure 3b). Simultaneous rotation of the R2
+ϵ and R2
−ϵ planes is also a symmetry of this configuration.
This corresponds to the U(1)ℏ symmetry of the 3d theory (see Table 4), the two parameters being roughly
related by ϵ ∼ ln ℏ.
...
...
NS5 NS5′
L D5s
N D3s
R1
R2
Figure 4. A cross-section of the brane configuration from Table 5.
convert the D1s, NS5s, and D3s to F1s, D5s, and D3s respectively. The Ω-deformed D5s give rise
to a gl(1|1) 4d holomorphic-topological Chern–Simons theory on Ry×R1×Eτ where Ry × R1 is
the topological plane and the elliptic curve provides the holomorphic direction. The D3 branes
create line operators in this theory [66, 67]. From [31] we know that crossing of line operators
in the topological plane of 4d Chern–Simons creates R-matrix for certain spin chains. In the
starting configuration of Table 5 this crossing corresponds to changing the R1-positions of two D5
branes as we move along in the Ry direction, ultimately swapping the two (see Figure 5). The
|i⟩ |j⟩
⟨k| ⟨l|
Line operators in 4d CS
or D5 branes from Table 5
Ry
R1
Figure 5. Line operators crossing in the topological plane of 4d Chern–Simons, computing the matrix
element ⟨k| ⊗ ⟨l|R |i⟩ ⊗ |j⟩ of an R-matrix. This translates to bending D5 branes (from Table 5) in the
Ry × R1 plane.
incoming and outgoing asymptotic states correspond to states in the spin chain, or according
to the Bethe/Gauge philosophy, vacua of the D3 brane theory. Without further analysis of the
string background, it is unclear how exactly to bend branes while preserving supersymmetry
or how to specify incoming and outgoing states. However, there is a natural supersymmetric
44 N. Ishtiaque, S.F. Moosavian and Y. Zhou
process in the field theory language that accomplishes something analogous. The locations of
the D5 branes in the R1 direction are real twisted masses for the chiral multiplets of the SQCD
that lives on the D3 branes. Swapping the R1-locations, m1 and m2, of two D5 branes then
correspond to giving y-dependent masses m1(y) and m2(y) such that
lim
y!−∞
(m1(y),m2(y)) = (m1,m2), lim
y!+∞
(m1(y),m2(y)) = (m2,m1). (4.10)
Such position-dependent masses can be made supersymmetric, creating a mass Janus interface
[4, 19, 42, 61]. The interface interpolates between two distinct chambers, as defined in (4.8). In
Section 5, we shall be more specific and identify suitable boundary conditions in the 3d theory, in
any chamber, that mimics supersymmetric vacua, and define mass Janus interpolating between
them.
4.4 Gauge-theoretic definition of elliptic stable envelope
The gauge-theoretic definition of elliptic stable envelopes as Janus partition functions first ap-
peared in [24, 40]. They define the Janus interfaces for 3d N = 2 supersymmetry but provide
boundary conditions for 3d N = 4 vacua, which give elliptic stable envelopes of 3d N = 4
Higgs branches. We take our previous discussion on the brane construction and in particular
the position dependent masses (4.10) as motivations to study Janus interfaces. In this section,
we briefly review the definition of the elliptic stable envelopes as Janus partition functions, and
in Section 5, we generalize the boundary conditions to correspond to the vacua of 3d N = 2
theories computing the elliptic stable envelopes for the 3d N = 2 Higgs branches.
Our starting point here is the 3d N = 2 SQCD on [y−, y+] × Eτ . For the moment, replace
the interval [y−, y+] with the infinite line (−∞,+∞). Suppose, for any chamber C, we have
a set of asymptotic boundary conditions {p} for the 3d N = 2 SQCD corresponding to its
Higgs branch vacua.17 We can construct a Janus interface interpolating between a vacuum p−
in a chamber C− at infinite past and a vacuum p+ in a chamber C+ at infinite future. Then,
our previous discussion on branes suggests that the expectation value of this interface with p−
and p+ boundary conditions at infinite past and future should give us a matrix element of an
R-matrix
RC+ C−(p+, p−) = ⟨p+| J (mC− ,mC+) |p−⟩ ,
where mC− and mC+ are two generic masses from the chambers C− and C+ respectively, and J
is the Janus interface interpolating between these two masses. A similar expression for the R-
matrix in terms of Janus interfaces was also used in [23] in the context of bosonic rational spin
chains.
As already mentioned in the introduction (equations (1.3), (1.4) and (1.5)), a more funda-
mental quantity than the R-matrix is the stable envelope. While the R-matrix corresponds
to changing masses across chambers, the stable envelope corresponds to changing masses from
some nonzero value to zero. The full R-matrix is then a composition of a stable envelope and
an inverse stable envelope (3.12), the first one taking the nonzero masses from one chamber
to zero, the second one taking the zero masses to a nonzero value in a different chamber. In
the gauge theory setup, a stable envelope then corresponds to a Janus interface interpolating
between nonzero and zero masses. We expect that due to the presence of nonzero equivariant
17In the presence of twisted masses, the classical Higgs branch of 3d N = 2 SQCD retracts down to the fixed
point set of the flavor torus action. This set is a disjoint union of connected components. Here p labels a connected
component, and by the boundary condition labeled by p we roughly refer to a brane-type boundary supported
on the attracting/repelling set of p. We shall describe in detail the fixed points, their attracting sets and the
corresponding boundary conditions in Sections 5.2.2, 5.2.3 and 5.3, respectively.
Elliptic Stable Envelopes and Dynamical R-Matrices 45
elliptic parameters, even in the massless side of the interface we can choose a basis of boundary
conditions labeled by the same p that labels the massive boundaries.18 Then the stable envelope
corresponds to the Janus partition function
StabC(p+, p−) = ⟨p+| J (mC, 0) |p−⟩ .
We want to compute these partition functions using supersymmetric localization. In practice,
localization on non-compact manifolds are technically subtle and it is more convenient to replace
the infinite line with the finite interval [y−, y+]. Then we need to find boundary conditions for
the boundaries at finite distances that are cohomologous to the asymptotic vacuum boundaries
with respect to our choice of localizing supercharge. We describe our choice of supercharge in
Appendix B.3 and in Section 5, we shall identify the vacua and various boundary conditions
equivalent to these vacua. For the massive boundary, we shall construct two types of boundary
conditions, the so-called thimbles DC(p) (see Section 5.3.1) and the enriched Neumann NC(p) (see
Section 5.3.2). For the massless boundary, we shall construct one type of boundary conditions
depending on a choice of polarization L(p) of the space of chirals BL(p)(p) (see Section 5.3.3). All
these boundary conditions are labeled by the connected components of the (flavor) torus fixed-
point set of the Higgs branch. We shall compute the matrix elements of the stable envelopes as
the partition function
StabC(p+, p−) =
〈
BL(p+)(p+)
∣∣J (mC, 0) |NC(p−)⟩ . (4.11)
Here we have the boundary conditions NC(p−) and BL(p+)(p+) at the two endpoints of the
finite interval [y−, y+] and a Janus interface interpolating between them. By keeping the masses
constant throughout most of the interval and varying them rapidly near y = 0 we can think
of the Janus interface as being located at y = 0. The setup of this computation is depicted in
Figure 6.
y−
|NC(p−)⟩
y = 0
J (mC, 0)
y+
⟨BL(p+),p+ |
mC ̸= 0 m = 0
Figure 6. The physical setup for the computation of elliptic stable envelopes. We start with an enriched
Neumann boundary condition NC(p−) at the past boundary y−. This boundary imitates the choice of the
vacuum p− at −∞. We insert a mass-Janus interface J (mC, 0) at y = 0 whose role is to change the real
masses from nonzero to zero as we cross over to y > 0. We then choose an exceptional Dirichlet boundary
condition BL(p+),p+
at y+. This future state is associated with the choice of a vacuum p+ at +∞ and
a polarization L(p+) of the space of the chirals. The interval partition function of this configuration gives
the matrix elements (4.11) of elliptic stable envelopes.
Our choice of localizing supercharge makes the 3d theory topological in the y-direction (B.10),
and therefore the length of the interval [y−, y+] will not be a parameter of the theory. In fact,
we shall find an effective description of the theory as a 2d N = (0, 2) theory on Eτ – the amount
of supersymmetry reduced to half by the half BPS boundaries and interfaces. The 3d interval
partition function will be given by the elliptic genus of this 2d theory.
18Note that, if we go down in dimension then the periods of the background connections become part of the
twisted masses (4.9).
46 N. Ishtiaque, S.F. Moosavian and Y. Zhou
Due to the presence of a mass-Janus interface along which real masses change, we have
two different effective descriptions of the theory on the two sides of the interface. The theory
living on [0, y+], where m = 0, is the full gauge theory with the gauge Lie algebra g = u(N).
On the other hand, the theory living on [y−, 0], where m ̸= 0, is the gauge theory describing
the effective theory of low energy (relative to m) excitations in the vacuum p−. This theory
depends on the choice of a chamber C for the mass m, which in turn determines the phase of
the theory. For generic masses, the adjoint scalar σ from the vector multiplet takes a generic
VEV at this boundary (according to (5.25)), and as a result gauge symmetry is broken down to
its commutant, the maximal abelian subgroup h = RN .
5 Boundaries and interfaces in 3d N = 2 SQCD
In this section, we look at the classical Higgs branch vacua of 3d N = 2 SQCDs and N = (0, 2)
boundary conditions that are cohomologous to these vacua. Our goal is to compute Janus
partition functions (4.11) of the 3d theory on space-time manifolds of the form [y−, y+] × Eτ .
We do so using supersymmetric localization using the supercharge (B.8). The supercharge
belongs to an N = (0, 2) subalgebra of the 3d N = 2 supersymmetry algebra, which is why
we shall look at boundary conditions preserving this subalgebra. We follow [40] to find such
boundary conditions that mimic the Higgs branch vacua of the theory. The main difference in
our case is that, unlike in the 3d N = 4 cases, we do not have a discrete set of completely massive
vacua even in the presence of generic twisted masses. This requires us to slightly modify the
notion of the so-called thimble boundaries so that they can be attached to extended subspaces
of the Higgs branch.
5.1 Vacua and BPS equations in 3d N = 2 Gauge theories
5.1.1 Vacuum equations
Consider a 3dN = 2 Lagrangian gauge theory with gauge groupG and flavor symmetry group F .
We minimize its potential (consisting of bosonic fields only) to find the classical moduli of vacua.
All terms of the action containing dynamical and background vector multiplet fields can be
packaged together in terms of a single vector multiplet for the group G′ := G × F . We denote
the fields of the G′-multiplet as
(
A, σ, λ, λ,D
)
. Part of the Lagrangian containing A, σ and D
only is
1
2
tr
(
1
2
FµνFµν +DµσDµσ + D2
)
− iζ trD. (5.1)
The first trace is from the Yang–Mills action and the second is the FI term. We are assuming
that G has a U(1) center and so there is a single real FI parameter ζ. Note that ζ corresponds
to the background value of the adjoint scalar in the vector multiplet for the U(1)top global
symmetry (4.6).
We consider chiral multiplets in the representation M of the gauge group G and anti-chiral
multiplets in the contragradient representation M∨. Let us denote the gauge invariant pairing
between M and M∨ as ⟨ , ⟩. Then, the part of the chiral multiplet Lagrangian containing only
bosonic fields from the chiral and vector multiplets is
−
〈
ϕ,DµDµϕ
〉
+
〈
σ · ϕ, σ · ϕ
〉
+ i
〈
ϕ,Dϕ
〉
+
〈
F,F
〉
, (5.2)
The equations of motion for the auxiliary fields D and F are
D = iζ − iµg, F = F = 0, (5.3)
Elliptic Stable Envelopes and Dynamical R-Matrices 47
where µg = ϕϕ is the real moment map of the G-action on M. Eliminating the auxiliary
fields from the Lagrangians (5.1) and (5.2) using the equations of motion we find the on-shell
Lagrangian containing bosonic fields only
tr
(
1
4
FµνFµν +
1
2
DµσDµσ
)
−
〈
ϕ,DµDµϕ
〉
+ tr(−ζ + µg)
2 +
〈
σ · ϕ, σ · ϕ
〉
.
The derivative terms are the kinetic terms and the non-derivative terms constitute the potential
that we want to minimize. The potential, being a sum of squares, is minimized when the
individual terms are zero
µg − ζ = 0, (5.4a)
(σg + σf) · ϕ = 0. (5.4b)
Here σg + σf = σ is the decomposition of σ into its dynamical and background components.
The classical Higgs branch of the theory is defined as the real symplectic quotient µ−1
g (ζ)/G.
The first vacuum equation (5.4a) is precisely this real moment map equation, and after imple-
menting the G-quotient, the second equation (5.4b) becomes the criteria for fixed points under
the infinitesimal action generated by σf. σf will be some element from the Cartan subalgebra t
of the flavor symmetry group. Nonzero σ = σg+σf generates mass terms for various chiral mul-
tiplets and W-bosons. Thus we come to the following characterization of the (classical) massive
vacua of a 3d N = 2 gauge theory: they correspond to the fixed point set in the Higgs branch
under the action of the maximal torus of the flavor symmetry group.
5.1.2 BPS equations
We look for field configurations left invariant by the localizing supercharge Q (B.8). We consider
the 3d N = 2 theory on the space-time I × Eτ , where I is an interval with real coordinate y
and Eτ is an elliptic curve with holomorphic coordinate z. A supercharge of the theory is
parameterized by 2 Dirac spinors ϵ and ϵ, as in (B.4), and our choice of Q is determined by
Q = Qϵ,ϵ, where ϵ = ϵ =
1√
2
(
1
1
)
.
We thus find the BPS equations by setting the supersymmetry variations for the fermions from
the vector (B.5) and the chiral (B.6) multiplets to zero for this specific choice of parameters.
Doing so, we find the BPS equations – the ones containing derivatives along the elliptic curve Eτ
Fzz = 0, (5.5a)
Dzσg = Dzσf = 0, (5.5b)
Dzϕ = Dzϕ = 0, (5.5c)
and the ones containing derivatives along I
DyAz = 0, (5.6a)
Dyσg + ζ + µg = 0, (5.6b)
Dyσf + iDF = 0, (5.6c)
Dyϕ+ (σg + σf) · ϕ = 0, (5.6d)
along with the conjugate equation for ϕ. We have used the equations of motion (5.3) to eliminate
the auxiliary fields Dg, F and F. The only constraint on the background auxiliary field DF is (5.6c)
and we use this equation as the definition of this particular background. The equations (5.5)
48 N. Ishtiaque, S.F. Moosavian and Y. Zhou
determine the fields along the elliptic curve Eτ and the equations (5.6) determine their flow in
the y-direction.
The connections in the above equations all contain both dynamical and background fields for
the gauge and the flavor symmetry groups respectively. According to (5.5a) all these connections
are flat and will be characterized by their holonomies around the cycles S1
A and S1
B of Eτ .
Equation (5.6a) further implies that these holonomies are constant along the flow in the y-
direction. σg and σf are holomorphic according to (5.5b) but since they are elements of real
Lie algebras, they have to be constant on Eτ . Similarly, ϕ and ϕ are holomorphic according
to (5.5c). However, in the path integral we would like to impose the reality condition that the
anti-chiral field ϕ is the complex conjugate of the chiral field ϕ. Then both ϕ and ϕ have to be
constant on Eτ as well. Having nonzero holonomies for the flat connections is only compatible
with constant VEVs for ϕ if the holonomies act trivially on ϕ
sx−1ℏ
1
2 zϕ = ϕ. (5.7)
Here s, x, ℏ and z are the group valued holonomies for the Cartan valued connections for the
gauge group G, flavor groups FA and U(1)ℏ, and the topological symmetry group U(1)top. None
of the chirals are charged under U(1)top so z acts as identity on all chirals. The equation (5.7)
is saying that the combined holonomies are invisible to the chirals, which is also referred to as
a screening condition.
There is no equation for Ay, which is gauge equivalent to the zero connection since the y-
direction is contractible. We are now left with the flow equations for σg (5.6b) and ϕ (5.6d),
which we shall address in Section 5.2.3.
5.2 Classical Higgs branch
We consider the classical Higgs branch, where the chiral multiplet scalars have VEVs in the
absence of twisted masses. The vacuum equation (5.4b) is then automatically satisfied be-
cause σg = σf = 0 and we find the classical Higgs branch to be (cf. (2.3))
MH = µ−1
g (ζ)/G, (5.8)
where G is the gauge group, µg is the real moment map of the G-action on the representation
space M of the chirals and ζ is a positive central element in the gauge Lie algebra. The
construction (5.8) is the real symplectic reduction of M by G MH = M//G. In 3d N = 2, the
space of chirals is always Kähler and so is its symplectic quotient by a real Lie group. If we are
only interested in the complex structure of the Higgs branch, we can equivalently define it as
a GIT quotient by the complexified gauge group instead19
MH = Ms/GC. (5.9)
Here Ms is the stable locus of M, which consists of all points of M whose GC orbits inter-
sect µ−1
g (ζ).
Twisted masses are turned on by giving t-valued nonzero VEV to the adjoint scalar σf in the
flavor vector multiplet. We then have the additional vacuum equation (σg + σf) · ϕ = 0 (5.4b).
This tells us that most of the Higgs branch is now lifted and only the fixed points of the T-action
in the Higgs branch remain as vacua of the theory.20
We work out some details of the construction of the classical Higgs branches and their flavor
fixed points for the specific theories defined by the quiver in Figure 3b.
19This is how the Higgs branch was defined in (2.1).
20Later in computations, we only turn on A-twisted masses and look for A-fixed points where T = A×U(1)ℏ.
Elliptic Stable Envelopes and Dynamical R-Matrices 49
5.2.1 Classical Higgs branches of 3d N = 2 SQCDs
The real moment map of the G-action on the space of chirals is dictated by the D-term in the
chiral multiplet Lagrangian (5.2). For the theory defined by Figure 3b with the representations
of the chirals as in (4.7), the moment map is21 µ = QQ† − Q̃†Q̃. And so the real moment map
equation becomes:
QQ† − Q̃†Q̃ − ζ = 0, (5.10)
where ζ ∈ R+ is the positive FI parameter. The Higgs branch is the quotient MH(N,L) :=
µ−1(ζ)/U(N), From this, we get the dimension of the Higgs branch
dimCMH(N,L) = 2NL−N2. (5.11)
Furthermore, since Q, Q̃ ∈ Hom
(
CL,CN
)
, we have rkµ ≤ L. On the other hand, the moment
map equation sets µg to ζ which has rank N . We thus get the constraint
N ≤ L. (5.12)
Notice from (5.11) that for odd values of N the complex dimension is odd and the resulting
space can not have a holomorphic symplectic form. This is in sharp contrast with the case of
3d N = 4 supersymmetry where the hyper-Kähler structure of Higgs branches equips them with
a holomorphic symplectic form.
For our purposes, we mainly need the complex structure of the Higgs branch instead of the
full Kähler structure. Let us look at an example.
Example 5.1 (N = 1, L = 2). For gauge group G = U(1) and flavor group FA = SU(2),
the Higgs branch is a 3-dimensional complex manifold. The space of the chirals is, M =
Hom
(
C2,C
)
⊕ Hom
(
C,C2
)
with holmorphic coordinates Qi and Q̃i for i = 1, 2. The moment-
map equation is given by∣∣Q1
∣∣2 + ∣∣Q2
∣∣2 − ∣∣Q̃1
∣∣2 − ∣∣Q̃2
∣∣2 − ζ = 0. (5.13)
Under a complexified gauge transformation by λ ∈ C× the coordinates transform as(
Q, Q̃
)
7!
