The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals Arising in Supersymmetric Gauge Theory
The purpose of this article is to demonstrate that i) the framework of elliptic hypergeometric integrals (EHIs) can be extended by input from supersymmetric gauge theory, and ii) analyzing the hyperbolic limit of the EHIs in the extended framework leads to a rich structure containing sharp mathemati...
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| description | The purpose of this article is to demonstrate that i) the framework of elliptic hypergeometric integrals (EHIs) can be extended by input from supersymmetric gauge theory, and ii) analyzing the hyperbolic limit of the EHIs in the extended framework leads to a rich structure containing sharp mathematical problems of interest to supersymmetric quantum field theorists. Both of the above items have already been discussed in the theoretical physics literature. Item i was demonstrated by Dolan and Osborn in 2008. Item ii was discussed in the present author's Ph.D. Thesis in 2016, wherein crucial elements were borrowed from the 2006 work of Rains on the hyperbolic limit of certain classes of EHIs. This article contains a concise review of these developments, along with minor refinements and clarifying remarks, written mainly for mathematicians interested in EHIs. In particular, we work with a representation-theoretic definition of a supersymmetric gauge theory, so that readers without any background in gauge theory - but familiar with the representation theory of semi-simple Lie algebras - can follow the discussion.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 043, 30 pages
The Hyperbolic Asymptotics
of Elliptic Hypergeometric Integrals
Arising in Supersymmetric Gauge Theory
Arash Arabi ARDEHALI
School of Physics, Institute for Research in Fundamental Sciences (IPM),
P.O. Box 19395-5531, Tehran, Iran
E-mail: a.a.ardehali@gmail.com
Received January 09, 2018, in final form April 29, 2018; Published online May 06, 2018
https://doi.org/10.3842/SIGMA.2018.043
Abstract. The purpose of this article is to demonstrate that i) the framework of ellip-
tic hypergeometric integrals (EHIs) can be extended by input from supersymmetric gauge
theory, and ii) analyzing the hyperbolic limit of the EHIs in the extended framework leads
to a rich structure containing sharp mathematical problems of interest to supersymmetric
quantum field theorists. Both of the above items have already been discussed in the theo-
retical physics literature. Item i was demonstrated by Dolan and Osborn in 2008. Item ii
was discussed in the present author’s Ph.D. Thesis in 2016, wherein crucial elements were
borrowed from the 2006 work of Rains on the hyperbolic limit of certain classes of EHIs.
This article contains a concise review of these developments, along with minor refinements
and clarifying remarks, written mainly for mathematicians interested in EHIs. In particu-
lar, we work with a representation-theoretic definition of a supersymmetric gauge theory, so
that readers without any background in gauge theory – but familiar with the representation
theory of semi-simple Lie algebras – can follow the discussion.
Key words: elliptic hypergeometric integrals; supersymmetric gauge theory; hyperbolic
asymptotics
2010 Mathematics Subject Classification: 33D67; 33E05; 41A60; 81T13; 81T60
1 Introduction
Elliptic hypergeometric integrals [14, 15, 41, 43] are multivariate (or matrix-) integrals of the
form
I(p, q) =
∫
F (p, q;x1, . . . , xrG) drGx,
taken over −1/2 ≤ x1, . . . , xrG ≤ 1/2, with rG the rank of some semi-simple matrix Lie group G.
The parameters p, q are often assumed to be complex numbers satisfying 0 < |p|, |q| < 1, but for
simplicity we take them in this article to be inside the open interval ]0, 1[ of the real line. The
title “elliptic hypergeometric integral” (EHI) comes from the fact that the integrand F involves
ratios of products of a number of elliptic gamma functions. The definition of the elliptic gamma
function can be found in equation (2.2), and explicit EHIs can be found in Section 3 below. For
a very brief introduction to EHIs see [34].
Extra complex parameters (denoted by ti in [41], for example) besides p, q are often considered
inside the arguments of the integrand F and the EHI I. Our EHIs here correspond to special
cases where all those extra parameters are taken to be some powers of the product pq, such
This paper is a contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications.
The full collection is available at https://www.emis.de/journals/SIGMA/EHF2017.html
mailto:a.a.ardehali@gmail.com
https://doi.org/10.3842/SIGMA.2018.043
https://www.emis.de/journals/SIGMA/EHF2017.html
2 A.A. Ardehali
that their so-called “balancing conditions” – as well as other constraints discussed in Section 3 –
are satisfied. We will comment on those extra parameters briefly in Section 3 and also in the
appendix.
Mathematicians’ interest in EHIs has been to a large extent due to the remarkable transfor-
mation identities [36, 41, 43] of the form∫
F (p, q;x1, . . . , xrG) drGx =
∫
F̃ (p, q; y1, . . . , yrG̃) drG̃y, (1.1)
that they exhibit. We formally allow rG̃ to be zero, in which case there is only a function
of p, q – and no integral – on the right-hand side; then (1.1) would be an integral evaluation. In
their magical flavor, the transformation identities of EHIs are somewhat analogous to, though
generally much more non-trivial than, the celebrated Rogers–Ramanujan identities featuring in
analytic number theory. A fruitful idea, beyond the scope of the present article, for studying
EHI transformations has been the application of Bailey transforms to EHIs [42, 47].
A major mathematical development in which EHIs played a key role is the elliptic generaliza-
tion [33, 36] of the Koornwinder–Macdonald theory of orthogonal polynomials [28]. Specifically,
in [33, 36] abelian biorthogonal functions were constructed whose biorthogonality relation is gov-
erned by the “Type II” EHI of [14, 15, 43]. This development, too, is beyond the scope of the
present article, and the interested reader is encouraged to consult [39] for a better perspective.
Theoretical physicists’ interest in EHIs started growing in 2008 when Dolan and Osborn [17]
showed that
• four-dimensional supersymmetric (SUSY) gauge theory provides a framework in which
the classes of EHIs known at the time arise as a particular partition function, called the
Romelsberger index [38], of some of the most famous models (namely SUSY QCD models
with gauge group G either unitary or symplectic);
• the transformation identities of the EHIs have a very natural interpretation in the physical
framework as the equality of the Romelsberger indices of a pair of electric-magnetic (or
Seiberg-) dual models.