(
λQ, λ−1Q̃
)
.
So a set of gauge invariant holomorphic functions on the Higgs branch is given by the matrix
elements of Q̃Q
Q̃Q =
(
Q̃1Q1 Q̃1Q2
Q̃2Q1 Q̃2Q2
)
.
Any gauge invariant polynomial function on the Higgs branch can be written in terms of these.
Since Q̃Q : C2 ! C2 factors through C, it has rank at most 1, consequently
det Q̃Q = 0. (5.14)
The equation (5.14) defines the singular conifold as a subspace of Hom
(
C2,C2
)
with singularity
at Q = Q̃ = 0 where the rank of Q̃Q is strictly less than 1. The Higgs branch MH(1, 2) is
a resolution of this singularity by the moment map equation (5.13). The moment map equation
requires a nonzero
∣∣Q1
∣∣2 + ∣∣Q2
∣∣2, which means that with ζ > 0, at det Q̃Q = 0 we have Q̃ = 0.
21Q and Q are a priori independent fields in Euclidean signature, but we have to choose some reality condition
on them to do our path integrals and we choose Q = Q†.
50 N. Ishtiaque, S.F. Moosavian and Y. Zhou
Then the moment map equation becomes
∣∣Q1
∣∣2 + ∣∣Q2
∣∣2 = ζ. This defines an S3 with radius
√
ζ,
which together with the U(1)-gauge identification implies that the singularity of the conifold is
replaced with S3/U(1) ≃ P1 of size determined by ζ.
The resolved complex geometry can be described as follows. For ζ > 0, the requirement(
Q1,Q2
)
̸= (0, 0) coming from the moment map equation says that the map Q : C2 ! C is
surjective. Due to the quotient by complex gauge transformations, only the kernel of this map
is physical. Different choices of kerQ ≃ C ⊆ C2 are parameterized by a P1. Q1 and Q2 are the
homogeneous coordinates on this P1. We can define the two local coordinates z := Q1
Q2 , z
′ := Q2
Q1 ,
to cover this P1. There is no constraint on Q̃ : C ! C2 and since we have already used up the
gauge transformations to classify Q, all choices of Q̃ : C! C2 are physically significant. A map
C ! C2 is characterized by the image of 1 which is unconstrained. Over the chart of P1 with
local coordinate z we can parameterize these images of 1 by
( Q̃1
Q2 ,
Q̃2
Q2
)
and over the chart with
coordinate z′, we can parameterize the same images by(
Q̃1
Q1
,
Q̃2
Q1
)
= z′
(
Q̃1
Q2
,
Q̃2
Q2
)
.
Overall, the maps
(
Q, Q̃
)
modulo complex gauge transformations form a rank 2 vector bundle
over P1 with transition function z′ as we change coordinate z 7! z′. Dualizing this transition
function we get the underlying space as the Higgs branch
MH(1, 2) = O(−1)⊕ O(−1)! P1. (5.15)
It is easy to generalize this example to arbitrary N and L ≥ N . First, as a consequence of
the moment map equation we have
Proposition 5.1. For ζ > 0, on the Higgs branch rkQ = N .
Proof. Suppose Q has rank less than N . Then QQ† has rank less than N and we can choose a
nonzero v ∈ kerQQ†. The moment map equation (5.10) then implies−Q̃†Q̃v = ζv. However, the
operator on the left-hand side is negative semi-definite and can not have a positive eigenvalue –
we have a contradiction. ■
Now let us consider the Higgs branch. Once again, a generating set of holomorphic functions
on the Higgs branch are given by the matrix elements of Q̃Q. Since Q̃Q factors through CN we
have the constraint rk Q̃Q ≤ N . This defines the determinantal variety, denoted by Det(N,L),
which has singularities where the rank of Q̃Q is strictly less than N . These singularities are
resolved by the real moment map equation.
Since Q : CL ! CN is surjective (see Proposition 5.1), modulo complexified gauge trans-
formation by GLN , Q is completely fixed by its kernel. Once Q is fixed, Q̃ : CN ! CL is
unconstrained, though the choices of Q̃ are fibered nontrivially over the choices of Q due to
the GLN quotient that acts on both the choice of Q and that of Q̃. The quotient of CL by
kerQ is parameterized by the Grassmannian Gr(N,L). We can remember the GLN action on Q
by the tautological principal GLN bundle KP over Gr(N,L). Then the choices of
(
Q, Q̃
)
such
that Q is surjective are parameterized by KP × Hom
(
CN ,CL
)
. Taking the gauge quotient, we
find the Higgs branch
MH(N,L) = KP ×GLN Hom
(
CN ,CL
)
= Hom
(
K,CL
)
,
where K and CL refer to the tautological vector bundle and a trivial bundle over Gr(N,L).
By construction, this is a vector bundle over Gr(N,L).22 The above space is also called the
desingularization of the determinantal variety Det(N,L) [83], and can be denoted by D̂et(N,L).
22Had we used the opposite sign ζ < 0 of the real FI parameter in the moment map equation (5.10), the Higgs
branch would be the dual bundle Hom
(
CL,K
)
.
Elliptic Stable Envelopes and Dynamical R-Matrices 51
The singular conifold (5.14) is the special case of a determinantal variety for N = 1 and L = 2,
and the resolved conifold (5.15) is the aforementioned resolution thereof. In all abelian cases,
we get
MH(1, L) = D̂et(1, L) = O(−1)⊕L ! PL−1.
5.2.2 Fixed points
Flavor fixed points in the Higgs branch are points of M satisfying the real moment map equation
where the flavor torus action can be undone by a gauge transformation. We are considering
the gauge and flavor symmetry group G = U(N) and A = U(1)L−1 ⊆ SU(L). Under an
infinitesimal G× A action, the chiral multiplets transform as Q ! Q + σQ − Qm, Q̃ ! Q̃ −
Q̃σ +mQ̃ where σ ∈ u(N), and
m =
m1
m2
. . .
mL
∈ a = RL, (5.16)
with the constraint m1 + · · · + mL = 0. Therefore,
(
Q, Q̃
)
is a fixed point if the following
equations admit solutions for σ
σQ−Qm = 0, Q̃σ −mQ̃ = 0. (5.17)
And
(
Q, Q̃
)
must satisfy the moment map equations (5.10) to be in the Higgs branch.
Let us denote by Qi and Q̃i for i = 1, . . . , L the columns and rows of Q and Q̃, respectively.
Then we can write (5.17) as
σQi = miQi, Q̃iσ = miQ̃i. (5.18)
These are eigenvalue equations for the 2L vectors Qi, Q̃i. We assume that the masses mi
are all distinct. There are then L different eigenvalue equations since Qi and Q̃i share an
eigenvalue. Since σ is an N × N matrix with N ≤ L, at most N out of these L eigenvalue
equations can produce nonzero eigenvectors. On the other hand, for ζ > 0, the moment map
equation (5.10) implies that rkQ = N (see Proposition 5.1). Consequently, a solution to the
eigenvalue equations (5.18) contains exactly N nonzero eigenvectors. The nonzero eigenvectors
are labeled by an inclusion
p : {1, . . . , N} ↪! {1, . . . , L}, (5.19)
such that the nonzero eigenvectors are Qp(a). Eigenvectors of hermitian operators with distinct
eigenvalues are mutually orthogonal. So, as an orthonormal basis for the N -dimensional vector
space spanned by Qp(a) we can choose
{ Qp(1)
∥Qp(1)∥ , . . . ,
Qp(N)
∥Qp(N)∥
}
. In this basis, the matrix Q satisfies
Qi
a = δip(a)
∥∥Qp(a)
∥∥ (no sum over a). (5.20)
Q̃i must be a multiple of Qi since they have the same eigenvalue
Q̃a
i = δ
p(a)
i eiφ
∥∥Q̃p(a)
∥∥. (5.21)
The phase is not fixed. Substituting (5.20) and (5.21) in the moment map equation (5.10), we
get
QQ† − Q̃†Q̃ = diag
(∥∥Qp(1)
∥∥2 − ∥∥Q̃p(1)
∥∥2, . . . ,∥∥Qp(N)
∥∥2 − ∥∥Q̃p(N)
∥∥2) = ζ1N .
52 N. Ishtiaque, S.F. Moosavian and Y. Zhou
Using gauge symmetry, we can fix all components of Q to be real and positive, then there is
a one-parameter family of solutions to the above moment map equation:
Qi
a = δip(a)
√
ζ + |λ|2, Q̃a
i = δ
p(a)
i λ, λ ∈ C. (5.22)
To summarize, we have the following characterization of A-fixed points of the Higgs branch
MH(N,L). There are
(
L
N
)
mutually disjoint connected components of the fixed point set,
labeled by a choice p of N numbers out of L (5.19).23 For such a choice p, the corresponding
component of the fixed point set in the Higgs branch is given by the gauge orbit of a reference
N -dimensional subspace F̂p ⊆ M which we take to be (no sum over a, b)
F̂p :=
{(
Q, Q̃
)
∈ M |
(
Qi
a, Q̃b
j
)
=
(
δip(a)
√
ζ + |λa|2, δp(b)j λb
)
, (λ1, . . . , λN ) ∈ CN
}
. (5.23)
We denote the image of F̂p in the complex Higgs branch (5.9) under the bundle projection
Ms π
−!Ms/GLN = MH (5.24)
by Fp := π
(
F̂p
)
. The projection π is one-to-one when restricted to F̂p.
Observe that the fixed points also satisfy the vacuum equations (5.4) in the case σf = σA, in
other words, when we turn on twisted masses only with respect to the flavor symmetry SU(L)
(the framing node of the quiver Figure 3b). At these vacua, the value of the vector multiplet
scalar σ is determined by (5.4b) (or equivalently (5.18)), which is
σp(m) :=
mp(1)
mp(2)
. . .
mp(N)
. (5.25)
Thus, the A-fixed point set corresponds to the residual classical Higgs branch in the presence of
the A-twisted masses
MA
H =
⊔
p
Fp = classical Higgs branch with A-twisted masses. (5.26)
At the vacuum p, we also get constraints on the holonomies for the flat connections around the
elliptic curve from the screening conditions (5.7). The nontrivial screening conditions come from
the chirals (5.22) that take nonzero VEVs, the conditions being
sax
−1
p(a)ℏ
1/2 = 1. (5.27)
We shall refer to the gauge holonomies satisfying this condition by s
(p)
a .
5.2.3 Attracting sets and Morse flow
Consider the definition of the attracting set of a fixed point x0 ∈ Fp (3.4)
AttrC(x0) =
{
x ∈ MH | lim
t!0
f(t) · x = x0 for all f ∈ C
}
.
Choose some generic m = diag(m1, . . . ,mL) ∈ a such that t 7! tm is a cocharacter in the
chamber C. Now, introduce the variable y = − ln t such that the limit t ! 0 corresponds to
23Only maps (5.19) modulo permutations of {1, . . . , N} lead to physically inequivalent vacua since the permu-
tations coincide with the action of the Weyl subgroup of the gauge group U(N).
Elliptic Stable Envelopes and Dynamical R-Matrices 53
the limit y ! ∞. We take the limit t ! 0 along the positive real axis so that y becomes
a non-negative real number. Then we can rewrite the definition of the above attracting set as
AttrC(x0) =
{
x ∈ MH | lim
y!∞
e−ym · x = x0
}
.
Define the trajectory of x in the Higgs branch, as we take the limit y !∞, as a path
x : [0,∞)!MH , x(y) := e−ym · x0.
This path satisfies the flow equation
∂yx+m · x = 0, (5.28)
with the asymptotic boundary condition limy!∞ x = x0. The definition of the above attracting
set can now be rewritten as
AttrC(x0) =
{
x(0) ∈ MH | ∂yx+m · x = 0, lim
y!∞
x = x0
}
. (5.29)
Any smooth path x(y) in the Higgs branch can be lifted to a smooth path in the stable lo-
cus Ms by undoing the projection (5.24) and non-uniquely choosing some point from each fiber
over x(y). Let x̂(y) be such a lift of x(y). The flow equation (5.28) for x lifts to the following
flow equation for x̂ Dyx̂ + (σ +m) · x̂ = 0. Here Dy involves some glN -connection and σ(y) is
some glN -valued function of y. These are the extra ingredients that will be lost once we take
the projection by the GLN -quotient. Let x̂0 ∈ F̂p be a lift of x0. We can constraint the lift x̂
by requiring that its asymptotic value, instead of being something arbitrary from the GLN -
orbit of x̂0, be exactly equal to x̂0 limy!∞ x̂(y) = x̂0. This completely breaks gauge symmetry
at y = ∞ and therefore the value of σ must also be constrained asymptotically to take the
value (5.25) limy!∞ σ(y) = σp(m). We can rewrite the definition of the attracting set (5.29)
once more by projecting x̂ as follows
AttrC(x0) =π
(
Flow+
C
(
x̂0
))
, (5.30)
where we have defined
Flow±
C (x̂0) :=
{
x̂(0) ∈ Ms | Dyx̂+ (σ +m) · x̂ = 0, lim
y!±∞
(x̂, σ) =
(
x̂0, σp(m)
)}
. (5.31)
This relates Morse flow in M to the attracting sets in the Higgs branch. The flows in M are
not completely fixed because both the connection Dy and σ can be changed by GLN gauge
transformations as long as we satisfy the limit condition. We can choose some convenient
representatives of these flows as follows.
Let us zoom in on the relation (5.30) at x0, i.e., we look at the tangent space to the attracting
set at x0
Tx0AttrC(x0) =π∗
({
x̂(0)∈Ms | Dyx̂+ (σp(m) + yDyσ|x0+m)·x̂ = 0, lim
y!∞
x̂ = x̂0
})
.(5.32)
We further use our gauge freedom to set Dy = ∂y and Dyσ
∣∣
x0
= 0. Small displacements
around x0 ∈ Ms can be parameterized by the chiral scalars Qi
a and Q̃a
i . These displacements
are scaled by the action of (σp(m) +m) as
(σp(m) +m) · Qi
a = (mp(a) −mi)Qi
a, (σp(m) +m) · Q̃a
i = −(mp(a) −mi)Q̃a
i . (5.33)
Solutions to the flow equation ∂yx̂+ (σp(m) +m) · x̂ = 0 are therefore given by
Qi
a(y) = e−y(mp(a)−mi)Qi
a(0), Q̃a
i (y) = ey(mp(a)−mi)Q̃a
i (0).
54 N. Ishtiaque, S.F. Moosavian and Y. Zhou
The linear combination x̂(y) = x̂0 + κaiQi
a(y) + κ̃iaQ̃a
i (y) then satisfies the same flow equation as
well as the limit condition limy!∞ x̂ = x̂0 if and only if the coefficients κ and κ̃ vanish for all
chirals with non-positive weights. Let us decompose the space of chirals into subspaces according
to the sign of their weights under the action (5.33)
M = MC;+(p)⊕M0(p)⊕MC;−(p), (5.34a)
where
MC;+(p) := SpanC
{
Qi
a, Q̃b
j | mp(a) −mi > 0, mp(b) −mj < 0
}
,
M0(p) := SpanC
{
Qi
a, Q̃b
j | mp(a) −mi = mp(b) −mj = 0
}
,
MC;−(p) := SpanC
{
Qi
a, Q̃b
j | mp(a) −mi < 0, mp(b) −mj > 0
}
.
We can now rewrite (5.32) as
Tx0AttrC(x0) ≃π∗ (MC;+(p)) . (5.35)
Here the vector space inside the projection on the right-hand side has its origin at x̂0. The
attracting set AttrC(x0) is generated by the action of f(t) = tm for some generic cocharacter f
in the chamber C. This action can be lifted to an action f̂ onM via an embedding AC ↪! HC×AC,
such that for x̂ ∈ M we have f̂(t) · x̂ = tσp(m)+m · x̂. We can then define the attracting set of x̂0
in Ms with respect to this action:
ÂttrC(x̂0) :=
{
x̂ ∈ M | lim
t!0
tσp(m)+m · x̂ = x̂0
}
.
The tangent space to this affine space at x̂0 is MC;+(p). The projection π is AC equivariant and
it is a surjection (5.35) on tangent spaces. Therefore, it is a surjection globally, i.e., ÂttrC(x̂0)
projects onto the full attracting set AttrC(x0). We can also immediately see that the attracting
set ÂttrC(x̂0) is nothing but {x̂0} ×MC;+(p). Therefore,
AttrC(x0) = π
({
x̂0
}
×MC;+(p)
)
.
Comparing with (5.30) we find a gauge equivalence between positive weight spaces in M and
Morse flow in the stable locus Ms that asymptotes some fixed point in F̂p. More specifically, by
varying x0 over Fp, we find
π
(
Flow+
C
(
F̂p
))
= AttrC(Fp) = π
(
F̂p ×MC;+(p)
)
. (5.36)
Replacing the limit y !∞ with y ! −∞, and following the same arguments we find
π
(
Flow−
C
(
F̂p
))
= RepC(Fp) = π
(
F̂p ×MC;−(p)
)
. (5.37)
Here Rep refers to the repelling set defined similarly to the attracting set (3.4) but with the
opposite limit t!∞.
5.3 Boundaries and interfaces
We discuss half-BPS boundary conditions that are equivalent, in localization computations, to
forcing the fields to reach any particular connected component of the solutions to the vacuum
equation (5.4). The choice of localizing supercharge is discussed in Appendix B.3, (B.8) in
particular. We discuss half-BPS boundaries preserving Q−, Q−. So these are N = (0, 2)
boundary conditions in the 3dN = 2 theory. We also discuss the mass Janus interface preserving
the same supersymmetry. Our discussion mostly follows similar boundary conditions for the 3d
N = 4 case considered in [40]. Our description of the thimble boundaries (see Section 5.3.1)
differs slightly from this reference due to the non-isolated nature of 3d N = 2 Higgs branch
vacua.
Elliptic Stable Envelopes and Dynamical R-Matrices 55
5.3.1 Thimble (exceptional Dirichlet), DC(p)
To find a boundary condition at y = y− that is cohomologous to a choice of vacuum, we
extend our space-time to negative infinity and consider the theory on (−∞ × y−] × Eτ . We
impose the constraint that the theory must reach the vacuum p at infinity. Then we observe
the supersymmetric evolution of the fields along the interval (−∞, y−] and see what values the
fields take at y = y−. More specifically, we solve the BPS equations (5.6) with the boundary
condition that at y = −∞ the fields satisfy the vacuum equation (5.4). Then the values of these
solutions at y = y− provide a past boundary condition for the theory at y >= y− that mimics
the chosen vacuum in BPS computations. Similar boundary conditions were constructed and
called thimble or exceptional Dirichlet boundary conditions in [22].
In this process, we observe that the BPS equation (5.6d) with the condition that at past
infinity the chiral and vector multiplet scalars satisfy the vacuum equations (5.4) is precisely
what defines the set Flow−
C
(
F̂p
)
(5.31). The choice of chamber and vacuum determine the Morse
function and the connected component of the vacuum configurations defining the boundary
condition. The BPS equation (5.6b) puts constraints on the evolution of σg but Flow−
C
(
F̂p
)
is
agnostic to such constraints. According to (5.37), the set Flow−
C
(
F̂p
)
is gauge equivalent to the
subspace F̂p ×MC;−(p) of M. So, we formulate a (0, 2) boundary condition of brane type with
support for the chiral multiplet scalars on F̂p ×MC;−(p).
Remark 5.1 (thimbles in N = 4 vs N = 2). Support of this brane (at the past boundary) on
the Higgs branch is the projection of F̂p × MC;−(p), which is the repelling set (5.37). In [22],
the thimble boundaries are supported on holomorphic Lagrangians in the hyperkähler Higgs
branches of 3d N = 4 theories. For 3d N = 2 theories, the Higgs branches are generally
only Kähler and the attracting/repelling sets (5.36), (5.37) are not supported on holomorphic
Lagrangians, they are just holomorphic.
Let us decompose the space of chirals further from (5.34a) based on where the Qs and the Q̃s
belong
MC;±(p) = RC;±(p)⊕ R̃C;±(p), M0(p) = R0(p)⊕ R̃0(p).
Since Qi
a and Q̃a
i have weights with opposite signs (see (5.33)), we have the following relations
among these subspaces
RC;±(p) = R̃∨
C;∓(p), R0(p) = R̃∨
0 (p).
We thus have a Lagrangian splitting of the chirals
M = LC(p)⊕ L⊥
C (p), LC(p) := MC;+(p)⊕R0(p),
L⊥
C (p) := MC;−(p)⊕ R̃0(p). (5.38)
F̂p is parameterized by arbitrary VEVs of fields from R̃0(p) (see (5.23)), therefore, chirals
from L⊥
C (p) parameterize the brane F̂p ×MC;−(p). So we shall put Dirichlet and Neumann
boundary conditions on the scalars from the chirals in LC(p) and L⊥
C (p) respectively. The bound-
ary conditions will be extended to the rest of the fields of these multiplets by the requirement
of preserving (0, 2) supersymmetry.
Gauge symmetry is broken at the vacuum by the VEVs of the chirals and as such the gauge
field receives Dirichlet boundary condition, the rest of the vector multiplet receiving boundary
conditions accordingly. The holonomy of the dynamical gauge field at the vacuum is given in
terms of the holonomy of the background gauge fields via the screening condition (5.27).
56 N. Ishtiaque, S.F. Moosavian and Y. Zhou
Altogether, the thimble boundary condition DC(p) involves:
Chirals in LC(p) : Dirichlet with VEVs according to vacuum p,
Chirals in L⊥
C (p) : Neumann,
Vector multiplet : Dirichlet with screening (5.27) of gauge holonomy.
(5.39)
Boundary ’t Hooft anomaly. Let f ∈ h, fx ∈ a and fℏ ∈ Rℏ be the curvatures of the gauge,
SU(L) flavor, and U(1)ℏ flavor symmetry at the boundary. The gauge symmetry is completely
broken at the boundary and f is a function of the flavor curvatures. Their dependence is fixed
by demanding that the nonzero VEVs at the boundary do not observe any flux at the boundary
f = f (p)(fx, fℏ) such that, w(f + fx + fℏ) = 0 for all w ∈ R0(p) . Contributions to the anomaly
polynomial come from two sources: 1) Chern–Simons terms created by integrating out mas-
sive fermions and the Chern–Simons coupling to background multiplet associated with U(1)top
symmetry, 2) purely from boundary conditions. In the present case, these contributions are
[40, 43]24
PCS
− (DC(p)) = 2 trMC;+(p)(fℏ(f + fx))− 2 tr(fftop), P ∂(DC(p)) = 2 trR0(p)(fℏ(f + fx)).
The subscript in PCS
− refers to the past boundary. The Chern–Simons anomaly depends on the
orientation of the boundary and therefore PCS
− = −PCS
+ . P ∂ is independent of the choice of past
or future. If we define
P (DC(p)) := PCS
− (DC(p)) + P ∂(DC(p)) = 2 trLC(p)(fℏ(f + fx))− 2 tr(fftop),
then the total anomaly polynomial associated to the thimble boundary is P (DC(p)) evaluated
at f = f (p)(fx, fℏ).
5.3.2 Enriched Neumann, N C(p)
Swap the boundary conditions on the chiral multiplets from (5.39). Meaning, the chirals in LC(p)
now receive Neumann, and those in L⊥
C (p) receive Dirichlet boundary conditions with zero VEV.
The gauge field in the vector multiplet receives the Neumann boundary condition. This implies
that the adjoint scalar σg from the vector multiplet gets Dirichlet [43, Section 2.5.2]. We set
the value of σg equal to the VEV σp(m) (5.25) corresponding to the vacuum p. This breaks the
gauge symmetry down to the maximal torus H = U(1)N . Additionally, we couple a 2d N = (0, 2)
theory ΥC(p) at the boundary by gauging some of its global symmetries with the boundary gauge
symmetry H. We shall characterize the theory ΥC(p) momentarily. Let us first summarize this
enriched Neumann boundary condition NC(p)
Chirals in LC(p) : Neumann,
Chirals in L⊥
C (p) : Dirichlet with zero VEVs,
Vector multiplet : Neumann,
Coupled to the boundary: 2d N = (0, 2) theory ΥC(p).
The relation between the thimble boundary conditions (5.39) and vacua are relatively straight-
forward. However, in computation, we shall use this new boundary condition NC(p) to repre-
sent the vacuum p. In [40, Section 6.3], Dedushenko–Nekrasov argue that both |DC(p)⟩ ⟨NC(p)|
and |NC(p)⟩ ⟨DC(p)| are projections onto the vacuum p in cohomology, where ⟨·| and |·⟩ refer to
24The trace without explicit representation is taken over the fundamental representation.
Elliptic Stable Envelopes and Dynamical R-Matrices 57
future and past boundary condition, respectively. A necessary condition for this to be true is
that these projectors are free of ’t Hooft anomaly
P±(DC(p))
∣∣
f=f (p)(fx,fℏ)
+ P∓(NC(p)) = 0. (5.40)
Here P+ and P− refer to anomaly polynomials of the future and past boundary conditions,
respectively. In this paper, we only describe the NC(p) boundary condition and show that this
anomaly condition is satisfied, beyond that, we take their equivalence as an ansatz. For more
details, see [40].
Boundary theory ΥC(p). At the boundary preserving N = (0, 2) supersymmetry, we
have U(1)N gauge symmetry. So we can couple a 2d N = (0, 2) theory with U(1)N global
symmetry by gauging it as long as the anomaly matching condition (5.40) is satisfied. It is
straightforward to write down a boundary theory for any chamber, but for the sake of explicit
formulas we shall focus here on the following two chambers
C1 := m1 < m2 < · · · < mL, C2 := m1 > m2 > · · · > mL. (5.41)
For C1, we propose the following field content and charges for the theory under various symmetry
groups applying the procedure of field content of ΥC1(p):
U(1)a U(1)ℏ U(1)top
Chiral1,a −1 1 0
Chiral2,a 0 L
2 − p(a) 1
Fermi1,a 1 L
2 − p(a) 1
Fermi2,a 0 1 0
.
We have one set of these fields for each a = 1, . . . , N , totaling in 2N Fermi and 2N chi-
ral multiplets. The U(1)a global symmetry is identified with a U(1) subgroup of the bulk
U(1)N ×U(1)L−1 ×U(1)ℏ symmetry such that the 2d U(1)a-fugacity is identified with the bulk
fugacity sax
−1
p(a)ℏ
1/2. The remaining symmetries are identified straightforwardly. The elliptic
genus of this theory is [17, 18]
W(ΥC1(p)) =
N∏
a=1
ϑ
(
sax
−1
p(a)ℏ
1
2
+L
2
−p(a)z
)
ϑ(ℏ)
ϑ
(
s−1
a xp(a)ℏ
1
2
)
ϑ
(
zℏ
L
2
−p(a)) . (5.42)
Here we are using the theta functions defined in the notation of [2]
ϑ(t; q) :=
(
t
1
2 − t−
1
2
) ∞∏∏∏
n=1
(
1− tqn
)(
1− t−1qn
)
. (5.43)
Some of its identities are
ϑ(1; q) = 0, ϑ
(
t−1; q
)
= −ϑ(t; q), ϑ
(
e2πit; q
)
= −ϑ(t; q),
ϑ
(
qkt; q
)
= q−
k2
2 (−t)−kϑ(t; q), k ∈ Z.
This theta function can be related to Jacobi’s theta function and the Dedekind eta function as
follows:
q
1
12ϑ(t; q) =
iθ1
(
ln t
2πi ;
ln q
2πi
)
η
( ln q
2πi
) .
We omit the argument q in the theta function when it is clear from the context.
58 N. Ishtiaque, S.F. Moosavian and Y. Zhou
For the chamber C2, we simply write down the elliptic genus
W(ΥC2(p)) =
N∏
a=1
ϑ
(
sax
−1
p(a)ℏ
− 1
2
−L
2
+p(a)z
)
ϑ(ℏ)
ϑ
(
s−1
a xp(a)ℏ
1
2
)
ϑ
(
zℏ−
L
2
+p(a)−1
) . (5.44)
The field content of the theory and their charges can be read off from this if needed. Notice
that when the screening conditions (5.27) are imposed the elliptic genera reduce to 1.
The choice of the boundary theory is defined only up to the addition of anomaly-free 2d
N = (0, 2) theories. This leads to some ambiguities regarding the overall partition function,
and consequently regarding the stable envelope. We can fix some of this ambiguity by imposing
certain normalization conditions on the stable envelopes (as was done in their definition in
Theorem 3.1). The normalization condition restricts our choice of boundary theory to those
without anomaly and whose elliptic genus reduces to 1 when the screening conditions (5.27) are
imposed. Even then the choice is not unique. Our choice differs minimally from the closely
related example of the T ∗Gr(N,L) Grassmannian theory with 3d N = 4 symmetry from [40,
Section 6.5.3].
Boundary ’t Hooft anomaly. The enriched Neumann boundary condition (5.3.2) receives
contributions to its anomaly polynomial from Chern–Simons terms, boundary conditions on
chirals and vectors, and the boundary theory
P (NC(p)) := PCS
− (NC(p)) + P ∂(NC(p)) + P (ΥC(p)).
The Chern–Simons anomalies only depend on the orientation of the boundary, PCS
− (NC(p)) =
PCS
− (DC(p)). Whereas, the boundary conditions on the chirals and the vector multiplet fields
are swapped between the thimble (5.39) and the enriched Neumann (5.3.2). So they contribute
with opposite signs: P ∂−(NC(p)) = −P ∂−(DC(p)). Since we have a gauge theory at the boundary,
the gauge anomalies from all the different sources must neatly cancel. In fact, the boundary
theory is defined to satisfy the constraint that only ’t Hooft anomalies survive and they match
the negative of the future thimble boundary anomaly (5.40) define ΥC(p) such that
P−(NC(p)) = −P+(DC(p))
∣∣
f=f (p)(fx,fℏ)
.
The anomaly of the elliptic genera (5.42) and (5.44) satisfy this condition for the chambers C1
and C2 (5.41) respectively. For reference, we note that in any chamber C, the anomaly of the
boundary theory must be
P (ΥC(p)) = −2(trLC(p)−2 trR0(p))
((
f − f (p)
)
fℏ
)
+ 2 tr
((
f − f (p)
)
ftop
)
.
It can be checked that the elliptic genera (5.42) and (5.44) contribute the appropriate anomalies
for their respective chambers.
5.3.3 Massless boundary condition, BL(p)(p)
Broadly speaking, stable envelope is a map from the cohomology of the flavor fixed points of
the Higgs branch to the cohomology of the Higgs branch itself (1.3). To compute this map, we
compute the partition function of the 3d theory on [y−, y+]×Eτ such that at the past boundary
we have twisted masses which eventually vanish at the future boundary (4.11). At the past
boundary, we can, in principle, use either the thimble (see Section 5.3.1) or the enriched Neumann
(see Section 5.3.2) boundary conditions both of which are labeled by connected components of
the fixed point set and provide a basis for the cohomology of the fixed point set. As vector
spaces, the cohomology of the Higgs branch is isomorphic to that of its fixed points. We,
therefore, look for a set of boundary conditions to impose at the future massless boundary
Elliptic Stable Envelopes and Dynamical R-Matrices 59
parameterized by connected components of the fixed point set as well, such that we can write
down the matrix elements of the stable envelope in terms of pairs of connected components. Such
a set of boundary conditions was proposed by Dedushenko–Nekrasov in [40]. These boundary
conditions are very similar to the thimble boundary conditions. However, since there are no
masses at this boundary, there is no choice of chamber and we can not choose a polarization, as
we did for the thimbles in (5.38), based on positive and negative masses. More specifically, there
are no obvious candidates for MC;±(p). The massless parts of the polarization, namely R0(p)
and R̃0(p) still make sense as they are chamber independent. The proposal of Dedushenko–
Nekrasov in this case is to arbitrarily choose a polarization of the chirals M = L(p)⊕L⊥(p), as
long as they satisfy (cf. (5.38))
R0(p) ⊆ L(p), R̃0(p) ⊆ L⊥(p). (5.45)
In all our computations, we make the following choice L(p) := R, L⊥(p) := R̃, which obviously
satisfies (5.45).
Once the choice of polarization is made, this boundary condition, which we call BL(p)(p), can
be described just as in (5.39):
Chirals inL(p) : Dirichlet with VEVs according to vacuum p,
Chirals inL⊥(p) : Neumann,
Vector multiplet : Dirichlet with screening (5.27) of gauge holonomy.
5.3.4 Janus interface, J (mC,0)
The transition from a massive boundary in the past labeled by a fixed point p to a massless
boundary in the future is achieved via a mass Janus interface. We assume that the masses in
the past belong to the chamber C. Take y− < 0 and y+ > 0 and put the Janus interface at y = 0
wrapping the elliptic curve Eτ .
We only vary the FA = SU(L) twisted masses in the theory across the Janus interface. We
do not turn on any twisted masses for the U(1)ℏ flavor symmetry. In principle, the mass Jauns
is implemented by giving a y-dependent background value to the adjoint scalar σfA in the flavor
vector multiplet. The background value interpolates between some generic mass m (5.16) in
the chamber C and zero. This is supersymmetric as long as we also turn on the auxiliary
field in the multiplet satisfying the BPS equation (5.6c) (with σℏ = Dℏ = 0 such that σf = σfA
and DF = DFA
). In particular, we can choose a background profile for the mass that is constant
on the intervals [y−, 0) and (0, y+] and changes rapidly near y = 0. Similar Janus interfaces
in 3d N = 4 theories were previously used to compute stable envelopes [40] and R-matrices
[23, 24] for bosonic spin chains.
The supersymmetric quantities we compute are supposed to be renormalization group invari-
ants and capture the infrared dynamics. We, therefore, do not lose anything by considering the
nonzero masses to be arbitrarily large. In this limit, we can integrate out the massive fields from
the interval [y−, 0). But these fields are all present on the other side (0, y+]. Thus, the fields
that acquire masses in the massive vacuum p, including chirals and W-bosons, terminate at the
interface when approaching from the massless side. This is implemented by a set of half-BPS
boundary conditions at y = 0 that were worked out in [40]. We are simply going to use their
prescription.
The chirals with positive and negative masses in the vacuum p belong to MC;+ and MC;−
respectively, according to our earlier definition of these spaces (5.34a). To characterize the
massive W-bosons, let us decompose the set ∆ of roots of the gauge Lie algebra into subsets
corresponding to the sign of their masses at the vacuum p. At p, the adjoint scalar in the vector
multiplet takes the VEV σp(m) given by (5.25). Therefore, the vector field associated to the
60 N. Ishtiaque, S.F. Moosavian and Y. Zhou
root α ∈ ∆ picks up a mass α(σp(m)). The decomposition of the roots can then be written
as ∆ = ∆+(p) ⊔∆0 ⊔∆−(p), where
∆+(p) := {α ∈ ∆|α(σp(m)) > 0}, ∆0 := {α ∈ ∆|α(σp(m)) = 0},
∆−(p) := {α ∈ ∆|α(σp(m)) < 0}.
For generic masses, always only the Cartan valued fields remain massless, so ∆0 = {0} in any
vacuum.
The interface is then implemented by the following boundary conditions on fields [40]:
Chirals in MC;+(p) : Dirichlet,
Chirals in MC;−(p) : Neumann,
Vectors in ∆+(p) : Dirichlet,
Vectors in ∆−(p) : Neumann.
(5.46)
From the definition of the Janus interface as a supersymmetric position-dependent background
profile for the masses, it is clear that it does not introduce any anomaly. However, in practice,
we implement the Janus interface effectively using the above-mentioned boundary conditions
and it is not obvious that they are anomaly free. After all, both the boundary conditions and
the Chern–Simons terms created by integrating out massive fermions contribute to anomaly at
the boundary. It is however a simple exercise to check that these two anomalies precisely cancel
each other, as was checked explicitly in [40].
6 Stable envelopes and R-matrix from gauge theory:
the computation
6.1 Elliptic stable envelope
We are now in a position to compute the matrix elements of the stable envelope of sl(1|1) elliptic
spin chains. For the purpose of BPS computations, we can shrink the interval [y−, y+] to a point,
resulting in an effective 2d N = (0, 2) theory on Eτ . Then the 3d N = 2 partition function
on [y−, y+]× Eτ is given by the elliptic genus of this effective 2d theory. To derive the field
content of this effective 2d theory, we need to look at the intervals [y−, 0] and [0, y+], in addition
to their union, and determine which 3d multiplets have compatible boundary conditions at both
ends of all intervals where it is defined. If any field living on [ystart, yend] has opposite boundary
conditions at ystart and yend then it vanishes in the effective theory. A 3d multiplet that has the
same boundary condition at both ends of an interval survives the collision of boundaries and
decomposes into 2d N = (0, 2) multiplets. Of these (0, 2) multiplets, the ones with Dirichlet
boundary conditions are simply frozen to their boundary values. The remaining (0, 2) multiplets
with Neumann boundary conditions, along with the theory ΥC1(p1) at the past massive bound-
ary, comprise the field content of the effective 2d theory. Note that throughout the computation
of the stable envelope, we use the specific chamber (5.41) to write down formulas.
The vector fields that remain massless in the vacuum p1 are part of the theory along the
entire interval [y−, y+] and they have opposite boundary conditions at
∣∣BL(p2)(p2)
〉
(Dirichlet)
and ⟨NC1(p1)| (Neumann). These vector fields, therefore, do not contribute to the elliptic genus.
All the vector fields receive Dirichlet boundary condition at
∣∣BL(p2)(p2)
〉
and all the vector fields
with positive mass in the vacuum p1 receive Dirichlet at the Janus interface. In the IR, these
vector multiplets with positive masses in p1 are only part of the massless theory along y > 0 and
therefore they survive. They decompose into 2d N = (0, 2) vector and adjoint chiral multiplets.
Dirichlet for 3d N = 2 vector multiplet imposes Dirichlet and Neumann on its constituent 2d
Elliptic Stable Envelopes and Dynamical R-Matrices 61
vector and 2d adjoint chiral multiplets respectively [43]. Contributions from these adjoint chirals
to the elliptic genus are
V(p1) :=
N∏
a,b=1
p1(a)>p1(b)
q−1/12
ϑ
(
sas
−1
b
) . (6.1)
These 2d chirals are not charged under any global symmetry group.