Since then, the physics community, often working together with mathematicians, started con-
tributing to the mathematical theory by studying new EHIs arising in SUSY gauge theory,
using SUSY dualities to conjecture new transformation identities, and sometimes also proving
the new identities. References [26, 29, 44, 46] are a few particularly clear demonstrations of the
fruitfulness of this interplay between physics and mathematics. References [9, 50] are examples
of several works in the other direction, using rigorous mathematics to shed light on dualities
in SUSY gauge theory. The relation between EHIs and SUSY gauge theory, and between the
transformation identities and SUSY duality, is briefly reviewed in Section 3 below.
The main focus of the present article is not the transformation properties of the EHIs
though, but their rich asymptotic behavior in the so-called hyperbolic limit, where p, q → 1
while log p/ log q is kept fixed. Defining b, β ∈ ]0,∞[ through
p = e−βb, q = e−βb
−1
,
we have
the hyperbolic limit: β → 0+, with b ∈ ]0,∞[ fixed. (1.2)
The title “hyperbolic” comes from the fact that in this limit the elliptic gamma functions reduce
to hyperbolic gamma functions; see Section 2 below for the details. EHIs also have nontrivial
“trigonometric”, “rational”, and “classical” limits, which we do not consider here; the interested
reader can learn more about these limits in [35].
The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals 3
Mathematicians’ interest in the hyperbolic limit of EHIs has been mainly because taking the
hyperbolic limit of a transformation identity like (1.1), one often arrives at a reduction of the
identity to the hyperbolic level:∫
Fh(b;x1, . . . , xrG) drGx =
∫
F̃h(b; y1, . . . , yrG̃) drG̃y,
where now Fh and F̃h involve hyperbolic (rather than elliptic) gamma functions, and the integrals
range over ]−∞,∞[.
While the study of the hyperbolic limit of EHIs goes back to [16, 48], rigorous asymptotic
estimates were obtained first by Rains in 2006 [35] for certain special classes of EHIs. (See also
[10, Section 5] where the results of [35] are used to further analyze the hyperbolic limit of certain
EHIs.) In this article we review the work in [2, 3], which used Rains’s machinery to analyze
the hyperbolic asymptotics of general EHIs arising in SUSY gauge theory. In Section 4 we first
present the conjecture in [2] for the most general case, stating that
I(b, β) ≈
(
2π
β
)rG ∫
drGx e−[EDK
0 (b,β)+V eff(x;b,β)]+iΘ(x;β), (1.3)
where ≈ means an O
(
β0
)
error after taking logarithms of the two sides. The symbol x denotes
the collection x1, . . . , xrG , and EDK
0 , V eff are real functions of order 1/β, while Θ is a real
function of order 1/β2; see equations (4.7)–(4.9) below for the explicit expressions. Next, we
will specialize to the (still rather large) class of non-chiral EHIs for which Θ = 0; this class
encompasses all the EHIs studied by Rains [35]. For non-chiral EHIs we present the precise
analysis performed in [3], and demonstrate that not only (1.3) is true, but that in fact
log I(b, β) = −
[
EDK
0 (b, β) + V eff
min(b, β)
]
+ dim hqu log
(
2π
β
)
+O
(
β0
)
, (1.4)
where hqu is the locus of minima of V eff(x; b, β) as a function of x, and V eff
min(b, β) the value of
V eff(x; b, β) on this locus. (Recall that the generically leading terms EDK
0 , V eff
min are of order 1/β.)
Finding the small-β asymptotics of I(b, β) thus involves a minimization problem for V eff as
a function of x. Interestingly, it turns out that V eff is a piecewise linear function of x; see the
plots in Figs. 2, 3, 4, and 7.
We have not been able to prove general theorems on the minimum value or the dimension
of the locus of minima of V eff , but have been able to address the minimization problem for
specific EHIs, on a case-by-case basis, using Rains’s generalized triangle inequalities [35] or
some variations thereof; see Section 5 for a few explicit examples.
The O
(
β0
)
term on the r.h.s. of (1.4) is where the hyperbolic reduction of I(b, β) resides. We
will not discuss this term in depth in the present article, and will only make brief remarks about
it in certain examples in the last two sections; other examples for which this term is explicitly
analyzed can be found in [2, 35].
Theoretical physicists’ interest in the hyperbolic limit of EHIs has been partly because the
hyperbolic reduction of the Romelsberger index I(b, β) of a 4d SUSY gauge theory1 often yields
the squashed three-sphere partition function ZS3(b) of the dimensionally reduced – hence 3d –
SUSY gauge theory; in other words the O
(
β0
)
term on the r.h.s. of (1.4) is often logZS3(b),
with ZS3(b) given in turn by a hyperbolic hypergeometric integral. This ties well with Rains’s
results for the hyperbolic reduction of the special classes of EHIs studied in [35]. The physics
intuition for the reduction is roughly as follows. The index I(b, β) can be computed [5] by the
path-integral of the SUSY gauge theory placed on Euclidean S3
b ×S1
β, where S3
b is the unit-radius
1We follow the common terminology and refer to specific “models” in the gauge theory framework as “theories”.
4 A.A. Ardehali
squashed three-sphere with squashing parameter b, and β is the circumference of the S1. The
β → 0 limit shrinks S1
β, hence leaving us with the dimensionally reduced theory on S3
b . This
reduction has been noticed quantitatively in some special cases [1, 18, 22, 24, 32, 45]. However,
the mathematical results of [2, 3] clarify that the reduction works in the nice way encountered
in [1, 18, 22, 24, 32, 45] – and so the above intuitive physical picture is correct – only when V eff
is minimized just at x = 0; this condition was satisfied in all the examples studied rigorously by
Rains [35] as well. When this condition is not satisfied, the hyperbolic reduction is more subtle.
See Sections 5.2 and 5.4 for two such more subtle examples.
There is an additional reason for the interest of the theoretical physics community in the
hyperbolic limit of EHIs. Interpreting S1
β in S3
b × S1
β as the Euclidean time circle of the back-
ground spacetime, we get an analogy with thermal quantum physics where the circumference β
of the Euclidean time circle becomes the inverse temperature2. The hyperbolic limit of the
EHI then corresponds to the high-temperature (or “Cardy”) limit of the index I(b, β). Since
the celebrated work of Cardy on the high-temperature asymptotics of 2d conformal field the-
ory (CFT) partition functions [11], the “Cardy asymptotics” of various quantum field theory
partition functions have been of interest in theoretical physics. In particular, for the special
cases where the underlying SUSY gauge theory describes a 4d CFT, the index I(b, β) encodes
the analytic combinatorics of the supersymmetric operators in the CFT [27]. The hyperbolic
asymptotics is then connected to the asymptotic degeneracy of the large-charge supersymmetric
operators. The counting of these operators can then have implications, through the AdS/CFT
correspondence [30], for heavy states of quantum gravity on anti-de Sitter spacetimes [2, 7, 27].