Next, let us look at all the 3d N = 2 chiral multiplets that survive with Dirichlet boundary
conditions. Multiplets receiving Dirichlet at |NC1(p1)⟩ are in L⊥
C1
(p1) = MC1;−(p1) ⊕ R̃0(p1)
((5.3.2) and (5.38)). But everything in MC1;−(p1) receives Neumann at the Janus (5.46) and
therefore vanishes in the effective 2d theory. Multiplets in R̃0(p1) extends unconstrained across
the Janus and intersects with the multiplets L(p2) receving Dirichlet at
〈
BL(p2)(p2)
∣∣. Similarly,
everything in MC1;+(p1) receives Dirichlet at the Janus and intersects with multiplets receiving
Dirichlet at
〈
BL(p2)(p2)
∣∣. Therefore, 3d chiral multiplets surviving the collision of boundaries
with Dirichlet are from
L(p2) ∩
(
R̃0(p1)⊕MC1;+(p1)
)
= R∩
(
R̃0(p1)⊕RC1;+(p1)⊕ R̃C1;+(p1)
)
= RC1;+(p1).
Dirichlet on a 3d chiral multiplet means Dirichlet and Neumann on its constituent 2d N = (0, 2)
chiral and fermi multiplets respectively. Fields with Dirichlet are frozen at their boundary values
in the effective 2d theory and the remaining fermi multiplets contribute to the elliptic genus
Mfe(p1) :=
N∏
a=1
L∏
i=1
i<p1(a)
q1/12ϑ
(
sax
−1
i ℏ1/2
)
.
Now we look for the 3d N = 2 chiral multiplets with Neumann boundary condition contributing
to the effective 2d theory. At |NC1(p1)⟩ all fields in LC1(p1) = MC1;+ ⊕ R0(p1) get Neumann.
Only R0(p1) passes through the Janus unconstrained. On the interval [0, y+] all fields with
negative masses in the vacuum p1 receive Neumann boundary condition at the Janus interface.
Taking the intersection of all these fields with the fields receiving Neumann at ⟨BL(p2)|, we find
L⊥(p2) ∩ (R0(p1)⊕MC1;−(p1)) = R̃ ∩
(
R0(p1)⊕RC1;−(p1)⊕ R̃C1;−(p1)
)
= R̃C1;−(p1).
Neumann on a 3d chiral multiplet means Neumann and Dirichlet on its constituent 2d N = (0, 2)
chiral and fermi multiplets respectively [43]. These 2d chirals contribute to the elliptic genus
Mch(p1) :=
N∏
a=1
L∏
i=1
i<p1(a)
q−1/12
ϑ
(
s−1
a xiℏ1/2
) .
Lastly, we have the contribution (5.42) of the boundary theory ΥC1(p1) to the elliptic genus.
Putting all these together, we find the (unnormalized) elliptic stable envelope
S̃tabC1(p) = SymSN
(V(p)Mfe(p)Mch(p)W(ΥC1(p))) . (6.2)
The symmetrization is over the gauge fugacities. In addition to the maximal torus U(1)N , the
Weyl group SN of U(N) also remains as gauge symmetry at the massive vacuum p1 and it acts
on the gauge fugacities by permuting them.
Let us define
fC1,m(s,x, ℏ, z) :=
(∏
i<m
ϑ(sxi)
)
ϑ
(
sxmzℏm−L)
ϑ
(
zℏm−L
) (∏
i>m
ϑ(sxiℏ)
)
. (6.3)
62 N. Ishtiaque, S.F. Moosavian and Y. Zhou
Then the stable envelope (6.2) becomes
S̃tabC1(p) =
q−(
N
2 )/12ϑ(ℏ̃)N∏
a,i ϑ
(
sax̃iℏ̃
) StabC1(p),
where
StabC1(p) := (−1)
∑
a #(i<p(a))
× SymSN
[(∏
a
fC1,p(a)(sa, x̃, ℏ̃, z̃)
)( ∏
p(a)>p(b)
1
ϑ
(
sas
−1
b
))], (6.4)
and we have also redefined the fugacities as
x̃i := x−1
i ℏ
1
2 , ℏ̃ := ℏ−1, z̃ := zℏ−
L
2 . (6.5)
These redefinitions and the specific form of the stable envelope in terms of the function (6.3) are
merely to facilitate comparison with analogous known formulas coming from BPS computations
in 3d N = 4 SQCDs [40] or from the theory of elliptic stable envelopes on complex symplectic
varieties [2] that correspond to bosonic XYZ spin chains.
Remark 6.1 (normalization of the stable envelopes). The overall factor in S̃tabC1(p) is in-
dependent of the choice of the chamber C and the vacuum p. This makes it ambiguous in the
context of defining stable envelopes since we can change it by normalizing all the stable envelopes
simultaneously. Only the relative normalization of the stable envelopes associated to different
vacua and different chambers is meaningful. We have therefore defined the normalized stable
envelope StabC1(p) without any leading factor. This normalized version is used in Section 3.4
to check that it satisfies all formal properties of elliptic stable envelopes for sl(1|1) spin chains.
Note however that, the overall sign from (6.4) is missing in (3.6) due to a slightly different choice
of normalization for the stable envelopes, see footnote 8.
In the expression (6.4), we have not yet fully imposed the boundary condition BL(p2)(p2).
We still need to impose the screening condition (5.27) associated to the fixed point p2. Doing so
completes the computation of the 3d N = 2 interval partition function, providing us with the
matrix elements of the stable envelope
StabC1(p2, p1) =
〈
BL(p2)(p2)
∣∣J (m, 0) |NC1(p1)⟩ = StabC1(p1)
∣∣
s=s(p2)
.
The specific values of the mass m is not important as long as it belongs to the chamber C1.
Remark 6.2 (SQED and comparison with 3d N = 4/bosonic stable envelopes). The difference
between the quiver (3b) we used to compute elliptic sl(1|1) stable envelope and the quiver used
in [40] to compute the elliptic sl(2) stable envelope is the lack of a self-loop at the gauge node in
the sl(1|1)-quiver. In field theory, this simply means the absence of the 3d N = 2 adjoint chiral
multiplet, which is necessarily a part of the 3d N = 4 vector multiplet. In the computation
of the last section, the 3d N = 4 vector multiplet contribution to the elliptic genus would
involve V(p) from (6.1) along with the contributions from this adjoint chiral. This is the only
difference between the sl(1|1) and sl(2) cases. Notice then, in the 1-magnon sector, i.e., for
SQEDs, there is no distinction between the two cases. Because the vector multiplet, as well as
any adjoint chiral multiplet, do not contribute anything to the elliptic genus in abelian theories.
Thus, the sl(1|1) stable envelopes and the R-matrix in the 1-magnon sector must coincide with
Elliptic Stable Envelopes and Dynamical R-Matrices 63
those of the sl(2) spin chains. More generally, the following two quivers will always lead to the
same stable envelopes, assuming that the hidden parts of the quivers coincide:
1
...
· · ·· · · ≃ 1
...
· · ·· · ·
Another way of saying this is that in a magnon sector with magnon number 1 assigned to
a simple root, the stable envelopes (and consequently the R-matrix) will remain invariant if we
change the parity of the simple root. The equivalence between even and odd gauge nodes in the
case of abelian gauge groups was also pointed out in Example 3.6 from a more mathematical
perspective.
6.2 R-matrix for elliptic sl(1|1) spin chains
From the perspective of integrable systems, we are interested in computing the R-matrix for
supersymmetric XYZ spin chains, not just the stable envelope. As stated in (3.12), R-matrix is
a composition of a stable envelope and an inverse stable envelope.
The action of an R-matrix on a spin chain preserves magnon numbers. As a result, an R-
matrix takes a block diagonal form, one block for each sector with fixed magnon numbers. We
focus on the sl(1|1) spin chain with two fundamental spins.25 The fundamental representation
of sl(1|1) is C1|1, where the lowest weight state is bosonic, and the only excited state is fermionic.
Thus, the tensor product of two such representations is four complex dimensional, consisting of
a bosonic one-dimensional subspace of magnon number 0, a fermionic two-dimensional subspace
of magnon number 1, and a bosonic one-dimensional subspace of magnon number 2. The R-
matrix also decomposes as
R =
R(0)
R(1)
R(2)
, (6.6)
where the superscript refers to the magnon number.
The gauge theory dual of a spin chain with two fundamental spins for sl(1|1) corresponds to
a 3d N = 2 SQCD with SU(2) flavor symmetry. Each magnon sector corresponds to a different
unitary gauge group, the rank of the gauge group is equal to the magnon number. The zero
magnon sector thus corresponds to a trivial gauge theory and accordingly, the R-matrix in the
0-magnon sector is always trivial
R
(0)
C2 C1
= 1. (6.7)
In the above formula, C1 and C2 are the two chambers for the Cartan of su(2). To compute
the R-matrix, we always need stable envelopes in two different chambers. We use two real
masses m1 and m2 to parameterize the Cartan subalgebra of su(2) with the tracelessness condi-
tion m1 +m2 = 0. There are then two chambers for the masses C1 = m1 < m2, C2 = m1 > m2.
We computed the stable envelope for C1 in (6.4). The stable envelope in the other chamber can
25The spin chain can be arbitrarily large, but the R-matrix always acts on the tensor product of two repre-
sentations. So for the purpose of computing the R-matrix it suffices to consider just two sites. While testing
Yang–Baxter we will have to extend the R-matrix to act trivially on a third site as well.
64 N. Ishtiaque, S.F. Moosavian and Y. Zhou
be computed following identical steps as in the previous section. Here we simply write down the
answer
StabC2(p) := (−1)
∑
a #(i>p(a))
× SymSN
[(∏
a
fC2,p(a)(sa,x, ℏ, z)
)( ∏
p(a)<p(b)
1
ϑ
(
sas
−1
b
))], (6.8)
where we have defined
fC2,m(s,x, ℏ, z) :=
(∏
i>m
ϑ(sxi)
)
ϑ
(
sxmzℏ1−m
)
ϑ
(
zℏ1−m
) (∏
i<m
ϑ(sxiℏ)
)
.
Note that, in this section, all fugacities are the newly defined ones from (6.5), but to avoid
clutter we omit the tildes.
Let us now specialize these formulas to the cases of gauge group U(1) and U(2) to compute
the stable envelopes and the R-matrix in the 1 and 2-magnon sectors.
1-magnon. The Higgs branch (see Example 5.1) has two fixed points p1 and p2, correspond-
ing to the two possible maps {1} ↪! {1, 2} mapping 1 to either 1 or 2 (cf. (5.19)). There is
a single gauge fugacity and two screening conditions (5.27) for the two fixed points. We can now
compute the four matrix elements of each of the two stable envelopes (6.4) and (6.8) in the two
chambers, we find
Stab
(1)
C1
=
(
ϑ
(
x−1
1 x2ℏ
)
0
ϑ(x1x
−1
2 zℏ−1)ϑ(ℏ)
ϑ(zh−1)
−ϑ
(
x1x
−1
2
)) ,
Stab
(1)
C2
=
(
−ϑ
(
x−1
1 x2
) ϑ(x−1
1 x2zℏ−1)ϑ(ℏ)
ϑ(zh−1)
0 ϑ
(
x1x
−1
2 ℏ
) )
.
The R-matrix is a simple matrix product
R
(1)
C2 C1
= Stab−1
C2
· StabC1
=
−1
ϑ
(
x−1
1 x2ℏ−1
)
ϑ(x−1
1 x2)ϑ(z)ϑ(zℏ−2)
ϑ(zℏ−1)2
ϑ(ℏ)ϑ(x−1
1 x2zℏ−1)
ϑ(zℏ−1)
ϑ(ℏ)ϑ(x1x−1
2 zℏ−1)
ϑ(zℏ−1)
−ϑ(x1x−1
2 )
. (6.9)
To simplify the upper left component, we have used the identity∑
cyclic
ϑ(AB)ϑ
(
AB−1
)
ϑ(C)2 = 0.
2-magnon. The gauge theory dual of the 2-magnon sector has gauge group U(2). There are
two gauge fugacities and a unique vacuum corresponding to the unique (modulo permutation)
map {1, 2} ↪! {1, 2}. The stable envelopes are just functions
Stab
(2)
C1
= −ϑ
(
x−1
1 x2ℏ
)
, Stab
(2)
C2
= −ϑ
(
x1x
−1
2 ℏ
)
.
As is the R-matrix,
R
(2)
C2 C1
= Stab−1
C2
StabC1 =
ϑ
(
x−1
1 x2ℏ
)
ϑ
(
x1x
−1
2 ℏ
) . (6.10)
Elliptic Stable Envelopes and Dynamical R-Matrices 65
Putting together (6.7), (6.9), and (6.10) as in (6.6), we find the complete R-matrix for the
elliptic sl(1|1) spin chain
RC2 C1 =
1 0 0 0
0 − ϑ(x−1
1 x2)ϑ(z)ϑ(zℏ−2)
ϑ(x−1
1 x2ℏ−1)ϑ(zℏ−1)2
ϑ(ℏ)ϑ(x−1
1 x2zℏ−1)
ϑ(x1x
−1
2 ℏ)ϑ(zℏ−1)
0
0
ϑ(ℏ)ϑ(x1x−1
2 zℏ−1)
ϑ(x1x
−1
2 ℏ)ϑ(zℏ−1)
− ϑ(x−1
1 x2)
ϑ(x−1
1 x2ℏ−1)
0
0 0 0
ϑ(x−1
1 x2ℏ)
ϑ(x1x
−1
2 ℏ)
. (6.11)
Here x1 and x2 are referred to as the spectral parameters associated to the two sites, z as the
dynamical parameter, and ℏ as the quantization parameter. This R-matrix solves the dynam-
ical Yang–Baxter equation for sl(1|1) spin chains (3.19). Note that, relative to the R-matrix
presented in (3.18), the signs in front of the two diagonal entries in the 1-magnon sector are
different in the above formula. This is due to the slightly different choice of normalization of
the stable envelopes, as pointed out in Remark 6.1.
7 2d and 1d avatars of elliptic stable envelopes and the R-matrix
This section is devoted to taking the 3d! 2d and 2d! 1d reduction of elliptic stable envelopes.
In Section 7.1, we take the 3d ! 2d reduction, which produces the K-theoretic stable envelopes
and the trigonometric R-matrix for sl(1|1). The 2d ! 1d reduction, which produces cohomo-
logical stable envelopes and the rational R-matrix, is the subject of Section 7.2. In the case
of cohomological stable envelopes and the rational R-matrix for sl(1|1), we compare our result
with that of Rimányi and Rozansky in [117] and find perfect agreement. The discussion in this
section is based on Section 3.9 and also the physical description in [40, Section 2.1].
7.1 The K-theory limit
In this section, we reduce the formulas for elliptic stable envelopes to two dimensions and give
a prediction for K-theoretic stable envelopes associated with Lie superalgebra sl(1|1).
Procedure for 3d ! 2d reduction. In Section 4.2, we explained the parameters in three-
dimensional theory on I×Eτ . One could shrink one of the cycles of Eτ , corresponding to τ ! 0,
to get a two-dimensional theory S1
A× I. The procedure of going from three to two dimensions is
that we shrink the B-cycles of Eτ . This limit could be better handled if we take our parameters
to take value in the S-transformed elliptic curve E− 1
τ
. Then, the two-dimensional limit is given
by τ ! 0, q−
2πi
τ ! 0. In this limit, the elliptic curve becomes nodal and the smooth locus is
isomorphic to C×. It turns out that the K-theoretic stable envelope depends on the choice of
a slope. This makes the behavior of elliptic Kähler parameters in the q ! 0 limit more subtle.
Defining the slope parameter as (3.22) such that s is generic (i.e., it belongs to an alcove). The
precise procedure to take the K-theory limit is explained in Section 3.9.
In the case of N = 2 SQCD, we need two results for deriving the K-theoretic stable envelopes:
(1) the behavior of this function in the limit q ! 0, that can be deduced from its very definition
(5.43) limq!0 ϑ(t; q) = t
1
2 − t−
1
2 .
(2) and that [2, equation (2)]
lim
q!0
(
ϑ(t1z; q)
ϑ(t2z; q)
)∣∣∣∣
z=qs
=
(
t1
t2
)⌊s⌋+ 1
2
. (7.1)
66 N. Ishtiaque, S.F. Moosavian and Y. Zhou
Finally, we note that all the holonomies introduced in Section 4.2 are still multiplicative but
are now valued in C×.
K-theoretic stable envelopes for sl(1|1). We now perform a 3d ! 2d reduction of the
elliptic stable envelope, which gives us the K-theoretic stable envelopes.
From the definition of the function fC1,m (6.3) and (7.1), we find that
lim
q!0
[
ϑ
(
saxp(a)ℏ−(L−p(a))z
)
ϑ
(
ℏ−(L−p(a))z
) ∣∣∣∣∣
z=qs
]
= (saxp(a))
⌊s⌋+ 1
2 .
Hence,
lim
q!0
fC1,m(s,x, ℏ, z) =
(
ℏ
#(i>p(a))
2
∏
i
(sxi)
1
2
)
(sxm)
⌊s⌋
∏
i<m
(
1− (sxi)
−1
) ∏
i>m
(
1− (sxiℏ)−1
)
=
(
ℏ
#(i>p(a))
2
∏
i
(sxi)
1
2
)
fC1,s,m(s,x, ℏ),
where we have defined
fC1,s,m(s,x, ℏ) := (sxm)
⌊s⌋
∏
i<m
(
1− (sxi)
−1
) ∏
i>m
(
1− (sxiℏ)−1
)
.
Furthermore,
lim
q!0
(∏
a>b
1
ϑ
(
sas
−1
b
)) =
∏
a>b
1(
sas
−1
b
) 1
2 −
(
sas
−1
b
)− 1
2
.
Finally, we know from (3.23) that we need to multiply the result by two factors: (1) the inverse
of the square root of the determinant of the partial polarization (det PolX)
− 1
2 for X = MH(N,L),
which is given by the product of fugacities of the quarks Q
(det PolX)
− 1
2 =
∏
a,i
1(
sax
−1
i ℏ1/2
) 1
2
.
However, we need to implement the redefinition of fugacities according to (6.5). Therefore,
(det PolX)
− 1
2 =
∏
a,i
1
(saxi)
1
2
,
(2) the square root of the determinant of the polarization of the fixed loci (det PolXA)
1
2 , which
in our case is trivial (det PolXA)+
1
2 = 1. Putting these results together, and using (3.23), we see
that26
StabC1,s(p) = (−1)
∑
a #(i<p(a))ℏ
1
2
∑
a #(i>p(a))
× SymSN
( N∏
a=1
fC1,s,p(a)(sa,x, ℏ)
)
·
(∏
a>b
1(
sas
−1
b
) 1
2 −
(
sas
−1
b
)− 1
2
) . (7.2)
26The K-theory limit of elliptic stable envelope for the resolved determinantal variety MH(N,L) has also been
computed in (3.26) using a different polarization as explained there. Compared to (3.26), the expression (7.2)
contains the extra factor (−1)
∑
a #(i<p(a))|, which is the result of a choice of different polarization in the BPS
computation of elliptic stable envelope.
Elliptic Stable Envelopes and Dynamical R-Matrices 67
By Lemma 3.2, this is the K-theoretic stable envelope for the Lie superalgebra sl(1|1). Notice
that it is manifestly a locally-constant function of s, which is a characteristic feature of K-
theoretic stable envelopes as a function of s.
Trigonometric R-matrix for sl(1|1). We can now perform the reduction of the sl(1|1)
elliptic R-matrix (6.11) to construct the sl(1|1) trigonometric R-matrix. Let us define the
function G(x) := x
1
2 − x−
1
2 . The result of the reduction is
RC2 C1(u) =
1 0 0 0
0 G(u)
G(u−1ℏ) u⌊s⌋+
1
2
G(ℏ)
G(u−1ℏ) 0
0 u−⌊s⌋− 1
2
G(ℏ)
G(u−1ℏ)
G(u)
G(u−1ℏ) 0
0 0 0 G(uℏ)
G(u−1ℏ)
. (7.3)
One can check that it satisfies the trigonometric version of Yang–Baxter equations given by
R12(u/v)R13(u/w)R23(v/w) = R23(v/w)R13(u/w)R12(u/v).
7.2 The cohomology limit
In the previous section, we have constructed the K-theoretic stable envelopes and the trigono-
metric R-matrix for Lie superalgebra sl(1|1) by performing a 3d ! 2d reduction. It is then
natural to consider a further 2d! 1d reduction to construct the cohomological stable envelopes
and the rational R-matrix for sl(1|1). Fortunately, the mathematical construction of cohomo-
logical stable envelope for Lie superalgebra sl(1|1) is available in the literature [117], and we
can compare our result. In this section, we perform this 2d ! 1d reduction, and we recover the
cohomological stable envelopes and the geometrical R-matrix of Rimányi and Rozansky.
Procedure for 2d ! 1d reduction. We need to know what happens by going from 2d
to 1d. In the 3d ! 2d, reduction, one considers the reduction of the theory on one of the
circles of the elliptic curve. Similarly, the 2d ! 1d result is obtained by the reduction of the
theory on the remaining circle. This amounts to the following substitution of gauge and flavor
holonomies27 (see Section 3.9 and [40, Section 2.1])
(sa, xi, ℏ) 7! lim
ϵ!0
(
eϵsa , eϵxi , eϵℏ
)
, (7.4)
On the other hand, the holonomy associated with the topological symmetry defines Käher
parameter in three dimensions. Upon 3d ! 2d reduction, this becomes a theta angle, i.e.,
the slope parameter, as explained in Section 7.1. By 2d ! 1d reduction, it turns out that this
parameter completely disappears [40, p. 16]. This is consistent with the fact that cohomological
stable envelopes are not dependent on extra choices like a slope. Therefore, one should (1)
use (7.4) to replace multiplicative holonomies with additive parameters in the ϵ ! 0 limit by
throwing away terms of order O
(
ϵ2
)
, and (2) discard all s-dependent factors. In this process, all
gauge and flavor holonomies become additive and C-valued.
Cohomological stable envelope for sl(1|1). We can now perform the reduction explicitly.
Different factors of (7.2) are reduced as follows
fC1,s,m ! fC1,m(s,x, ℏ) :=
∏
i<m
(s+ xi)
∏
i>m
(s+ xi + ℏ),
27The precise statement in going from K-theoretic stable envelope to the cohomological one is given in (3.25).
68 N. Ishtiaque, S.F. Moosavian and Y. Zhou
and ∏
a>b
1(
sas
−1
b
) 1
2 −
(
sas
−1
b
)− 1
2
!
∏
a>b
1
sa − sb
.
Therefore, according to Lemma 3.2, we end up with the cohomological stable envelope28
StabC1(p) = (−1)
∑
a #(i<p(a))SymSN
[(
N∏
a=1
fC1,m(s,x, ℏ)
)
·
(∏
a>b
1
sa − sb
)]
. (7.5)
Rational R-matrix for sl(1|1). We can now perform the reduction of the sl(1|1) trigono-
metric R-matrix (7.3) to construct the sl(1|1) rational R-matrix. The result is
RC2 C1(u) =
1 0 0 0
0 u
ℏ−u
ℏ
ℏ−u 0
0 ℏ
ℏ−u
u
ℏ−u 0
0 0 0 ℏ+u
ℏ−u
. (7.6)
This satisfies the rational Yang–Baxter equations given by
R12(u− v)R13(u− w)R23(v − w) = R23(v − w)R13(u− w)R12(u− v).
Let us finally compare our result with the construction in the mathematical literature [117]. For
this matter, we first give a synopsis of [117].
Brief recap of the work of Rimányi and Rozanski. Consider an index with values
(00), (01), (10), or (11). Depending on the choice of r, we have
r = 00: Ceven ⊕ Ceven, r = 10: Ceven ⊕ Codd, r = 01: Codd ⊕ Ceven,
r = 11: Codd ⊕ Codd,
which corresponds to sl(2, 0), sl(1, 1), sl(1|1), and sl(0|2), respectively. We are only interested
in r = (10) (or equivalently r = (01)).29 Analog to the classical case, there is a super-stable
map Stab
(r)
π , which depends on a permutation π ∈ SL. The definition of this map involves the
triple (N,L, π) and are defined if certain classes κ
(r)
p,π := Stab
(r)
π (1p)
30 exist and satisfy certain
axioms with p denoting the injective map (5.19), that uniquely determine Stab
(r)
π if it exists [117,
Definition 4.1]. The explicit forms of the stable envelopes are then given in terms of certain ra-
tional functions W
(r)
l;N,L(s,x, ℏ) that are called super-weight functions and are the super-analog
of weight functions defined by Tarasov and Varchenko [121, 142]. Here, x := {s1, . . . , sN}
and x := {x1, . . . , xL} are Chern roots and are equivariant parameters, respectively. The super-
weight functions are defined as [117, Section 5.2]
W (10)
p (s,x, ℏ) := SymSN
(
U (10)
p (s,x; ℏ)
)
, (7.7)
28The cohomology limit of K-theoretic stable envelope for the resolved determinantal variety MH(N,L) has also
been computed in (3.27) using a different polarization as explained there. Compared to (3.27), the expression (7.5)
contains the extra factor (−1)
∑
a #(i<p(a))|, which is the result of a choice of different polarization in the BPS
computation of elliptic stable envelope.
29The case r = (00) corresponds to sl(2) spin chains, which is explored in [23, 88].
30In [117], these classes are denoted as κ
(r)
I,π, where I = {I1, . . . , IN} ⊆ {1, . . . , L}. I is determined by the
injective map p : {1, . . . , N} ! {1, . . . , L}. We prefer to use the map p to have a uniform notation throughout the
paper.
Elliptic Stable Envelopes and Dynamical R-Matrices 69
where31
U (10)
p (s,x, ℏ) =
N∏
a=1
p(a)−1∏
i=1
(sa − xi + ℏ)
L∏
i=p(a)+1
(−sa + xi)
·
(∏
a>b
1
(sa − sb)
)
. (7.8)
More generally, one can define the superweight functions for any permutation π ∈ SL through
W (10)
p,π (s,x; ℏ) :=W
(10)
π−1(p)
(s, π(x), ℏ), (7.9)
where π(x) = {xπ(1), . . . , xπ(L)}. Consider the set of all subsets of {1, . . . , L} consisting of N
elements. Each such subset is determined by an injective map p : {1, . . . , N} ! {1, . . . , L}.
We denote the set of all such injective maps as pN . For any p, p′ ∈ pN , we denote the poly-
nomial function32 obtained by substituting sa = xp′(a)
33 by W
(10)
p,π (xJ ,x, ℏ). Then, the tu-
ple
(
W
(10)
p,π (xp′ ,x, ℏ)
)
p′∈pN
, is a class in the equivariant cohomology HT (Gr(N,L)), which we
denote as
[
W
(10)
p,π
]
[117, Proposition 7.1], and satisfies the axioms for the class κ
(r)
p,π [117, The-
orem 7.3]. Hence by [117, Definition 4.1], κ
(10)
p,π = Stab
(10)
π (1p) =
[
W
(10)
p,π
]
. Therefore, the stable
envelope maps the identity class to the class
[
W
(10)
p,π
]
, and as a result is given by the superweight
functions W
(10)
p,π (s,m, ℏ). Furthermore, the geometric R-matrix for r = (10) is given by [117,
Section 8.2]
RRR(u) =
1 0 0 0
0 u
ℏ−u
ℏ
ℏ−u 0
0 ℏ
ℏ−u
u
ℏ−u 0
0 0 0 ℏ+u
ℏ−u
. (7.10)
Recovering the result of Rimányi and Rozansky. We can now compare our re-
sults to that of [117]. First consider a permutation π such that {x1, . . . , xp(a), . . . , xL} !
{xL, . . . , xp(a), . . . , x1}. Then,
W (10)
p,π (s,−x, ℏ) = (−1)
∑
a #(i<p(a))
× SymSN
N∏
a=1
( ∏
i<p(a)
(sa + xi)
∏
i>p(a)
(sa + xi + ℏ)
)
·
(∏
a>b
1
(sa − sb)
) .
Comparing this expression to (7.5), we find that StabC1(p) = W
(10)
p,π (s,−x, ℏ). Furthermore, by
comparing (7.10) to (7.6), we find that RC2 C1(u) = RRR(u). This verifies our construction of
cohomological stable envelopes for sl(1|1).
A Equivariant elliptic cohomology
In this appendix, we give a brief overview of equivariant elliptic cohomology, we only list the
essential ingredients that are used in this paper. We basically follow the presentation in [109,
Section 1.4, Appendices A and B]. The general theory on equivariant elliptic cohomology can
be found in [58, 59, 60, 62, 87, 126],
31The explicit form of U
(r)
l (s;x; ℏ) for other values of r can be found in [117, Section 5].
32The fact that Wp,π(xp′ ;x; ℏ) is a polynomial function is proven in [117, Proposition 6.1].
33This is the additive analog of the screening condition sa ! s
(p)
a that we used to construct matrix elements of
elliptic stable envelopes.
70 N. Ishtiaque, S.F. Moosavian and Y. Zhou
Let q be a nonzero complex number such that |q| < 1, then take the elliptic curve E = C×/qZ.
For simplicity, we choose the elliptic curve E to be general enough such that it has no complex
multiplication, i.e., End(E) = Z.
Fix a compact Lie group G and denote its complexification by GC, then the G-equivariant
elliptic cohomology is a functor
{pairs of G-spaces}! {graded super schemes over EG} (X, ∂X) 7! EllG(X, ∂X),
where EG is the moduli scheme of semistable principal GC-bundles of trivial topological type on
the dual elliptic curve E∨. For a torus T , ET = E⊗ZCochar(T ), and for a compact Lie group G
with maximal torus T , it is known that [56] EG ∼= ET /W , whereW is the Weyl group of G which
acts on Cochar(T ) naturally. When G is simple and simply-connected, it is known that [86] EG
is isomorphic to the weighted projective space P(1, g1, . . . , gr), where gi are coefficients in the
decomposition θ∨ =
∑
i giα
∨
i , of the dual of highest root into simple coroots. The map EllG is
functorial with respect to the change of groups, i.e., for a group homomorphism G! G′, there
is a natural transformation EllG ! EllG′ . In the following, we focus on the case when ∂X = ∅
and X is a complex algebraic variety.
The graded super scheme means the structure sheaf O•
EllG(X) =
⊕
d∈Z Od
EllG(X) is graded
and graded-commutative. It is known that each homogeneous piece Od
EllG(X) is a coherent sheaf
on EG. Moreover, EllG is two-periodic Od
EllG(X) = Od+2
EllG(X) ⊗ ω, ω = T ∗
0E. Since the stable enve-
lope is in degree zero, we will focus on O0
EllG(X), which is the structure sheaf of a scheme, and it
is finite over EG [60]. In the later discussions, we use the same notation EllG(X) to denote the
degree zero piece.
A.1 Chern class
If H is another compact Lie group, and P ! X is a G-equivariant principal H-bundle, then P
induces a map c : EllG(X)! EllH(pt) = EH , called the Chern class map. For a vector bundle V
of rank r, which is equivalent to a principal GLr-bundle, the Thom line bundle associated to V
is defined as
Θ(V ) := c∗O(DΘ), DΘ = {0}+ Sr−1E ⊂ SrE = EGLr .
Θ(V ) inherits a canonical section ϑ(V ) from the effective divisor DΘ. Θ: V ! Θ(V ) descends
to a group homomorphism KG(X)! Pic(EllG(X)), this follows from
Θ(V ) = Θ(V1)⊗Θ(V1) for short exact sequence 0! V1 ! V ! V2 ! 0.
Note that the canonical section simply multiplies: ϑ(V ) = ϑ(V1)ϑ(V2). We also have Θ
(
V ∨) ∼=
Θ(V ), such that the canonical section picks up a sign ϑ
(
V ∨) = (−1)rkV ϑ(V ).
For vector bundles V1, V2 of ranks r1, r2, Θ(V1 ⊗ V2) is the pullback of O(DΘ) on S
r1r2E via
the composition of maps
EllG(X)
c1×c2−! Sr1E× Sr2E m
−! Sr1r2E,
where m is induced from the multiplication of elliptic curve
m : ({x1, . . . , xr1}, {y1, . . . , yr2}) 7! {xi + yj}.
In particular, for G = 1 and considering the tensor product of line bundles, the above gives rise
to the formal group law of non-equivariant elliptic cohomology.
Elliptic Stable Envelopes and Dynamical R-Matrices 71
A.2 Gysin map
For a proper G-equivariant map f : X ! Y , assume that f factors as a regular embedding
i : X ↪! Z and a smooth projection p : Z ! Y , and that both i and p are G-equivariant, then
there exists a distinguished element
f⊛ ∈ HomOEllG(Y )
(f∗Θ(Tf ),OEllG(Y )),
where f∗ : EllG(X)! EllG(Y ) is the induced map between elliptic cohomologies, and Tf is the
relative tangent bundle. We call f⊛ the Gysin map.
If Tf equals to f∗V in KG(X) for some V ∈ KG(Y ), then we denote by [X] the sec-
tion Γ(EllG(Y ),Θ(−V )) induced by the Gysin map f⊛ : f∗OEllG(X) ! Θ(−V ) precomposed
with the canonical map OEllG(Y ) ! f∗OEllG(X).
For example, let N ! X be a G-equivariant vector bundle and let i : X ↪! N be the
zero section, then f∗ : EllG(X) ! EllG(N) is an isomorphism of schemes due to the homotopy
invariance of elliptic cohomology, and i⊛ ∈ Γ(EllG(X),Θ(N)) is the section ϑ(N).
A.3 Supports
For a section α of a coherent sheaf F on EllG(X), and a G-invariant open subset j : U ↪! X,
we say that α is supported on X \U if j∗(α) = 0 in EllG(U). Define the support supp(α) to be
the intersection of G-invariant closed subset that α is supported on.
The Gysin map can be defined for compactly supported sections. Namely for a G-equivariant
map f : X ! Y such that it factors as a regular embedding followed by a smooth projection,
then there exists a distinguished element
f⊛ ∈ HomOEllG(Y )
(f∗Θ(Tf )c,OEllG(Y )),
where Θ(Tf )c ⊂ Θ(Tf ) is the subsheaf of sections α such that f |supp(α) is proper.
A.4 Correspondences
Consider the diagram
X2 ×X1
X2 X1.
pt
p2 p1
q2 q1
Assume that X1 is smooth, then for a pair of line bundles Li ∈ Pic(EllG(Xi)), and for any
section α ∈ Γ
(
EllG(X2 × X1),L2 ⊠
(
L ∨
1 ⊗ Θ(TX1)
))
, such that supp(α) is proper over X2, α
induces a map
q1∗L1 q2∗L2,
p2⊛(αp∗1(·))
in Coh(EG).
72 N. Ishtiaque, S.F. Moosavian and Y. Zhou
A.5 Degree of a line bundle
For a line bundle L on an abelian variety A, we define its degree degL to be its image in
the Néron–Severi group NS(A) = Pic(A)/Pic0(A). It is known that NS(A) is isomorphic to
the subgroup of homomorphisms f ∈ Hom
(
A,A∨) such that f = f∨, and the isomorphism
is given by34 L 7!
(
ϕL : x 7! x∗L ⊗ L −1
)
. Let T be a torus, then Hom
(
ET ,E
∨
T
)
is iso-
morphic to Char(T )⊗2 ⊗Z End(E). As we have assumed in the beginning, E is chosen such
that End(E) = Z, so the Néron–Severi group NS(ET ) is isomorphic to S2Char(T ). Explicitly,
any µ ∈ Char(T ) gives rise to a map ϕµ : ET ! E, then deg ϕ∗µO(DΘ) = µ⊗ µ ∈ S2Char(T ). So
for V =
∑
µ Vµ · µ ∈ KT (pt), we have degΘ(V ) =
∑
µ(dimVµ)µ⊗ µ.
B Notations and conventions for supersymmetry
In this appendix, we collect the notations and conventions regarding supersymmetry used in the
computation of elliptic stable envelopes in Section 6.
B.1 Conventions for spinors
For a two component SU(2) spinor ϵ, we write its components as ϵα where α = 1, 2. We
use α, β, . . . as spinor indices. Our spinors will be anti-commuting. Let εαβ and εαβ be the
totally antisymmetric tensors satisfying ε12 = ε21 = 1. These tensors are used to raise and lower
spinor indices ϵα = εαβϵ
β, ϵα = εαβϵβ. Let γ
µ for µ = 1, 2, 3 be the Pauli matrices
[(
γ1
) β
α
]
=
(
1
1
)
,
[(
γ2
) β
α
]
=
(
−i
i
)
,
[(
γ3
) β
α
]
=
(
1
−1
)
.
We also define the charge conjugation matrix
C =
[
Cαβ
]
:= −iγ2 =
(
−1
1
)
.
In other words, Cαβ = −εαβ. Some properties of the charge conjugation matrix are
C2 = −1, CT = C−1 = −C, Cγµ = −(γµ)TC. (B.1)
Contraction of spinors ϵ and λ are defined as
ϵλ := ϵαλα, (B.2)
i.e., we use the NE-SW convention for contracting indices on anti-commuting spinors. We can
think of two-component spinors as column vectors, with lowered indices, e.g., ϵ = ( ϵ1ϵ2 ). Then
we can write the spinor bilinear (B.2) in matrix notation as35
ϵλ = ϵTCλ. (B.3)
Using the properties (B.1) of the charge conjugation matrix, we can derive some symmetry
properties of various spinor bilinears:
ϵγµ1 · · · γµnλ = (−1)
n(n+1)
2 λγµ1 · · · γµnϵ.
34This fact can be derived from Theorem 2 together with the remark after that in [92, Section 20].
35There should not be any confusion between when the matrix notation is being used in spinor bilinears, since
in the matrix notation there will always be the transpose of a spinor to the left (as in the right-hand side of (B.3)),
otherwise there will not be any transpose (as in the left-hand side of (B.3)).
Elliptic Stable Envelopes and Dynamical R-Matrices 73
B.2 3d N = 2 supersymmetry
The supersymmetry algebra has four supercharges which we can denote as Qα, Qα with α being
a spinor index. A generic supercharge is a linear combination of these
Qϵ,ϵ := ϵαQα + ϵαQα, (B.4)
where the coefficients ϵ, ϵ are two components spinors. The variation of a field in the theory
caused by the supercharge Qϵ,ϵ will be denoted δϵ,ϵ.
For a Lie group G, the corresponding 3d N = 2 vector multiplet consists of a vector field A,
two scalars σ and D and two spinors λ and λ. All these fields are valued in the Lie algebra g of G.
We adopt the convention that elements of real Lie algebras are hermitian. Their supersymmetry
variations are
δϵ,ϵAµ =
i
2
(
ϵγµλ− λγµϵ
)
, δϵ,ϵσ =
1
2
(
ϵλ− λϵ
)
,
δϵ,ϵλ = −1
2
γµνFµνϵ− Dϵ+ iγµDµσϵ, δϵ,ϵλ = −1
2
γµνFµνϵ+ Dϵ− iγµDµσϵ,
δϵ,ϵD = − i
2
ϵγµDµλ− i
2
Dµλγ
µϵ+
i
2
[ϵλ, σ] +
i
2
[
λϵ, σ
]
. (B.5)
Given a representation R of G we can add an R-valued chiral multiplet containing two scalars ϕ
and F, and a spinor ψ. We also have the anti-chiral multiplet containing scalars ϕ, F, and
spinor ψ valued in the dual representation R∨. These fields transform under supersymmetry as
δϵ,ϵϕ = ϵψ, δϵ,ϵϕ = ϵψ, δϵ,ϵψ = iγµϵDµϕ+ iϵσ · ϕ+ ϵF,
δϵ,ϵψ = iγµϵDµϕ− iϵσ · ϕ+ ϵF, δϵ,ϵF = ϵ
(
iγµDµψ − iσ · ψ − iλ · ϕ
)
,
δϵ,ϵF = ϵ
(
iγµDµψ + iσ · ψ − iλ · ϕ
)
. (B.6)
Here σ· is the action of σ on whatever follows in the corresponding representation.