Because of this interest in the Cardy asymptotics of I(b, β), there had been other physical
studies of the subject prior to [2, 3]. In particular, the leading Cardy asymptotics of I(b, β) was
proposed in a well-known paper [13] to be given by log I(b, β) ≈ −EDK
0 (b, β). The mathematical
results of [2, 3] clarified that this relation is modified, as in (1.4), for non-chiral EHIs with
V eff
min(b, β) 6= 0. A physical understanding of this modification due to nonzero V eff
min is recently
achieved in [12].
In the final section we mention some of the open problems of physical interest concerning the
hyperbolic limit of EHIs.
2 The required special functions and their asymptotics
The special functions and some of their useful properties
For complex a, q such that 0 < |q| < 1, we define the Pochhammer symbol as
(a; q) :=
∞∏
k=0
(
1− aqk
)
.
Often the notation (a; q)∞ is used for the above function; we are following Rains’s convention [35]
of omitting the ∞.
The Pochhammer symbol is related to the Dedekind eta function via
η(τ) = q1/24(q; q), (2.1)
with q = e2πiτ . The eta function has an SL(2,Z) modular property that will be useful for us:
η(−1/τ) =
√
−iτη(τ).
2The analogy with thermal physics is actually not quite precise. In the path-integral computation [5] one must
use a supersymmetric (i.e., periodic) spin connection around S1
β , whereas in thermal quantum physics the spin
connection is anti-periodic around the Euclidean time circle. Nevertheless, as in [2, 3] we keep employing the
analogy because it helps a useful import of intuition from thermal physics.
The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals 5
The elliptic gamma function (first introduced by Ruijsenaars in [40]) can be defined for
Im(τ), Im(σ) > 0 as
Γ(x;σ, τ) :=
∏
j,k≥0
1− z−1pj+1qk+1
1− zpjqk
, (2.2)
with z := e2πix, p := e2πiσ, and q := e2πiτ . The above expression gives a meromorphic function of
x, σ, τ ∈ C. For generic choice of τ and σ, the elliptic gamma has simple poles at x = l−mσ−nτ ,
with m,n ∈ Z≥0, l ∈ Z.
We sometimes write Γ(x;σ, τ) as Γ(z; p, q), or simply as Γ(z). Also, the arguments of elliptic
gamma functions are frequently written with “ambiguous” signs (as in Γ(±x;σ, τ)); by that we
mean a multiplication of several gamma functions each with a “possible” sign in the argument
(as in Γ(+x;σ, τ)× Γ(−x;σ, τ)). Similarly Γ
(
z±1
)
:= Γ(z; p, q)× Γ
(
z−1; p, q
)
.
The hyperbolic gamma function (first introduced in a slightly different form by Ruijsenaars
in [40]) can be defined, following Rains [35], via
Γh(x;ω1, ω2) := exp
(
PV
∫
R
e2πixw
(e2πiω1w − 1)(e2πiω2w − 1)
dw
w
)
. (2.3)
The above expression makes sense only for 0 < Im(x) < 2 Im(ω), with ω := (ω1 +ω2)/2. In that
domain, the function defined by (2.3) satisfies
Γh(x+ ω2;ω1, ω2) = 2 sin
(
πx
ω1
)
Γh(x;ω1, ω2),
which can then be used for an inductive meromorphic continuation of the hyperbolic gamma
function to all x ∈ C. For generic ω1, ω2 in the upper half plane, the resulting meromorphic
function Γh(x;ω1, ω2) has simple zeros at x = ω1Z≥1 + ω2Z≥1 and simple poles at x = ω1Z≤0 +
ω2Z≤0.
For convenience, we will frequently write Γh(x) instead of Γh(x;ω1, ω2), and Γh(x±y) instead
of Γh(x+ y)Γh(x− y).
The hyperbolic gamma function has an important property that can be easily derived from
the definition (2.3):
Γh(−Re(x) + i Im(x);ω1, ω2) =
(
Γh(Re(x) + i Im(x);ω1, ω2)
)∗
, (2.4)
with ∗ denoting complex conjugation.
We also define the non-compact quantum dilogarithm ψb (cf. the function eb(x) in [19]; ψb(x) =
eb(−ix)) via
ψb(x) := e−iπx
2/2+iπ(b2+b−2)/24Γh(ix+ ω;ω1, ω2), (2.5)
where ω1 := ib, ω2 := ib−1, and ω := (ω1 + ω2)/2. For generic choice of b, the zeros of ψb(x)±1
are of first order, and lie at ±
((
b + b−1
)
/2 + bZ≥0 + b−1Z≥0
)
. Upon setting b = 1 we get the
function ψ(x) of [20], i.e., ψb=1(x) = ψ(x).
An identity due to Narukawa [31] implies the following important relation between ψb(x) and
the elliptic gamma function [4]:
Γ(x;σ, τ) =
e2iπQ−(x;σ,τ)
ψb
(
2πix
β + b+b−1
2
) ∞∏
n=1
ψb
(
−2πin
β −
2πix
β −
b+b−1
2
)
ψb
(
−2πin
β + 2πix
β + b+b−1
2
)
= e2iπQ+(x;σ,τ)ψb
(
−2πix
β
− b+ b−1
2
) ∞∏
n=1
ψb
(
−2πin
β −
2πix
β −
b+b−1
2
)
ψb
(
−2πin
β + 2πix
β + b+b−1
2
) , (2.6)
6 A.A. Ardehali
where
Q−(x;σ, τ) = − x3
6τσ
+
τ + σ − 1
4τσ
x2 − τ2 + σ2 + 3τσ − 3τ − 3σ + 1
12τσ
x
− 1
24
(τ + σ − 1)
(
τ−1 + σ−1 − 1
)
,
Q+(x;σ, τ) = Q−(x;σ, τ) +
(
x− τ+σ
2
)2
2τσ
− τ2 + σ2
24τσ
,
and
σ =
iβ
2π
b, τ =
iβ
2π
b−1.
In the special case where b = 1, the expressions in (2.6) are corollaries of Theorem 5.2 of [20].
Therefore equation (2.6) can be regarded as an extension of (and in fact was derived in [4] in
an attempt to extend) Theorem 5.2 of [20].