By computing the commutator [δϵ,ϵ, δη,η] on fields, we find the (anti)commutation relations
between the supercharges{
Qα,Qβ
}
= −iγµαβDµ − iεαβσ · . (B.7)
In a gauge theory, we always have a dynamical vector multiplet for the gauge group. So the
last term above is an infinitesimal gauge transformation by the adjoint scalar σ of this vector
multiplet. But we can also introduce background vector multiplets for flavor symmetry groups.
If G is the gauge group and F the flavor group with Lie algebras g and f, respectively, then we can
treat the vector multiplet from (B.5) as (g⊕f)-valued. Thus all the fields of the multiplet become
sums of g and f-valued fields. In particular, we write σ = σG+σF according to this decomposition
and the supersymmetry algebra (B.7) contains a gauge and a flavor symmetry transformation by
the dynamical σG and the background σF respectively. We only turn on background values for
scalar fields, namely σF and DF . Furthermore, these values are constrained by the requirement
to preserve some amount of supersymmetry, i.e., to solve the BPS equations. More on BPS
equations in Section 5.1.2.
B.3 Localizing supercharge
Suppose the elliptic curve Eτ in our 3d space-time I × Eτ has a holomorphic volume form dz
and two 1-cycles, S1
A and S1
B defined by:
∮
S1
A
dz = 1,
∮
S1
B
dz = τ . Let y, θA and θB be the real
coordinates on I, S1
A and S1
B, respectively. We take the orientation defining volume form on M
to be dy ∧ dθA ∧ dθB. This induces the volume forms dy ∧ dθA on Σ := R× S1
A and dθA ∧ dθB
74 N. Ishtiaque, S.F. Moosavian and Y. Zhou
on Eτ . We use holomorphic coordinates w and z on Σ and Eτ respectively such that the volume
forms in real and complex coordinates are related by dy∧dθA = i
2dw∧dw, dθA∧dθB = i
2dz∧dz.
The real and complex coordinates are related by w = y + iθA, z = θA + iθB. We denote the
derivatives with respect to y, θA and θB by ∂y, ∂A and ∂B, respectively. They are related to
the derivatives with respect to the complex coordinates as ∂w = 1
2(∂y − i∂A), ∂z =
1
2(∂A − i∂B).
Similarly for the covariant derivatives Dw, Dz, Dy, DA and DB.
We can relabel our 3d N = 2 supercharges Qα, Qα to adjust to the complex structures of
either Σ or Eτ . If we define
Q+ :=
1√
2
(Q1 + Q2), Q− :=
1√
2
(Q1 − Q2), Q+ :=
1√
2
(
Q1 + Q2
)
,
Q− :=
1√
2
(
Q1 − Q2
)
,
then the subscripts ± refer to the chirality on Eτ and the algebra (B.7) reflects the complex
structure of Eτ{
Q+,Q+
}
= 2Dz,
{
Q−,Q−
}
= 2Dz,
{
Q+,Q−
}
= i(Dy − σ),{
Q−,Q+
}
= i(Dy + σ).
Alternatively, if we define
q+ :=
Q1√
2
, q− := −Q2√
2
, q+ :=
Q1√
2
, q− := −Q2√
2
,
then the subscripts ± refer to the chirality on Σ and the algebra (B.7) reflects the complex
structure of Σ
{q+, q+} = iDw, {q−, q−} = −iDw, {q+, q−} =
i
2
(DB − σ),
{q−, q+} =
i
2
(DB + σ).
The localization results used in this note are with respect to the supercharge
Q := Q− + Q− = q+ + q− + q+ + q−, (B.8)
which satisfies
Q2 = 2Dz. (B.9)
Furthermore, translation in the y-direction is Q-exact{
Q,Q+ + Q+
}
= 2iDy. (B.10)
From the point of view of both Σ and Eτ , the 3d N = 2 algebra appears as the 2d N = (2, 2)
algebra. On the elliptic curve, the supercharge Q belongs to the N = (0, 2) subalgebra. On Σ,
the supercharge Q does not belong to any proper subalgebra of (2, 2) but rather it can be related
to certain deformations of both the A-model and the B-model BRST operators, as we find below.
Dimensional reduction. By compactifying the S1
B direction of the 3d space-time, we land
on a 2d N = (2, 2) theory on Σ with four supercharges Q±, Q± that descend from the 3d
supercharges q±, q±, respectively. The 2d N = (2, 2) supersymmetry has the usual A-model
Elliptic Stable Envelopes and Dynamical R-Matrices 75
and B-model supercharges QA := Q+ + Q−, QB := Q+ + Q−, that are nilpotent up to gauge
transformations.36 The remaining supersymmetry in the A and B-model can be labeled as
GA+ := −iQ+, GA− := iQ−, GB+ := −iQ+, GB− := iQ−.
For any vector field V = V+∂w + V−∂w, we define the linear combination ιV G := V+G+ + V−G−
and we have the anti-commutation relations{
QA, ιV G
A
}
=
{
QB, ιV G
B
}
= V+Dw + V−Dw = LV + gauge transformation.
Ω-deforming the A and B-model supercharges with respect to a vector field V means changing
the BRST operator from QA and QB to QA + ιV G
A and QB + ιV G
B such that the new BRST
operator squares to the space-time transformation generated by V(
QA + ιV G
A
)2
=
(
QB + ιV G
B
)2
= LV + gauge transformation.
We observe that the localizing supercharge (B.8), upon reduction to 2d, descends to an Ω-
deformed supercharge both in the A and in the B-model
Q
3d ! 2d
−−−−−! Q+ + Q− + Q+ + Q− = QA + iGA+ − iGA− = QB + iGB+ − iGB−.
These Ω-deformations are defined with respect to the vector field i∂w − i∂w = ∂A, which rotates
the S1
A circle. So, the 3d localizing supercharge can be thought of as a lift of the S1
A-rotating
Ω-deformed supercharge from either the A or the B-model on Σ.
If we further compactify the S1
A direction and reduce to a 1d theory on I, we get an N = 4
supersymmetric gauged quantum mechanics. The localizing supercharge Q descends to a nilpo-
tent (up to gauge transformation) supercharge. Nilpotency in 1d follows from the fact that there
is no Dy on the right-hand side of (B.9), which is the only remaining translation in 1d.
C sl(1|1) rational R-matrix from the A-model localization
The aim of this appendix is to compute the geometric rational R-matrix for sl(1|1) superspin
chain from its natural theory from the perspective of the Bethe/Gauge correspondence, i.e.,
a 2d gauge theory corresponding to the sl(1|1) spin chains with fundamental representations,
and using the A-model localization formula of Closset, Cremonesi and Park [28]. The compu-
tation here is inspired by insights and observations in [23, 97, 98], and its sl(2) counterpart has
been performed in [23]. Unlike the construction in the main body of this work, the computation
in this appendix should be considered an interesting observation rather than being based on
a systematic framework. Furthermore, in this section, we assume that the explicit form of coho-
mological stable envelopes is known. Of course, by knowing the cohomological stable envelopes,
the computation of R-matrix is a simple matrix manipulation. However, viewing the R-matrix
elements as the two-point functions of certain (normalized) A-model observables would provide
an alternative route and may lead to a better understanding of the Bethe/Gauge correspondence.
C.1 Construction of geometric rational R-matrix
In Section 7, we have successfully constructed cohomological stable envelopes and the geomet-
ric R-matrix associated with sl(1|1) by starting from their 3d counterparts and performing
a 3d! 2d! 1d reduction, corresponding to reducing the three-dimensional gauge theory, living
on I×Eτ , on the two cycles of the elliptic curve Eτ . This is expected since the interval partition
36After compactifying S1
B the component of a gauge field in this direction becomes an adjoint scalar field in
the 2d theory and the derivative DB becomes a gauge transformation generated by this field.
76 N. Ishtiaque, S.F. Moosavian and Y. Zhou
function of the 3d theory reduces to the interval partition function of the corresponding 1d
quantum mechanical system after discarding all instanton corrections coming from particles
wrapping S1
A and S1
B. Before presenting the setup and the computation of the R-matrix, let us
make a series of remarks.
Remark C.1 (2d Gauge theory). The first remark is the theory we use for localization com-
putation. This is the gauge theory that Bethe/Gauge correspondence naturally associates to
an sl(1|1) spin chain in a fundamental representation. This gauge theory is constructed in
Example 4.1. For the N -magnon sector of a spin chain with L sites, the 2d gauge theory is
a U(N) gauge theory with N = (2, 2) supersymmetry coupled to a pair of SU(L) × U(1)ℏ
fundamental/anti-fundamental chiral multiplets.
Remark C.2 (relation to the dimensional reduction of 3d Gauge theory of Section 4.2). A nat-
ural question is the relation between the gauge theory we use for the localization computation
of this section and the 3d!2d dimensional reduction of the 3d theory of Section 4.2 used for
the computation of elliptic stable envelopes. The dimensional reduction of the 3d theory of
Section 4.2 is again a 2d gauge theory with an N = (2, 2) supersymmetry. However, this
is a different 2d theory, although with the same gauge and matter content. To see the rela-
tion, it is expected [40] that we should start from a 4d theory37 with N = 1 supersymmetry
on I × S1
A × S1
B × S1
C,
38 where S1
C is an extra circle. The dimensional reduction of the 4d theory
on S1
C would be the 3d theory of Section 4.2.39 On the other hand, the dimensional reduction
of the 4d theory on S1
A × S1
B gives a 2d theory on I × S1
C, which is the gauge theory that the 2d
gauge theory we use for the localization of this section. This is precisely the gauge theory that
the Bethe/Gauge correspondence assigns to an sl(1|1) spin chain with fundamental representa-
tions. Furthermore, the 2d gauge theory on I × S1
C can be thought of as a theory living on the
sphere from which two cigars have been capped off as shown in Figure 7.
S1
C
=⇒
I
S1
C
=⇒
I
S1
C
Figure 7. The cylinder I × S1
C can be thought as starting from the sphere (left), stretching (middle),
and capping off two cigars (right).
The hierarchy of theories we have considered in this work is shown in Figure 8.
Remark C.3 (identification of parameters). As we have seen in Section 6.1, the elliptic stable
envelopes depend on the holonomies for the gauge s, flavor (x, ℏ) and the topological symme-
try z along the cycle of the elliptic curve. Furthermore, we constructed cohomological stable
envelopes by reducing the elliptic ones to 1d upon which the holonomy z disappears and all
37The reason we need to start in 4d is that such a theory is a natural arena for the quantum version of
equivariant elliptic cohomology. The BRST cohomology of a 4d analog of the A-model supercharge is expected
to be equivalent to quantum equivariant elliptic cohomology of the Higgs branch of the 4d theory.
38Alternatively, we can think of the 4d theory on S2 × S1
A × S1
B. We can then think of S2 as I × S1
C to which
two cigars are capped. Reduction of the theory on S1
A × S1
B gives rise to a theory on S2.
39The reason that this 3d theory is enough for the purpose of the computation of the R-matrix is that the latter
is insensitive to the deformed non-perturbative corrections coming from the extra circle S1
C or equivalently, the
deformed ring structure of the quantum equivariant elliptic cohomology.
Elliptic Stable Envelopes and Dynamical R-Matrices 77
T4d on:
I
× × ×
S1
A S1
B S1
C
Re
duc
tio
n on
S
1
A
T ′
3d on:
I
× ×
S1
B S1
C
R
ed
u
ct
io
n
o
n
S
1 B
T ′′
3d on:
I
× ×
S1
A S1
C
Reduction onS 1
C
T ′′′
3d on:
I
× ×
S1
A S1
B
R
ed
u
ct
io
n
on
S
1 B
T ′
2d on: ×
S1
C
R
ed
u
ct
io
n
on
S
1 A
T ′′
2d on:
I
×
S1
C
R
ed
u
ction
on
S
1B
T ′′′
2d on:
I
×
S1
A
R
ed
u
ction
on
S
1A
T ′′′
1d on:
I
Figure 8. Since the order of dimensional reduction should not matter, one can identify T ′
2d and T ′′
2d
and it is the theory we use in this appendix. T ′′′
3d is the theory we used for the computation of elliptic
stable envelopes in Section 6.1.
other holonomies become complex twisted masses of the 2d theory on I × S1
C. We thus need
to compare the quantities in the theory T ′′′
1d (the theory used in deriving (7.5)) and T ′
2d (the
theory we use in this appendix) to be able to write the formulas in terms of parameters of the
latter (see Figure 8). First note that in the theory T ′′′
1d ,
s :=
∮
A
Ah
4d + i
∮
B
Ah
4d,
78 N. Ishtiaque, S.F. Moosavian and Y. Zhou
where Ah
4d is the four-dimensional gauge field for the Cartan of the gauge Lie algebra h = R⊕N .
Hence, they will be identified with σC := diag(σC1 , . . . , σ
C
N ), the complex scalar in the u(N)
vector multiplet, in the theory T ′
2d. Furthermore, in the theory T ′′′
1d ,
x :=
∮
A
Aa
4d + i
∮
B
Aa
4d, ℏ :=
∮
A
Aℏ
4d + i
∮
B
Aℏ
4d,
where Aa
4d is the four-dimensional gauge field for the Cartan of the flavor symmetry a =
u(1)⊕L−1,40 and Aℏ
4d is the four-dimensional gauge field for the Rℏ flavor symmetry. There-
fore, they are identified with mC :=
(
mC
1 , . . . ,m
C
L
)
and ℏC, the complex twisted masses, in the
theory T ′
2d. Hence, we identify the parameters as follows:
sa ! σCa , a = 1, . . . , N, xi ! mC
i , i = 1, . . . , L, ℏ ! ℏC. (C.1)
Let us emphasize that say mC
i has nothing to do with the (expectation value of the) quantity
defined in (4.9), which is the complex scalar in the vector multiplet of the theory T ′′′
2d (see the
right portion of Figure 8), while
(
σCa ,m
C
i , ℏC
)
in (C.1) are parameters of the theory T ′
2d (or
equivalently T ′′
2d). On the other hand, there are real masses in the theory T ′′′
1d whose order
determines a chamber C in the theory T ′′′
1d ,
mi =
∮
C
Aa
4d, i = 1, . . . , L. (C.2)
From the perspective of T ′
2d, these are holonomies of the 2d gauge field41 along S1
C. This
completes the identification of parameters. Furthermore, for convenience in the computation,
we consider a shift in the parameters42
sa ! sa, a = 1, . . . , N, xi ! xi +
ℏ
2
, i = 1, . . . , L. (C.3)
Remark C.4 (using the A-model localization). Note that the stable envelope we have con-
structed is written in the stable basis, which is a basis for the equivariant quantum cohomology
of the Higgs branch of the gauge theory. The quantum equivariant cohomology consists of the
vector space of the underlying classical equivariant cohomology together with a deformed ring
structure [140, 148]. On the other hand, for an A-twisted theory, the equivariant quantum co-
homology of the target space is the cohomology of the ordinary A-model supercharge, which
by definition is the twisted chiral ring of the gauge theory [65, 140, 148]. Therefore, the full
Bethe–Gauge correspondence would have to invoke the cohomology of A-model supercharge, as
has been argued in [40, 102, 103]. Since the stable envelopes provide a map between quantum
equivariant cohomology43 of the fixed-point loci44 and the target, one would like to construct
a basis for the twisted chiral ring which coincides with the stable basis for the equivariant
quantum cohomology. Therefore, cohomological stable envelopes (7.5) are the corresponding
A-model observables. As it is known, these generators are functions of the complex scalar in the
vector multiplet of the 2d theory [28].
40Note that x1 + · · ·+ xL = 0.
41Note that the gauge field in the theory T ′
2d comes from the components of the gauge field in 4d along I ×S1
C.
42These shifts are not essential and as we mentioned are done just for the convenience in the computations and
coping with existing literature.
43Stable envelope map preserves the ring structure of the quantum equivariant cohomology in some weak
sense [88, Theorem 7.2.1].
44This is the fixed-point loci of the flavor-symmetry action on the corresponding Higgs branch.
Elliptic Stable Envelopes and Dynamical R-Matrices 79
Remark C.5 (identifying the A-model observables). The A-model observables O
(
σC
)
are func-
tions of complex scalars σC in the vector multiplet associated with the gauge invariance [28].
In the computation of the R-matrix, we use particular A-model observables obtained from the
stable envelopes using the identification (C.1). Considering the identification (C.1) and the
shifts (C.3), and using (7.7) and (7.8), the cohomological stable envelopes take the following
form:
StabC(p) = SymSN
N∏
a=1
( ∏
i<p(a)
(
+σCa −mC
i +
ℏC
2
) ∏
i>p(a)
(
−σCa +mC
i +
ℏC
2
))
×
(∏
a>b
1
σCa − σCb
)]
. (C.4)
This is the form of the cohomological stable envelopes for sl(1|1) that we use in the following.
Note that these observables can be inserted anywhere on S2 and the final result is independent
of the location of insertions [28, Section 3.3].
Remark C.6 (Coulomb vs. Higgs branch localization schemes). For GLSMs, there are two
different localization schemes depending on the choice of the localizing action: the Higgs-branch
localization and the Coulomb-branch localization [16, 28]. These localization schemes are ex-
pected to give rise to the same final results but the explicit details and the localization procedure
could be very different. The A-localization computation of this section is an example of the
Coulomb-branch localization.
Remark C.7 (R-matrix from the A-model localization). From the mathematical definition, it
is expected that the R-matrix is related to the two-point function of stable envelopes [23]. As we
briefly recall in Section C.2, the A-model localization computations involve a summation over
all instanton sectors. However, R-matrix is insensitive to these non-perturbative corrections. As
such, the computation of R-matrix reduces to the computation of a two-point function of stable
envelopes in the zero-flux sector. We will further elaborate on the construction below.
Remark C.8 (interpretation in terms of N = 4 quantum mechanics). Since the computation of
R-matrix is the restriction of certain two-point functions to zero-flux section, we can interpret
the result in the corresponding quantum mechanics. This theory is obtained by the dimensional
reduction of our 2d theory on I×S1
C. The insertion of A-model observables on S2 leads to certain
boundary conditions in quantum mechanics, as illustrated in Figure 9. Therefore, the two-point
function of A-model observables in the zero-flux sector is the interval partition function of the
corresponding N = 4 quantum mechanics in the presence of these boundary conditions.
Having summarized all the necessary ingredients, we can now compute the R-matrix. In this
construction, we use the cohomological stable envelopes (C.4).
Rational R-matrix from Gauge theory. The geometric definition of R-matrix (3.12) (and
its trigonometric and rational versions) depends on two pieces of data: (1) the cohomological
stable envelope and its inverse, and (2) a choice of the chamber for the cohomological stable
envelope and another (in general different) chamber for its inverse. Therefore, it is natural to
consider the configuration of Figure 10 of the 2d gauge theories on S2.