The required asymptotic estimates
Throughout this article we take the parameter β to be real and strictly positive. Therefore by
β → 0 we always mean β → 0+.
We say f(β) = O(g(β)) as β → 0, if there exist positive real numbers C, β0 such that for
all β < β0 we have |f(β)| < C|g(β)|. We say f(x, β) = O(g(x, β)) uniformly over S as β → 0,
if there exist positive real numbers C, β0 such that for all β < β0 and all x ∈ S we have
|f(x, β)| < C|g(x, β)|.
We use the symbol ∼ when writing the all-orders asymptotics of a function. For example,
we have
log
(
β + e−1/β
)
∼ log β as β → 0,
because we can write the left-hand side as the sum of log β and log
(
1 + e−1/β/β
)
, and the latter
is beyond all-orders in β.
More precisely, we say f(β) ∼ g(β) as β → 0, if we have f(β) − g(β) = O(βn) for any
(arbitrarily large) natural n.
The only unconventional piece of notation is the following: we will write f(β) ' g(β) if
log f(β) ∼ log g(β) (with an appropriate choice of branch for the logarithms). By writing
f(x, β) ' g(x, β) we mean that log f(x, β) ∼ log g(x, β) for all x on which f(x, β), g(x, β) 6= 0,
and that f(x, β) = g(x, β) = 0 for all x on which either f(x, β) = 0 or g(x, β) = 0.
With the above notations at hand, we can asymptotically analyze the Pochhammer symbol
as follows. The “low-temperature” (T := β−1 → 0) behavior is trivial:(
e−β; e−β
)
' 1 as 1/β → 0.
The “high-temperature” (T−1 = β → 0) asymptotics is nontrivial. It can be obtained using
the SL(2,Z) modular property of the eta function, which yields
log η
(
iβ
2π
)
∼ −π
2
6β
+
1
2
log
(
2π
β
)
as β → 0.
The above relation, when combined with (2.1), implies
log
(
e−β; e−β
)
∼ −π
2
6β
+
1
2
log
(
2π
β
)
+
β
24
as β → 0. (2.7)
The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals 7
For the hyperbolic gamma function, Corollary 2.3 of [35] implies that when x ∈ R
log Γh(x+ rω;ω1, ω2) ∼ −iπ
[
x|x|/2 + (r − 1)ω|x|+ (r − 1)2ω2/2 +
(
b2 + b−2
)
/24
]
, (2.8)
as |x| → ∞, for any fixed real r, and fixed b > 0.
Combining (2.8) and (2.5) we find that for fixed Re(x) and fixed b > 0
logψb(x) ∼ 0 as β → 0, for Im(x) = −1/β
with an exponentially small error, of the type e−1/β.
The above estimate can be combined with (2.6) to yield the following estimates in the hy-
perbolic limit (1.2):
Γ(x;σ, τ) '
e2iπQ−(x;σ,τ)
ψb
(
2πix
β + b+b−1
2
) , for − 1 < Re(x) ≤ 0,
' e2iπQ+(x;σ,τ)ψb
(
−2πix
β
− b+ b−1
2
)
, for 0 ≤ Re(x) < 1,
(2.9)
with σ = iβ
2π b, τ = iβ
2π b
−1, and with the range of Re(x) explaining our subscript notations for Q+
and Q−. The relation (2.9), combined with (2.5), demonstrates the reduction of the elliptic
gamma function to the hyperbolic gamma function in the limit (1.2).
As a result of (2.9), for x ∈ R we have the following relations in the hyperbolic limit:
Γ
(
−x+
(
τ + σ
2
)
r;σ, τ
)
' e2iπQ−(−{x}+( τ+σ
2
)r;σ,τ)
ψb
(
−2πi{x}
β − (r − 1) b+b
−1
2
) ,
Γ
(
x+
(
τ + σ
2
)
r;σ, τ
)
' e2iπQ+({x}+( τ+σ
2 )r;σ,τ)ψb
(
−2πi{x}
β
+ (r − 1)
b+ b−1
2
)
, (2.10)
with {x} := x − bxc. The above estimates are first obtained in the range 0 ≤ x < 1, and then
extended to x ∈ R using the periodicity of the l.h.s. under x→ x+ 1.
3 Elliptic hypergeometric integrals
from supersymmetric gauge theory
3.1 How a SUSY gauge theory with U(1) R-symmetry gives an EHI
For the purpose of the present article, we take the following essentially representation theoretic
data to defines a 4d supersymmetric gauge theory with U(1) R-symmetry:
i) a gauge group G, which we take to be a semi-simple matrix Lie group of rank rG, denote
its root vectors by Dynkin labels α = (α1, . . . , αrG), while denoting the set of all the roots
by ∆G;
ii) a finite number of chiral multiplets χj (with j = 1, . . . , nχ), to each of which we associate
an R-charge rj ∈ ]0, 2[, and a finite-dimensional irreducible representation Rj of G, whose
weight vectors we denote by ρj := (ρj1, . . . , ρ
j
rG), while denoting the set of all the weights
of Rj by ∆j .
Note that even though we have as many αs as dimG and as many ρjs as dimRj , we are not
using further indices to label individual αs and ρjs among these.
We further demand the following anomaly cancellation, or “consistency”, conditions:∑
j
∑
ρj∈∆j
ρjl ρ
j
mρ
j
n = 0, for all l, m, n, (3.1a)
8 A.A. Ardehali∑
j
(rj − 1)
∑
ρj∈∆j
ρjl ρ
j
m +
∑
α∈∆G
αlαm = 0, for all l, m. (3.1b)
We can summarize our definition as follows.
Definition 3.1. A SUSY gauge theory with U(1) R-symmetry is a collection of the following
data satisfying the relations (3.1): a semi-simple matrix Lie group G, and a finite number nχ of
pairs {R1, r1}, . . . , {Rnχ , rnχ}, where Rj are finite-dimensional irreducible representations of G
while rj are real numbers inside ]0, 2[. We denote the roots of G by α, the weights of Rj by ρj ,
and the set of all the weights of Rj by ∆j .
Although the field theory formulation of a SUSY gauge theory is beyond the scope of the
present article, for the readers familiar with that formulation we add that
i) the “field content” of a SUSY gauge theory described as above is: a massless vector
multiplet (containing the gauge field and its fermionic super-partner gaugino fields) trans-
forming in the adjoint representation of G, a finite number nχ of massless chiral multiplets
(containing Weyl fermions and their super-partner complex scalars) transforming in Rj
of G, and for each of the chiral multiplets a CP-conjugate multiplet, with R-charge −rj ,
transforming in R̄j ;
ii) the constraints (3.1) are respectively the conditions for cancellation of the gauge3 and
U(1)R-gauge2 anomalies of the field theory3;
iii) since in field theory one should also specify the interactions of various fields, it would be
more precise from that perspective to say that Definition 3.1 does not single out a unique
SUSY gauge theory, but describes a universality class of SUSY gauge theories compatible
with the specified data.
Our next definition bridges SUSY gauge theory to EHIs, through the Romelsberger index
[27, 38] (also referred to as “the 4d supersymmetric index” [37], or “the 4d superconformal
index” when applied to superconformal field theories [27]).
Definition 3.2. The Romelsberger index of a SUSY gauge theory with U(1) R-symmetry (as
in Definition 3.1) is given by
I(b, β) :=
(p; p)rG(q; q)rG
|W |
∫ ( rG∏
k=1
dzk
2πizk
) ∏
j
∏
ρj∈∆j
Γ
(
(pq)rj/2zρ
j)
∏
α+∈∆G
Γ(z±α+)
. (3.2)
Here, p = e−βb, q = e−βb
−1
, and we take β, b ∈ ]0,∞[, so p, q are real numbers in ]0, 1[. Our
symbolic notation zρ
j
should be understood as z
ρj1
1 × · · · × z
ρjrG
rG . The α+ are the positive roots
of G, and |W | is the order of the Weyl group of G. The integral is over the unit torus in the
space of zk, or alternatively over xk ∈ [−1/2, 1/2] for xk defined through zk = e2πixk . By zα we
mean zα1
1 × · · · × z
αrG
rG .
In our notation the three-dimensional representation of SU(3), for example, has weights
(ρ1, ρ2) = (1, 0), (0, 1), (−1,−1), and the positive roots of SU(3) are α+ = (1,−1), (2, 1), (1, 2).
3The gauge-U(1)2
R and gauge-gravity2 anomalies vanish automatically because we are focusing on semi-simple
gauge groups (cf. equation (4.11) below); upon extending the framework to compact G, their cancellation should
be demanded as extra consistency conditions besides (3.1). The gauge2-gravity anomalies cancel between CP-
conjugate Weyl fermions.
The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals 9
Also, the parameters b, β are related to Rains’s parameters in [35] via ω1 = ib, ω2 = ib−1, and
v = β
2π .
The expression (3.2) is regarded in the physics literature as the outcome of a combinatorial
(Hamiltonian) [3, 17] or a path-integral (Lagrangian) [5] computation, the starting point being
more physical definitions for the Romelsberger index in both cases. For our purposes here
though, it is more convenient to take (3.2) as the definition. See [37] for a recent review of the
index from a more physical perspective.
One of the simplest examples of SUSY gauge theories with U(1) R-symmetry is the SU(2)
supersymmetric QCD (SQCD) with three flavors: the gauge group G is SU(2); there are three
“quark” chiral multiplets with R-charge 1/3 in the fundamental representation of SU(2), so
R1 = R2 = R3 = � and ρ1
1, ρ
2
1, ρ
3
1 = ±1; there are also three “anti-quark” chiral multiplets
with R-charge 1/3 in the anti-fundamental representation of SU(2), so R4 = R5 = R6 = �̄ and
ρ4
1, ρ
5
1, ρ
6
1 = ∓1. Since the positive root of SU(2) is α+ = 2, and its Weyl group has order 2, the
expression (3.2) ends up being in this case
INc=2,Nf=3(b, β) =
(p; p)(q; q)
2
∫ 1/2
−1/2
dx
Γ6
(
(pq)1/6z±1
)
Γ
(
z±2
) . (3.3)
This is a special case of the elliptic beta integral of Spiridonov [41], the first of the species of
EHIs to have been discovered.
In the previous sentence we said “a special case”, because, as alluded to in the introduction,
EHIs often depend on extra parameters (ti in [41], for example). We are focusing for simplicity
on special cases where all these parameters are taken to be powers of pq, such that their “ba-
lancing conditions”, as well as the constraints (3.1) following from their expression as in (3.2),
are satisfied. Introducing those parameters back corresponds to turning on flavor fugacities –
or flavor chemical potentials – in the physical picture. We briefly comment on the incorporation
of flavor fugacities in the appendix.
Dolan and Osborn [17] realized that the Romelsberger index of SU(Nc) SQCD with Nf flavors
(which has gauge group SU(Nc), and has 2Nf chiral multiplets of R-charge 1 −Nc/Nf , half of
them in the fundamental and the other half in the anti-fundamental representation of the gauge
group) corresponds to the EHI denoted I
(m)
An
in [36], with n = Nc − 1 and m = Nf − Nc − 1.
The Sp(2N) gauge theory with 2Nf chiral multiplets of R-charge 1 − (N + 1)/Nf in the 2N
dimensional fundamental representation gives rise to the EHI denoted I
(m)
BCn
in [36], with n = N
and m = Nf − N − 2. This is enough reason to claim that the expression (3.2) provides
a legitimate extension of the framework of EHIs. In summary, every supersymmetric gauge theory
with a U(1) R-symmetry defined as above, gives what may be called an elliptic hypergeometric
integral.
For brevity, we sometimes drop the adjective “with U(1) R-symmetry”, but by a SUSY gauge
theory we mean a SUSY gauge theory with U(1) R-symmetry throughout this article; the latter
is the appropriate framework for EHIs, as explained above.
The general expression (3.2) appears for instance in [5]. There, the constraints (3.1) were
assumed, but the condition 0 < rj < 2 was not.
Remark 3.3. The assumption 0 < rj guarantees that the poles of the gamma functions in the
integrand of the EHI (3.2) are avoided, so I(b, β) is a continuous real function in the domain
b, β ∈ ]0,∞[.
That I(b, β) is real follows from dividing the integral to two pieces, one over x1 ∈ [−1/2, 0],
xi>1 ∈ [−1/2, 1/2], the other over x1 ∈ [0, 1/2], xi>1 ∈ [−1/2, 1/2], and then arguing that the
two pieces are complex conjugates of each other because under x → −x the integrand goes to
its complex conjugate. That I(b, β) is continuous on b, β ∈ ]0,∞[ follows from the continuity of
the integrand when 0 < rj .