J (mC, 0) and J (0,mC′) are certain supersymmetric Janus interfaces.45 Colliding these in-
terfaces gives rise to a single interface: J (mC, 0)J (0,m′
C′) = J (mC,m
′
C′). Setting m′
C′ = mC
and noting that J (mC,mC) is equivalent to no interface, we see that J −1(mC, 0) = J (0,mC).
Therefore, we consider the configuration of Figure 11 for the computation of R-matrix.
45Through (C.2) and going to 1d by compactification on S1
C, we can think of these interfaces as real-mass Janus
interfaces in the corresponding 4d N = 4 quantum mechanics. The presence of these types of interfaces leads to
an exact deformation of the theory [40, Section 4.2]. Therefore, the precise shape of the profile of the parameters
does not matter and only their asymptotic values affect the computations. This is the only fact we need.
80 N. Ishtiaque, S.F. Moosavian and Y. Zhou
OL(σ
C) OR(σ
C)
(a) The sphere on which the 2d A-twisted
gauge theory is defined. Two observables
OL(σ
C) and OR(σ
C) are inserted on sphere.
OL(σ
C) OR(σ
C)
(b) The deformation of the sphere to a geometry involving
a long cylinder and two capped cigar-like geometries.
|OL(σ
C)⟩ |OR(σ
C)⟩
(c) Doing a path integral in the presence of
OL(σ
C) and OR(σ
C) defines the correspond-
ing state on the boundary circles. We can
then cap off the cigar-like ends.
BL BR
(d) Compactification of the 2d theory in the cylinderical re-
gion would lead to a 1d quantum mechanics and the boundary
states become boundary conditions BL and BR in the quantum
mechanics.
Figure 9. The 2d ! 1d reduction of the 2d N = (2, 2) theory to a quantum mechanical system with
boundary conditions prescribed by the observables OL(σ) and OR(σ). The subscripts just mean left and
right. Restricting to the zero-flux sector, the two-point function on the sphere becomes the partition
function of the quantum mechanics with certain boundary conditions.
J (mC, 0) J (0,mC′)
O(σC) O′(σC)
Figure 10. The configuration of two Janus interfaces for computing the rational R-matrix. O
(
σC) and
O′(σC) are certain A-model observables.
The effect of the presence of the Janus interface J (mC,m−C) is that the parameters belong
to the two opposite chambers C and−C on the two sides of the interface.
The next ingredient is the formula for the computation of the two-point functions of GLSM
observables on sphere S2 in the zero-flux sector. In the case of theory T ′
2d, it is given by [28]
(see Appendix C.2.2 for the derivation)
⟨OO′⟩S2,0 =
1
N !
∮
JK
dσCZ[O,O′]
(
σC), (C.5)
where the subscript 0 denotes the zero-flux sector, σC =
(
σC1 , . . . ,σ
C
N
)
,
dσC = dσC1 ∧ · · · ∧ dσCN , Z[O,O′]
(
σC) = Z
(
σC)O(σC)O′(σC),
and
Z
(
σC) ≡ N∏
a=1
L∏
i=1
1(
σCa −mC
i + 1
2ℏC
)(
−σCa +mC
i + 1
2ℏC
) · N∏
a̸=b
(
σCa − σCb
)
. (C.6)
Elliptic Stable Envelopes and Dynamical R-Matrices 81
J (mC,m−C)
O(σC) O′(σC)
Figure 11. The configuration for the computation of R-matrix obtained by colliding the two interfaces
in Figure 10.
The contribution to the integral is determined by the JK-pole prescription and comes from the
contribution of chiral multiplets. The location of the poles is given by (C.26).
The A-model observables relevant for the computation of R-matrix are constructed from (C.4)
as follows. First of all these observables depend on the pair (N,L) and also the injective
map p. Furthermore, we can consider a permutation π : {1, . . . , L}! {1, . . . , L} which leads to
a permutation of complex masses and the real parameters (C.2), and hence determines a choice
of chamber. We denote these observables as Op,π,N,L. It turns out that the relevant two-point
functions are [23]46〈
Op,π;N,L
(
σC
)
O⋆
p′,π′;N,L
(
σC
)〉
S2,0
, (C.7)
where
Op,π;N,L ∝ StabC(p), (C.8)
where the chamber C is determined through the permutation π, and ⋆ operation corresponds
to replacing σCa ! −σCa and mC
i ! −mC
i . This operation corresponds to the longest permu-
tation π : {1, . . . , L}! {L, . . . , 1} in (7.9). We comment below (see Remark C.9) on how to
determine the proportionality constant in (C.8).
We are now in a position to compute the rational R-matrix for sl(1|1). For this purpose, it
suffices to restrict to L = 2, which through (5.12), implies N = 0, 1, 2, corresponding to the
vacuum, one-magnon, and two-magnon sectors of the spin chain. The R-matrix of the vacuum
sector is trivially one. We can compute the R-matrix in the one- and two-magnon sectors as
follows.
Remark C.9 (normalization of observables). Let us make a curious observation before pro-
ceeding to details. In the following computations, we demand that the observables Op,π;N,L are
orthonormal, meaning ⟨Op,π;N,LO⋆
p,π;N,L⟩ = 1 and ⟨Op,π;N,LO⋆
p′,π;N,L⟩ = 0. If we do not stress on
orthonormality of these observables, then we end up with a 4×4 matrix that all the components
of the middle 2 × 2 diagonal block47 are multiplied by
(
ℏC
)−1
and the bottom 1 × 1 diagonal
block48 is multiplied by
(
ℏC
)−2
. Such a matrix does not satisfy the Yang–Baxter equation and as
such is not an R-matrix at all. In the case of sl(2) spin chains considered in [23, Section 6.1], the
analogous observables are already normalized and there is no need for a rescaling. The reason
is a certain contribution of the adjoint chiral multiplet that takes care of the normalization. In
the present case, there is no adjoint chiral multiplet and as such we need this rescaling.
46In [23, Section 6.2], this type of two-point functions has been interpreted as an inner product in the corre-
sponding N = 4 quantum mechanics, where the ⋆ operation provides a notion of “complex conjugation”.
47For an R-matrix, this 2× 2 block is the one-magnon sector R-matrix.
48For an R-matrix, this 1× 1 block is the two-magnon sector R-matrix.
82 N. Ishtiaque, S.F. Moosavian and Y. Zhou
R-matrix in the one-magnon sector. The one-magnon sector is described by set-
ting N = 1, hence there are two possible choices of the injective map, which we call p and
p′: p(1) = 1 or p′(1) = 2. There are also two possible choices of permutation, which we
denote by π and π′: π({1, 2}) = {1, 2} and π′({1, 2}) = {2, 1}. There are two complex
twisted masses: mC =
(
mC
1 ,m
C
2
)
with mC
1 + mC
2 = 0, and there is only one effective com-
plex twisted mass mC ≡ mC
1 = −mC
2 . Similarly, there are two real parameters, defined in (C.2),
with m1 +m2 = 0, and a single effective parameter m ≡ m1 = −m2. Correspondingly, there
are two chambers C = {m > 0} and −C = {m < 0}. These chambers correspond to the
permutations π and π′. We then have
Op,π;1,2
(
σC
)
=
(
ℏC
) 1
2
(
−σC +mC
2 +
ℏC
2
)
,
Op,π′;1,2
(
σC
)
=
(
ℏC
) 1
2
(
+σC −mC
2 +
ℏC
2
)
= O⋆
p,π;1,2
(
σC
)
,
Op′,π;1,2
(
σC
)
=
(
ℏC
) 1
2
(
+σC −mC
1 +
ℏC
2
)
,
Op′,π′;1,2
(
σC
)
=
(
ℏC
) 1
2
(
−σC +mC
1 +
ℏC
2
)
= O⋆
p′,π;1,2
(
σC
)
.
The prefactor
(
ℏC
) 1
2 is chosen to guarantee the orthonormality of the observables Op,π;N,L, which
can be seen using (C.5) as follows〈
Op,π;1,2
(
σC
)
O⋆
p,π;1,2
(
σC
)〉
S2,0
= 1,
〈
Op,π;1,2
(
σC
)
O⋆
p′,π;1,2
(
σC
)〉
S2,0
= 0,〈
Op′,π;1,2
(
σC
)
O⋆
p′,π;1,2
(
σC
)〉
S2,0
= 1,
〈
Op′,π;1,2
(
σC
)
O⋆
p,π;1,2
(
σC
)〉
S2,0
= 0.
All these observables only depend on a single permutation π, which means that real masses are
chosen in a single chamber, say C. Physically this means that there is no Janus interface, and
these are just orthonormality conditions on the observables. Furthermore,〈
Op,π′;1,2
(
σC
)
O⋆
p,π;1,2
(
σC
)〉
S2,0
= − mC
2 −mC
1
mC
2 −mC
1 + ℏC
,
〈
Op,π′;1,2
(
σC
)
O⋆
p′,π;1,2
(
σC
)〉
S2,0
=
ℏC
mC
2 −mC
1 + ℏC
,
〈
Op′,π′;1,2
(
σC
)
O⋆
p,π;1,2
(
σC
)〉
S2,0
=
ℏC
mC
2 −mC
1 + ℏC
,
〈
Op′,π′;1,2
(
σC
)
O⋆
p′,π;1,2
(
σC
)〉
S2,0
= − mC
2 −mC
1
mC
2 −mC
1 + ℏC
.
These correlation functions involve two observables and in the presence of a Janus interface
J (mC,m−C), the observables in the two sides of the interval depend on two different permuta-
tions π and π′. Hence, we find the one-magnon sector R-matrix constructed from the quantum
mechanics
Rgt,(1) ≡
− mC
2−mC
1
mC
2−mC
1+ℏC
ℏC
mC
2−mC
1+ℏC
ℏC
mC
2−mC
1+ℏC − mC
2−mC
1
mC
2−mC
1+ℏC
. (C.9)
Remark C.10. By comparing this with [23, equation (6.7)], we see that the one-magnon sector
R-matrix is the same for sl(2) and sl(1|1). This is expected since the statistics of the excitation
do not play any role in the one-magnon. We have seen this above but in a different disguise:
the stable envelopes have been the same as we have explained in Remark 6.2.
Elliptic Stable Envelopes and Dynamical R-Matrices 83
Finally,
〈
Op,π;1,2
(
σC
)
O⋆
p,π′;1,2
(
σC
)〉
S2,0
= − mC
2 −mC
1
mC
2 −mC
1 − ℏC
,
〈
Op,π;1,2
(
σC
)
O⋆
p′,π′;1,2
(
σC
)〉
S2,0
= − ℏC
mC
2 −mC
1 − ℏC
,
〈
Op′,π;1,2
(
σC
)
O⋆
p,π′;1,2
(
σC
)〉
S2,0
= − ℏC
mC
2 −mC
1 − ℏC
,
〈
Op′,π;1,2
(
σC
)
O⋆
p′,π′;1,2
(
σC
)〉
S2,0
= − mC
2 −mC
1
mC
2 −mC
1 − ℏC
,
that regarding (C.9) is just the inverse
(
Rgt,(1)
)−1
of Rgt,(1), as it is expected
(
Rgt,(1)
)−1(
mC
1 ,m
C
2
)
=
− mC
2−mC
1
mC
2−mC
1−ℏC − ℏC
mC
2−mC
1−ℏC
− ℏC
mC
2−mC
1−ℏC − mC
2−mC
1
mC
2−mC
1−ℏC
.
Let us now compute the correlation functions (C.7) for the gauge theory corresponding to the
two-magnon sector.
R-matrix in the two-magnon sector. The two-magnon sector is described by set-
ting N = 2. There is only one possible choice of the injective map p({1, 2}) = {1, 2}. As
the Abelian case, there are two possible choices of permutation, which we denote by π and π′:
π({1, 2}) = {1, 2} and π′({1, 2}) = {2, 1}. The complex masses and the real parameters are as
in the Abelian case. We have
(
σC =
(
σC1 , σ
C
2
)
)
Op,π;2,2
(
σC
)
= ℏC
((
−σC1 +mC
2 + ℏC
2
)(
σC2 −mC
1 + ℏC
2
)
σC2 − σC1
)
+
(
σC1 ! σC2
)
,
and Op,π′;2,2
(
σC
)
= O⋆
p,π;2,2
(
σC
)
. Again, the prefactor ℏC is determined by demanding orthogo-
nality. Using (C.5), we see that〈
Op,π;2,2
(
σC
)
O⋆
p,π;2,2
(
σC
)〉
S2,0
= 1,
which means that Op,π;2,2
(
σC
)
is normalized. Furthermore,
〈
Op,π′;2,2
(
σC
)
O⋆
p,π;2,2
(
σC
)〉
S2,0
= −m
C
2 −mC
1 − ℏC
mC
2 −mC
1 + ℏC
.
This gives the geometric R-matrix in the two-magnon sector
Rgt,(2) = −m
C
2 −mC
1 − ℏC
mC
2 −mC
1 + ℏC
. (C.10)
Finally,
〈
Op,π;2,2
(
σC
)
O⋆
p,π′;2,2
(
σC
)〉
S2,0
= −m
C
2 −mC
1 + ℏC
mC
2 −mC
1 − ℏC
,
which is just the inverse of (C.10).
84 N. Ishtiaque, S.F. Moosavian and Y. Zhou
Putting (C.9) and (C.10) together, the R-matrix of sl(1|1) superspin chain constructed
through the Bethe–Gauge correspondence is
Rgt =
1 0 0 0
0 − mC
2−mC
1
mC
2−mC
1+ℏC
ℏC
mC
2−mC
1+ℏC 0
0 ℏC
mC
2−mC
1+ℏC − mC
2−mC
1
mC
2−mC
1+ℏC 0
0 0 0 −mC
2−mC
1−ℏC
mC
2−mC
1+ℏC
.
If we identify u! mC
1 −mC
2 and ℏ! ℏC, we see from (7.10) that Rgt(u) = RRR(u). Therefore,
our A-model localization successfully reproduces the geometric R-matrix of sl(1|1) superspin
chain, as it is expected from the Bethe/Gauge correspondence.
C.2 Further details on two-point functions of 2d GLSM observables
Here, we will collect concepts and formulas for the computation of two-point function of Ω-
deformed A-model observables. We follow [28] to which we refer for all the details.
C.2.1 Brief recap of Closset–Cremonesi–Park localization formula
Consider a 2d N (2, 2) GLSM with rank(g) = r gauge Lie algebra g, whose Cartan subalgebra we
denote by h, in the Ω-deformed sphere background S2
Ω.
49 We denote the corresponding groups
by G and H and the complexified Lie algebras by gC and hC, respectively. Different topological
sectors are labeled by magnetic flux n = (n1, . . . , nr) ∈ Γg∨ ≃ Zr ⊂ ih, where
Γg∨ ≡ {n | ρ(n), ∀ρ ∈ Γg},
and ρ(n) is given by the canonical pairing of dual vector spaces and electric charges take value
in Γg, the weight lattice. Supersymmetric observables belong to the cohomology of left- and
right-moving supercharges Q and Q̃. In the presence of Ω-deformation, we have δQσ = δ
Q̃
σ = 0
when the vector field V , that generates the rotation isometry, vanishes. Hence such operators are
functions of σ, the complex scalar field in the vector multiplet of the theory. Furthermore, they
can only be inserted at the north and south poles of S2
Ω. When we turn off the Ω background,
they can be inserted anywhere on the sphere. The two-point function of such operators, which
we denote by ON (σN ) and OS(σS) for N and S denoting the north and south poles, are given
by [28]〈
ON (σN )OR(σR)
〉
S2
Ω
=
(−1)#
|W |
∑
n∈Γg∨
qn
∮
JK
Z1-loop
n (σ, ϵ)ON
(
σ − ϵn
2
)
OS
(
σ +
ϵn
2
)
, (C.11)
where |W | denotes the order of the Weyl groupW of g, # denotes the number of chiral multiplets
with R-charge 2,50 σ is the coordinate on M̃ ≡ hC ≃ Cr, which is the cover of M ≡ M̃ /W , the
49We do not need the details of this background, which can be found in [28, Section 2.1].
50The prefactor (−1)# is a sign ambiguity which is irrelevant for the rest of our discussion since we do not have
a chiral multiplet of R-charge +2. The prescription to fix this sign ambiguity, which we have stated here, is given
in [28, Section 4.5].
Elliptic Stable Envelopes and Dynamical R-Matrices 85
Coulomb-branch moduli space, and ϵ is the Ω-deformation parameter. Z1-loop
n (σ) is the 1-loop
contribution to the partition function from the nth topological sector, which is give by
Z1-loop
n (σ) = Zvector
n (σ; ϵ)
∏
i
Zchirali
n (σ; ϵ),
where chirali denotes the ith chiral multiplet. These contributions could be written explicitly
using
Zp(x, ϵ) ≡
p
2
−1∏
q=− p
2
+1
(x+ ϵq), p > 1,
1, p = 1,
|p|
2∏
q=− |p|
2
(x+ ϵq)−1, p < 1.
(C.12)
For our purpose, we only need to know Z0(x, ϵ = 0) and Z2(x, ϵ = 0) for n = 0. From (C.12),
we have
Z2(x, ϵ = 0) = x, Z0(x, ϵ = 0) =
1
x
. (C.13)
We then have
Zvector
n (σ, ϵ) =
∏
α∈∆
Z2−α(n)(α(σ), ϵ),
Zchirali
n (σ, ϵ) =
∏
ρi∈Ri
ZqRi −ρi(n)(ρi(σ) +mi, ϵ).
(C.14)
In this formula, ∆ denotes the set of roots, ρi labels the weights of representation Ri of G in
which the ith chiral multiplet transforms, and mi and qRi denote the twisted mass and U(1)V
R-symmetry charge of the ith chiral multiplet. Defining the notation
(
WI ,mI , q
R
I
)
=
{(
ρi,mi, q
R
i
)
, I = (i, ρi),
(α, 0, 2), I = α.
(C.15)
We can write the 1-loop partition function as follows:51
Z1-loop
n (σ, ϵ) =
∏
I
ZWI(n)−qRI
(WI(σ) +mI , ϵ). (C.16)
The parameter q in (C.11) is related to the complexified FI parameters, defined as follows:
ηI ≡ θI
2π
+ iζI ∈ C, 1 ≤ I ≤ dim(c∗C),
where θI and ζI are the theta angle and real FI parameters. They belong to the dual of the
complexified center c∗C ⊂ g∗C and could be thought of as elements of h∗C through the embed-
ding c∗C ↪! h∗C. We denote the image of the physical FI parameters ζI under this embedding
by ζa, a = 1, . . . , r. Then, qn ≡ exp
(
2πi
∑
I η · n
)
, where nI denotes the flux in the free Ith U(1)
51Considering Figure 9, we only need the zero-flux sector of the GLSM observables in the computation of
interval partition function of quantum mechanics, as we will see momentarily.
86 N. Ishtiaque, S.F. Moosavian and Y. Zhou
part of the center of G. Note that n ∈ ih and hence η · n is given by the canonical pairing
between hC and h∗C.
Finally, we need to specify the contour of integration, which is given by the JK contour [21, 68,
137]. The poles of Z1-loop
n (σ, ϵ) are the loci of intersections of s ≥ r hyperplanes
{
Hk1
I1 , . . . ,H
ks
Is
}
,
where
Hk
I ≡
{
σ
∣∣∣∣WI(σ) = −mI −
(
k +
qRI − WI(n)
2
)
ϵ
}
, k ∈
[
0,WI(n)− qRI
]
int
, (C.17)
and [a, b]int denotes the set of integers between a and b. There are two types of singularities:
the non-degenerate singularities corresponding to the intersection of exactly r hyperplanes and
the degenerate ones corresponding to the intersection of s > r hyperplanes. This complex-
codimension-r singular loci is denoted as M̃ sing
n ⊂ M̃ . Note that Hk
I is a singular locus of Z1-loop
n
only when I labels a chiral multiplet field. Hk
α labels the codimension-1 poles with zero residue.