10 A.A. Ardehali
The further constraint rj < 2 is imposed to make I(b, β) still better-behaved.
Remark 3.4. The assumption 0 < rj < 2 allows using the estimates
1
Γ(z)
' 1− z, and Γ
(
(pq)rj/2z
)
' 1,
as 1/β → 0, for fixed x ∈ R and fixed b ∈ ]0,∞[, both valid uniformly over x ∈ R, so that we
get a universal “low-temperature” asymptotics for the Romelsberger index (3.2):
I(b, β) ' 1
|W |
∫
drGx
∏
α+
((
1− zα+
)(
1− z−α+
))
= 1 (3.4)
as 1/β → 0, for fixed b ∈ ]0,∞[. The equality on the r.h.s. results from the Weyl integral
formula. Such asymptotics are expected for partition functions of gapped quantum systems,
whose only state contributing significantly to the partition function at low-enough temperatures
is the vacuum state having unit Boltzmann factor. So the asymptotics (3.4) is a nice property
for the EHI to have.
We mention in passing that despite the anomaly cancellation conditions (3.1), we may still
have non-zero ’t Hooft anomalies, which do not lead to inconsistencies or R-symmetry violations
in the quantum gauge theory. A careful discussion of such ’t Hooft anomalies is beyond the
scope of the present article; the interested reader is referred to [45].
3.2 How SUSY dualities lead to transformation identities for EHIs
The SU(2) SQCD theory with three flavors, whose index appeared in (3.3), has a magnetic (or
Seiberg-) dual description as a theory of 15 chiral multiplets with R-charge 2/3 without a gauge
group (hence rG̃ = 0). Equality of the indices computed from the two descriptions implies
(p; p)(q; q)
2
∫ 1/2
−1/2
dx
Γ6
(
(pq)1/6z±1
)
Γ
(
z±2
) = Γ15
(
(pq)1/3
)
.
This is a special case (with flavor fugacities suppressed) of Spiridonov’s elliptic beta integral
formula [41], the first of the EHI transformation identities to have been discovered.
The SU(Nc) SQCD theory with Nf flavors described above, has a Seiberg dual description
with G̃ = SU(Nf −Nc), with Nf magnetic quark chiral multiplets in the fundamental of G̃ along
with Nf magnetic anti-quark chiral multiplets in the anti-fundamental of G̃, and N2
f magnetic
“mesons” in the trivial representation of G̃; the magnetic quark and anti-quark multiplets have
R-charge Nc/Nf , while the magnetic mesons have R-charge 2(1−Nc/Nf ). The equality of the
indices computed from the two descriptions implies the transformation identity [36]
I
(m)
An
(
(pq)(m+1)/2(m+n+2); (pq)(m+1)/2(m+n+2); p, q
)
= Γ
(
(pq)(m+1)/(m+n+2)
)(m+n+2)2
· I(n)
Am
(
(pq)(n+1)/2(m+n+2); (pq)(n+1)/2(m+n+2); p, q
)
.
Again, note that for simplicity we are suppressing flavor fugacities; in the language of [36] we
are focusing on the special case where all ti and ui are set equal to each other, hence – from
their balancing condition – equal to (pq)(n+1)/2(m+n+2).
Similarly, the transformation identity for I
(m)
BCn
(p, q) can be arrived at from the SUSY duality
for the corresponding Sp(2N) theory mentioned above. These and similar instances of the
relation between SUSY dualities and EHI transformation were discovered in [17]. See [44, 46]
for a more thorough discussion of these matters.
The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals 11
We would like to emphasize that currently no systematic procedure is known for deriving the
electric-magnetic dual of a given SUSY gauge theory with U(1) R-symmetry. The most system-
atic method available is to write down the Romelsberger index of the given gauge theory, and to
hope that somewhere in the mathematics literature there is a transformation identity discovered
for it; from the transformation identity one then reads the field content of the magnetic dual the-
ory. Of course, achieving a general systematic procedure for transforming the master EHI (3.2)
would change the story. Although it might be too much to hope for, such a procedure would
allow for the first time a systematic approach to the derivation of electric-magnetic dualities
between SUSY gauge theories with U(1) R-symmetry.
4 The rich structure in the hyperbolic limit
We now attempt to understand the asymptotics of the master EHI (3.2) in the hyperbolic limit:
β → 0+ with b > 0 fixed.
4.1 A conjecture for the general case
The following uniform estimate over x ∈ R [2] can be used for a preliminary investigation of the
β → 0 limit of (3.2) (cf. [35, Proposition 2.12]):
log Γ
(
(pq)r/2z
)
= i
2π3
3β2
κ(x) +
2π2
β
(
b+ b−1
2
)
(r − 1)ϑ(x)
− π2
3β
(
b+ b−1
2
)
(r − 1) +O
(
β0
)
for r ∈ ]0, 2[. (4.1)
As in [35], we have defined the continuous, positive, even, periodic function
ϑ(x) := {x}(1− {x})
(
= |x| − x2 for x ∈ [−1, 1]
)
. (4.2)
We have also introduced the continuous, odd, periodic function
κ(x) := {x}(1− {x})(1− 2{x})
(
= 2x3 − 3x|x|+ x for x ∈ [−1, 1]
)
. (4.3)
These functions are displayed in Fig. 1.
The estimate (4.1) can be derived from the second line of (2.6), but we need the following
fact: for fixed r ∈ ]0, 2[ and fixed b > 0, as β → 0 the function logψb
(
−2πi{x}
β + (r− 1) b+b
−1
2
)
is
uniformly bounded over (x ∈) R. It suffices of course to establish this fact in the “fundamental
domain” x ∈ [0, 1[. To obtain the uniform bound, divide this interval into [0, N0β] and [N0β, 1[,
with N0 chosen as follows. Since ψb
(
−2πiN + (r − 1) b+b
−1
2
)
→ 1 as N → ∞, there is a large
enough N0, so that for all N > N0 we have ψb
(
−2πiN+(r−1) b+b
−1
2
)
≈ 1, with an error of say .1.
With this choice of N0 it is clear that logψb
(
−2πix
β + (r − 1) b+b
−1
2
)
is uniformly bounded over
[N0β, 1[ (for all β smaller than 1/N0). On the other hand, since logψb
(
−2πix + (r − 1) b+b
−1
2
)
is continuous, it is guaranteed to be uniformly bounded on the compact domain [0, N0]; re-
scaling x → x
β this implies the uniform bound on logψb
(
−2πix
β + (r − 1) b+b
−1
2
)
over [0, N0β],
and we are done. Note that for logψb
(
−2πi{x}
β + (r− 1) b+b
−1
2
)
to not diverge at x ∈ Z, we need
r
(
b+b−1
2
)
/∈ bZ≤0 + b−1Z≤0 and (r − 2)
(
b+b−1
2
)
/∈ bZ≥0 + b−1Z≥0; our constraint r ∈ ]0, 2[ takes
care of these.
The estimate (4.1) can not, however, be applied to the elliptic gamma functions in the
denominator of (3.2); these would require an analog of (4.1) which would apply when r = 0.
12 A.A. Ardehali
Figure 1. The even, piecewise quadratic function ϑ(x) (on the left) and the odd, piecewise cubic func-
tion κ(x) (on the right). Both are continuous and periodic, and their fundamental domain can be taken
to be [−1/2, 1/2].
This analog, which is valid uniformly over compact subsets of R avoiding an O(β) neighborhood
of Z, reads (cf. [35, Proposition 2.12])
log
(
1
Γ(z±1)
)
=
4π2
β
(
b+ b−1
2
)
ϑ(x)− 2π2
3β
(
b+ b−1
2
)
+O
(
β0
)
. (4.4)
Note that the above relation would follow from a (sloppy) use of (4.1) with r = 0, but unlike (4.1)
the above estimate is not valid uniformly over R. For real x in the dangerous neighborhoods of
size O(β) around Z, the following slightly weaker version of (4.4) applies (cf. [35, Corollary 3.1]):
1
Γ(z±1)
= O
(
exp
[
4π2
β
(
b+ b−1
2
)
ϑ(x)− 2π2
3β
(
b+ b−1
2
)])
as β → 0. (4.5)
A stronger estimate in this region can be obtained by relating the product on the l.h.s. to
a product of theta functions, and then using the modular property of theta functions. The
weaker estimate above suffices for our purposes though.
Let us recall the asymptotic relation (2.7) from Section 2:
log
(
e−β; e−β
)
∼ −π
2
6β
+
1
2
log
(
2π
β
)
+
β
24
as β → 0,
where∼ indicates asymptotic equality to all orders. Combining this with (4.1) and (4.4), a sloppy
simplification of the EHI in (3.2) now yields
I(b, β) ≈
(
2π
β
)rG ∫
hcl
drGx e−[EDK
0 (b,β)+V eff(x;b,β)]+iΘ(x;β). (4.6)
We have denoted the unit hypercube xi ∈ [−1/2, 1/2] by hcl, because in the path-integral
picture the range of integration can be interpreted as the “classical” moduli-space of the gauge
field holonomies – aka “Wilson loops” – around S1
β. We have also defined
EDK
0 (b, β) :=
π2
3β
(
b+ b−1
2
)dimG+
∑
j
(rj − 1) dimRj
, (4.7)
V eff(x; b, β) :=
4π2
β
(
b+ b−1
2
)
Lh(x), (4.8)
Θ(x;β) :=
8π3
β2
Qh(x). (4.9)
The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals 13
The real functions Qh(x) and Lh(x) are defined by
Qh(x) :=
1
12
∑
j
∑
ρj∈∆j
κ(〈ρj · x〉),
Lh(x) :=
1
2
∑
j
(1− rj)
∑
ρj∈∆j
ϑ(〈ρj · x〉)−
∑
α+∈∆G
ϑ(〈α+ · x〉), (4.10)
where 〈 · 〉 denotes the dot product.
Note that in (4.6) we are claiming that the EHI is approximated well with the integral of
its approximate integrand. This is far from obvious: while the estimate (4.1) for the gamma
functions in the numerator is valid uniformly over the domain of integration, the estimate (4.4)
for the denominator gamma functions is uniform only over compact subsets of hcl that avoid
an O(β) neighborhood of the Stiefel diagram
Sg =:
⋃
α+
{x ∈ hcl|〈α+ · x〉 ∈ Z}.
Let’s denote this neighborhood by S(β)
g . Intuitively speaking, we expect the estimate (4.5),
which applies also on S(β)
g , to guarantee that our unreliable use of (4.4) over this small region
modifies the asymptotics at most by a multiplicative O
(
β0
)
factor4; in absence of unforseen
cancellations due to integration, this factor is not O(β). Since the errors of the estimates used
in deriving (4.6) from (3.2) are also multiplicative O
(
β0
)
, we may hope the two sides of the
symbol ≈ in (4.6) to be equal, asymptotically as β → 0, up to a multiplicative factor of order β0
(but not order β). For non-chiral theories with Θ = Qh = 0 this intuitive argument can be made
more precise, as we do below. We leave the validity of (4.6) for the general case, with possibly
nonzero Qh, as a conjecture.
Conjecture 4.1. For a general Romelsberger index I(b, β) as in Definition 3.2, the estima-
te (4.6) is valid, asymptotically in the hyperbolic limit, up to an O
(
β0
)
error upon taking loga-
rithms of the two sides.
Studying the small-β behavior of the multiple-integral on the r.h.s. of (4.6) is now a (rather
nontrivial) exercise in standard asymptotic analysis. We have not been able to carry this analysis
forward for the general case with Θ 6= 0, so we will shortly restrict attention to non-chiral
theories which have Θ = 0. But before that, we comment on some important properties of the
functions Qh and Lh introduced above.
The real function Qh appearing in the phase Θ(x;β) is piecewise quadratic, because the cubic
terms in it cancel thanks to the anomaly cancellation condition (3.1a):
∂3Qh(x)
∂xl∂xm∂xn
=
∑
j
∑
ρj∈∆j
ρjl ρ
j
mρ
j
n = 0.
Moreover, we have the identity∑
ρj∈∆j
ρjl = 0, (4.11)
4A stronger version of (4.5) implies that the expression (4.10) for Lh should be corrected on S(β)
g . However,
the correction is of order one only in an O(e−1/β) neighborhood of Sg. In particular, the corrected Lh diverges
on Sg, because the integrand of the EHI vanishes there.
14 A.A. Ardehali
which follows from considering the action of a Weyl reflection, with respect to the hyperplane
perpendicular to some simple root αs ∈ ∆G, on
〈 ∑
ρj∈∆j
ρj ·αs
〉
; the reflection only permutes the
weights ρj , but negates αs, and the completeness of the simple roots αs as a basis for the weight
space establishes (4.11). Then we learn that Qh is stationary at the origin:
∂Qh(x)
∂xl
|x=0 =
1
12
∑
j
∑
ρj∈∆j
ρjl = 0.