The definition of JK residues at σ∗ ∈ M̃ sing
n depends on an additional parameter η ∈ ih∗, which
is defined as follows. The charge WI , defined in (C.15), belongs to ih∗ and hence any set of
charges {WI1 , . . . ,WIr} generates a cone
Con[WI1 , . . . ,WIr ] ⊂ ih∗. (C.18)
The set of all such cones can be divided into dimension-r chambers that are separated by
codimension-1 walls. The chamber to FI parameters ξa ∈ ih∗, a = 1, . . . , r belong to deter-
mines the phase of the GLSM [91, 147]. The effective FI parameter ζUV,I
eff at the infinity (i.e.,
when |σ|!∞) of Coulomb branch M is defined to be
ζUV,a
eff ≡ ζa +
ba
2π
lim
R!∞
lnR, ba ≡
∑
i
TrRi(t
a) =
∑
I
WI .
ta denotes the generator of the ath u(1) ⊂ g,52 and the last equality follows from embed-
ding c∗ ⊂ ih∗. Hence, phases of the theory on the Coulomb branch depend on the cham-
ber C ⊂ ih∗ that the parameter ζUV
eff belong to. This can be stated as follows: ζUV
eff belongs
to a chamber C ⊂ ih∗ if ζ + b
2π lnR ∈ C, ∀R ≥ R∗, for some R∗, which always exists for prac-
tical purposes [28]. When b ̸= 0, there could be two cases: 1) either b belongs to a definite
chamber C; In this case, ζ is irrelevant and the theory has only one phase determined by the
chamber C, or 2) b belongs to a wall where chambers C1, . . . , Cp meet; In this case, one can turn
on ζ and depending on its value, ζUV
eff could belong to any of the chambers Ci, i = 1, . . . , p, which
in turn determines the phase of the theory. When b = 0, and hence the IR theory is a CFT,
then ζUV
eff = ζ could be taken to be in any chamber C ⊂ ih∗. We then set
ξ ≡ ζFIeff . (C.19)
The JK residues can now be defined as follows. If a singular point σ∗ comes from the intersection
of s ≥ r hyperplanes
{
Hk1
I1 , . . . ,H
ks
Is
}
, we denote the set ofH-charges as W (σ∗) ≡ {WI1 , . . . ,WIs}.
The JK residues at σ∗ = {σ∗,1, . . . , σ∗,r} is then give by
JK-Res[W (σ), ξ]
∏
Wj∈W r
dσ1 ∧ · · · ∧ dσr
W1(σ) · · ·Wr(σ)
∣∣∣∣∣∣
σ=σ∗
=
1
|det(Wi)|
, ξ ∈ Con[W1, . . . ,Wr],
0, ξ /∈ Con[W1, . . . ,Wr],
(C.20)
where W r is any set among
(
s
r
)
sets of r h-charges in W (σ∗). This concludes our discussion of
the necessary results from [28].
52ta is the image of the Ith u(1) generator of the center of g∗ under the embedding c∗ ↪! h∗.
Elliptic Stable Envelopes and Dynamical R-Matrices 87
C.2.2 Explicit computation of two-point functions
We would like to compute the contribution of vector and chiral multiplets to the correlation
functions (C.7), given in (C.11) or more compactly in (C.16), explicit. For the fields in the
vector multiplet, I = (i, ρi) ≡ (ab, ρab), as (C.15) ρab = αab = ea − eb, a ̸= b, where αab denotes
the abth root of u(N), associated to the generator Eab, and {e1, . . . , eN} is the standard basis
for RN .53 Therefore,
WI
(
σC
)
= ρab
(
σC
)
= σCa − σCb , a ̸= b,
WI(n) = ρab(n) = na − nb, a ̸= b. (C.21)
Similarly, for chiral fields in the fundamental (+) and antifundamental (−) representations, the
weights are (I = (i, ρi) ≡ (a, ρa,±) as in (C.15))
ρa,± = ±ea, a = 1, . . . , N, (C.22)
and hence for the chiral multiplets54
(
Q i
a , Q̃ a
i
)
, we have
Q i
a : WI
(
σC
)
+mI = ρa,+
(
σC
)
+mQ i
a
= σCa −mC
i + ℏC,
Q̃ a
i : WI
(
σC
)
+mI = ρa,−
(
σC
)
+mQ̃ a
i
= −σCa +mC
i + ℏC.
Using (C.13) and (C.14), the contribution of vector multiplet is55
Zvector
2 (σ, ϵ = 0) =
N∏
a,b=1
a̸=b
(
σCa − σCb
)
. (C.23)
Similarly, setting qRQ = qR
Q̃
= 0, we obtain the contribution of chiral multiplets
ZQ
0 (σC; ϵ = 0) =
N∏
a=1
L∏
i=1
1(
σCa −mC
i + ℏC
2
) ,
ZQ̃
0 (σC; ϵ = 0) =
N∏
a=1
L∏
i=1
1(
−σCa +mC
i + ℏC
2
) . (C.24)
Putting together (C.23) and (C.24) and using (C.11), we thus see that the zero-flux sector
contribution to the sphere partition function can be written explicitly as56 (n = 0 in (C.11))
Z ≡ ⟨1⟩S2,0 =
1
N !
∮
JK
dσC1 ∧ · · · ∧ dσCNZ
(
σC1 , . . . , σ
C
N
)
,
where
Z
(
σC1 , . . . , σ
C
N
)
≡
N∏
a=1
L∏
i=1
1(
σCa −mC
i + 1
2ℏC
)(
−σCa +mC
i + 1
2ℏC
) · N∏
a̸=b
(σCa − σCb ). (C.25)
53For αab = ea − eb for a ̸= b, we consider the one-dimensional vector space generated by Eab. This space is an
eigenspace of the Cartan subalgebra with eigenvalue αab. These eigenspaces together with the Cartan subalgebra
generate u(N).
54We use the same notation for chiral multiplets of the theory T ′′
2d in Figure 8 as the ones for T ′′′
3d and its
dimensional reductions to 2d and 1d.
55Note that the contribution of vector multiplet is the same as an adjoint chiral with R-charge +2 [28]. Hence,
we have taken qRvector = 2.
56As we mentioned above, there is no overall (−1)# sign since there is no chiral multiplet of R-charge +2 in
our construction.
88 N. Ishtiaque, S.F. Moosavian and Y. Zhou
The factor 1/N ! comes from the order of the Weyl-group factor in (C.11). This proves (C.6).
Substituting (C.25) in (C.11) gives (C.5). The poles of JK prescription for the positive (effective)
FI parameter (ξ > 0, where ξ is defined in (C.19)) are given by
JK Poles : σCa = mC
i − ℏC
2
, a = 1, . . . , N, i = 1, . . . , L, (C.26)
coming from the intersection of hyperplanes Hk
I in (C.17) associated with fundamental chiral
multiplets Q i
a . Note that we need to take the contribution from all poles, i.e., all pairs (a, i).
In the following, we provide further details on this choice of JK poles in the computation of
R-matrix.
Jeffrey–Kirwan poles for the computation of R-matrix. Let us give the details of the
choice (C.26) of JK poles necessary for the computation of R-matrix in C.1. Although we have
stated the poles in (C.26), we give more details on this choice using the formalism of [28], as
reviewed in Section C.2. As we have explained there, the choice of JK poles depends on the
choice of the (effective) FI parameter ξ, defined in (C.19).
For the 1-magnon sector, we have a U(1) gauge theory with SU(2) fundamental Qi and
antifundamental Q̃i flavors. Using (C.22) and (C.17), the equations of hyperplanes (C.17) are
given by
HkQ1
IQ1
=
{
σC
∣∣∣∣σC = mC
1 − ℏC
2
−
(
kQ1 − n
2
)
ϵ
}
, kQ1 ∈ [0, n]int,
HkQ2
IQ2
=
{
σC
∣∣∣∣σC = mC
2 − ℏC
2
−
(
kQ2 − n
2
)
ϵ
}
, kQ2 ∈ [0, n]int,
HkQ̃1
IQ̃1
=
{
σC
∣∣∣∣−σC = −mC
1 − ℏC
2
−
(
kQ̃1 +
n
2
)
ϵ
}
, kQ̃1 ∈ [0, n]int,
HkQ̃2
IQ̃2
=
{
σC
∣∣∣∣−σC = −mC
2 − ℏC
2
−
(
kQ̃2 +
n
2
)
ϵ
}
, kQ̃2 ∈ [0, n]int,
with mC
1 +mC
2 = 0. The weight diagram is
(0, 0) ρQiρ
Q̃i
We only need to consider the zero-flux sector n = 0. From (C.18), we see that there are two
“cones”
Con(WIQ1 ,WIQ2 ) = Con(+1,+1), Con(WIQ̃1
,WIQ̃2
) = Con(−1,−1).
Using (C.20) and taking ξ > 0 and using (C.20), we see that only the poles coming from H
kQ1
Q1
and H
kQ2
Q2
, which is the same as (C.26).
For the 2-magnon sector, the gauge group is U(2) and the SU(2) × U(1)ℏ matter multiplet
consists of Q i
a and Q̃a
i with q
R
Q = qR
Q̃
= 0. (C.22), and (C.21), we can write the equations for
singular hyperplanes (C.17) associated to the fundamental chiral multiplets
HkQ11
IQ 1
1
=
{(
σC1 , σ
C
2
) ∣∣∣∣σC1 = mC
1 − ℏC
2
−
(
kQ11 −
n1
2
)
ϵ
}
, kQ11 ∈ [0, n1]int,
HkQ12
IQ 2
1
=
{(
σC1 , σ
C
2
) ∣∣∣∣σC1 = mC
2 − ℏC
2
−
(
kQ12 −
n1
2
)
ϵ
}
, kQ12 ∈ [0, n1]int,
HkQ21
IQ 1
2
=
{(
σC1 , σ
C
2
) ∣∣∣∣σC2 = mC
1 − ℏC
2
−
(
kQ21 −
n2
2
)
ϵ
}
, kQ21 ∈ [0, n2]int,
Elliptic Stable Envelopes and Dynamical R-Matrices 89
HkQ22
IQ 2
2
=
{(
σC1 , σ
C
2
) ∣∣∣∣σC2 = mC
2 − ℏC
2
−
(
kQ22 −
n2
2
)
ϵ
}
, kQ22 ∈ [0, n2]int.
Similarly, the hyperplanes associated to the antifundamental chiral multiplets are
H
kQ̃11
IQ̃ 1
1
=
{(
σC1 , σ
C
2
) ∣∣∣∣−σC1 = −mC
1 − ℏC
2
−
(
kQ̃11 +
n1
2
)
ϵ
}
, kQ̃11 ∈ [0, n1]int,
H
kQ̃12
IQ̃ 2
1
=
{(
σC1 , σ
C
2
) ∣∣∣∣−σC1 = −mC
2 − ℏC
2
−
(
kQ̃12 +
n1
2
)
ϵ
}
, kQ̃12 ∈ [0, n1]int,
H
kQ̃21
IQ̃ 1
2
=
{(
σC1 , σ
C
2
) ∣∣∣∣−σC2 = −mC
1 − ℏC
2
−
(
kQ̃21 +
n2
2
)
ϵ
}
, kQ̃21 ∈ [0, n2]int,
H
kQ̃22
IQ̃ 2
2
=
{(
σC1 , σ
C
2
) ∣∣∣∣−σC2 = −mC
2 − ℏC
2
−
(
kQ̃22 +
n2
2
)
ϵ
}
, kQ̃22 ∈ [0, n2]int.
We only need the zero-flux sector n1 = n2 = 0. To specify the JK poles, we take ξ to be in
the first quadrant, and using (C.18) and (C.20), we see that we need to take the contribution
from the hyperplanes forming the following cones associated to Q i
a Con(WIQ i
1
,WIQ i
2
), i = 1, 2,
which contain ξ. These poles are as given in (C.26).
Acknowledgements
We thank Kevin Costello, Yuan Miao, Hiraku Nakajima, Tadashi Okazaki, Surya Raghavendran,
Junya Yagi, Masahito Yamazaki, and Zijun Zhou for useful discussions. We thank Davide
Gaiotto, and Andrei Okounkov for reading the draft and sending us valuable feedback. Special
thanks go to Mykola Dedushenko for explaining some parts of his work with Nikita Nekrasov
and for providing extensive commentary on an earlier version of this work. We thank the Banff
International Research Station for generously hosting us during the last stage of this project. N.I.
is supported by the Huawei Young Talents Program Fellowship at IHES. The work of S.F.M. is
funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) funding
number SAPIN-2022-00028, and also in part by the Alfred P. Sloan Foundation, grant FG-
2020-13768. S.F.M would like to thank Davide Gaiotto for his support during the visit to the
Perimeter Institute. Kavli IPMU is supported by World Premier International Research Center
Initiative (WPI), MEXT, Japan.
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1 Introduction
1.1 Background and motivation
1.2 Summary of the results
1.3 Future directions
1.4 Organization of the paper
1.5 Glossary of notations
I Stable envelopes: beyond symplectic varieties
2 Classical Higgs branches of 3d N=2 gauge theories
2.1 Example: GIT quotients
2.2 Example: quiver varieties
3 Elliptic stable envelopes for partially-polarized varieties
3.1 Chambers and attracting sets
3.2 Attractive line bundle
3.3 Universal line bundle, Kähler torus and resonant locus
3.4 Elliptic stable envelope
3.5 Triangle lemma and duality
3.6 Abelianization
3.7 R-matrices and dynamical Yang–Baxter equations
3.8 R-matrices for quiver varieties
3.9 K-theory and cohomology limit
II The solution to dYBE for sl(1|1) from 3d N=2 SQCD
4 Stable envelopes and R-matrix from gauge theory: the setup
4.1 The Bethe/Gauge correspondence
4.2 3d N=2 SQCD and its parameters
4.3 Branes and the Bethe/Gauge correspondence
4.4 Gauge-theoretic definition of elliptic stable envelope
5 Boundaries and interfaces in 3d N=2 SQCD
5.1 Vacua and BPS equations in 3d N=2 Gauge theories
5.1.1 Vacuum equations
5.1.2 BPS equations
5.2 Classical Higgs branch
5.2.1 Classical Higgs branches of 3d N=2 SQCDs
5.2.2 Fixed points
5.2.3 Attracting sets and Morse flow
5.3 Boundaries and interfaces
5.3.1 Thimble (exceptional Dirichlet), DC(p)
5.3.2 Enriched Neumann, NC(p)
5.3.3 Massless boundary condition, BL(p)(p)
5.3.4 Janus interface, J(mC, 0)
6 Stable envelopes and R-matrix from gauge theory: the computation
6.1 Elliptic stable envelope
6.2 R-matrix for elliptic sl(1|1) spin chains
7 2d and 1d avatars of elliptic stable envelopes and the R-matrix
7.1 The K-theory limit
7.2 The cohomology limit
A Equivariant elliptic cohomology
A.1 Chern class
A.2 Gysin map
A.3 Supports
A.4 Correspondences
A.5 Degree of a line bundle
B Notations and conventions for supersymmetry
B.1 Conventions for spinors
B.2 3d N=2 supersymmetry
B.3 Localizing supercharge
C sl(1|1) rational R-matrix from the A-model localization
C.1 Construction of geometric rational R-matrix
C.2 Further details on two-point functions of 2d GLSM observables
C.2.1 Brief recap of Closset–Cremonesi–Park localization formula
C.2.2 Explicit computation of two-point functions
References
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| id | nasplib_isofts_kiev_ua-123456789-212645 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T19:16:00Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ishtiaque, Nafiz Moosavian, Seyed Faroogh Zhou, Yehao 2026-02-09T09:31:59Z 2024 Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence. Nafiz Ishtiaque, Seyed Faroogh Moosavian and Yehao Zhou. SIGMA 20 (2024), 099, 95 pages 1815-0659 2020 Mathematics Subject Classification: 81R12; 81T60; 55N34 arXiv:2308.12333 https://nasplib.isofts.kiev.ua/handle/123456789/212645 https://doi.org/10.3842/SIGMA.2024.099 We generalize Aganagic-Okounkov's theory of elliptic stable envelopes, and its physical realization in Dedushenko-Nekrasov's and Bullimore-Zhang's works, to certain varieties without holomorphic symplectic structure or polarization. These classes of varieties include, in particular, classical Higgs branches of 3d = 2 quiver gauge theories. The Bethe/gauge correspondence relates such a gauge theory to an isotropic/elliptic superspin chain, and the stable envelopes compute the -matrix that solves the dynamical Yang-Baxter equation (dYBE) for this spin chain. As an illustrative example, we solve the dYBE for the elliptic (1|1) spin chain with fundamental representations using the corresponding 3d = 2 SQCD whose classical Higgs branch is the Lascoux resolution of a determinantal variety. Certain Janus partition functions of this theory on × for an interval and an elliptic curve compute the elliptic stable envelopes, and in turn the geometric elliptic -matrix, of the anisotropic (1|1) spin chain. Furthermore, we consider the 2d and 1d reductions of elliptic stable envelopes and the -matrix. The reduction to 2d gives the K-theoretic stable envelopes, and the trigonometric -matrix, and a further reduction to 1d produces the cohomological stable envelopes and the rational -matrix. The latter recovers Rimányi-Rozansky's results that appeared recently in the mathematical literature. We thank Kevin Costello, Yuan Miao, Hiraku Nakajima, Tadashi Okazaki, Surya Raghavendran, Junya Yagi, Masahito Yamazaki, and Zijun Zhou for useful discussions. We thank Davide Gaiotto and Andrei Okounkov for reading the draft and sending us valuable feedback. Special thanks go to Mykola Dedushenko for explaining some parts of his work with Nikita Nekrasov and for providing extensive commentary on an earlier version of this work. We thank the Banff International Research Station for generously hosting us during the last stage of this project. N.I. is supported by the Huawei Young Talents Program Fellowship at IHES. The work of S.F.M. is funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) funding number SAPIN-2022-00028, and also in part by the Alfred P. Sloan Foundation, grant FG2020-13768. S.F.M would like to thank Davide Gaiotto for his support during the visit to the Perimeter Institute. Kavli IPMU is supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence Article published earlier |
| spellingShingle | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence Ishtiaque, Nafiz Moosavian, Seyed Faroogh Zhou, Yehao |
| title | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence |
| title_full | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence |
| title_fullStr | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence |
| title_full_unstemmed | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence |
| title_short | Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical -Matrices for Superspin Chains from the Bethe/Gauge Correspondence |
| title_sort | elliptic stable envelopes for certain non-symplectic varieties and dynamical -matrices for superspin chains from the bethe/gauge correspondence |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212645 |
| work_keys_str_mv | AT ishtiaquenafiz ellipticstableenvelopesforcertainnonsymplecticvarietiesanddynamicalmatricesforsuperspinchainsfromthebethegaugecorrespondence AT moosavianseyedfaroogh ellipticstableenvelopesforcertainnonsymplecticvarietiesanddynamicalmatricesforsuperspinchainsfromthebethegaugecorrespondence AT zhouyehao ellipticstableenvelopesforcertainnonsymplecticvarietiesanddynamicalmatricesforsuperspinchainsfromthebethegaugecorrespondence |