It is easy to verify that Qh(x) has a continuous first derivative. Also, Qh(x) is odd under
x→ −x, and vanishes at x = 0; these properties follow from the fact that the function κ(x)
defined in (4.3) is a continuous odd function of its argument. As a result of its oddity, Qh(x) iden-
tically vanishes if the set of all the non-zero weights of all the chiral multiplets in the theory
consists of pairs with opposite signs; a SUSY gauge theory satisfying this condition is called
non-chiral. The EHIs studied by Rains in [35] correspond to non-chiral gauge theories, and thus
for them Qh = 0. For an EHI with non-zero Qh see [2]; Fig. 5 of that work shows the plot of Qh
for that example.
When all xi are small enough, so that the absolute value of all the arguments of the κ functions
in Qh are less than 1, we can use κ(x) = 2x3−3x|x|+x to simplify Qh. The resulting expression –
which equals Qh for xi small enough – can then be considered as defining a function Q̃S3(x) for
any xi ∈ R. Explicitly, we have
Q̃S3(x) = −1
4
∑
j
∑
ρj∈∆j
〈ρj · x〉|〈ρj · x〉|,
with no linear term thanks to the anomaly cancellation condition (3.1a), and no cubic term
because of equation (4.11). In particular, Q̃S3 is homogeneous.
The real function Lh, which we will refer to as the Rains function of the gauge theory,
determines the effective potential V eff(x; b, β). It is piecewise linear ; the quadratic terms in it
cancel thanks to the anomaly cancellation condition (3.1b):
∂2Lh(x)
∂xl∂xm
=
∑
j
(rj − 1)
∑
ρj∈∆j
ρjl ρ
j
m +
∑
α∈∆G
αlαm = 0.
Also, Lh is continuous, is even under x → −x, and vanishes at x = 0; these properties follow
from the properties of the function ϑ(x) defined in (4.2). This function has been analyzed by
Rains [35] in the context of the EHIs associated to SU(N) and Sp(N) SQCD theories. For the
rank two cases considered in [35], this function is plotted in Figs. 2 and 3.
When all xi are small enough, such that the absolute value of the argument of every ϑ function
in Lh is smaller than 1, we can use ϑ(x) = |x| − x2 to simplify Lh. The resulting expression –
which equals Lh for small xi – can then be considered as defining a function L̃S3(x) for any
xi ∈ R. Explicitly, we have
L̃S3(x) =
1
2
∑
j
(1− rj)
∑
ρj∈∆j
|〈ρj · x〉| −
∑
α+∈∆G
|〈α+ · x〉|. (4.12)
Note that there is no quadratic term in L̃S3 , thanks to the consistency condition (3.1b). In
particular, L̃S3 is homogeneous.
The content of this subsection first appeared in [2].
The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals 15
Figure 2. The Rains function of SU(3) SQCD—referred to as the A1 SU(3) theory in [2] – with Nf > 3
flavors.
4.2 The answer for non-chiral theories
In this subsection we explain how the analysis of [2] was improved in [3] for the special class of
non-chiral theories.
Definition 4.2. A non-chiral SUSY gauge theory with U(1) R-symmetry is a SUSY gauge theory
with U(1) R-symmetry (as in Definition 3.1) in which the non-zero weights in ∪j∆j come in
pairs with opposite signs. We denote the positive weights therein by ρ+.
The SQCD theories discussed in Section 3 are examples of non-chiral theories. We refer to
the corresponding EHIs as non-chiral EHIs. All the EHIs studied by Rains in [35] are non-chiral.
In non-chiral theories Θ = Qh = 0, so the analysis of the hyperbolic limit simplifies.
In this subsection we use estimates more precise than the ones in the previous subsection.
Therefore the symbol ' will make an appearance.
The asymptotics of the integrand of (3.2) can be obtained from the estimates in (2.10). With
the aid of (2.7) and (2.10) we find the β → 0 asymptotics of I as
I(b, β) ' 1
|W |
(
2π
β
)rG
e−E
DK
0 (b,β)W0(b)eβEsusy(b)
∫
hcl
drGx e−V
eff(x;b,β)W (x; b, β), (4.13)
with
Esusy(b) =
1
6
(
b+ b−1
2
)3
TrR3 −
(
b+ b−1
2
)(
b2 + b−2
24
)
TrR,
known as the supersymmetric Casimir energy (see, e.g., [8]), and the expressions
TrR := dimG+
∑
j
(rj − 1) dimRj ,
TrR3 := dimG+
∑
j
(rj − 1)3 dimRj , (4.14)
known as the U(1)R-gravity2 and U(1)3
R ’t Hooft anomalies of the SUSY gauge theory. (The
trace is over the chiral fermions in the field theory formulation: in the vector multiplet the
16 A.A. Ardehali
Figure 3. The Rains function of Sp(4) SQCD with Nf > 3.
gaugino has R-cahrge 1, and in a chiral multiplet with R-charge rj the fermion has R-charge
rj − 1.) We have also defined W0(b), and the real function W (x; b, β) via
W0(b) =
∏
j
∏
ρj=0
Γh(rjω), (4.15)
W (x; b, β) =
∏
j
∏
ρj+
ψb
(
−2πi
β {〈ρ
j
+ · x〉}+ (rj − 1) b+b
−1
2
)
ψb
(
−2πi
β {〈ρ
j
+ · x〉} − (rj − 1) b+b
−1
2
)
×
∏
α+
ψb
(
−2πi
β {〈α+ · x〉}+ b+b−1
2
)
ψb
(
−2πi
β {〈α+ · x〉} − b+b−1
2
) . (4.16)
In (4.15), the second product is over the zero weights of Rj (the adjoint representation, for
example, has rG such weights), and ω is defined as ω := i
(
b+ b−1
)
/2. The ρj+ in (4.16) denote
the positive weights of Rj .
That W (x; b, β) is real follows from (2.4) and (2.5).
Our claim in (4.13) that the matrix-integral is approximated well with the integral of its
approximate integrand is justified because the estimates we have used inside the integrand are
uniform and accurate up to exponentially small corrections of the type e−1/β.
Now, from (2.4) it follows that W0(b) is a real number; it is moreover nonzero and finite,
because with the assumption rj ∈ ]0, 2[ the arguments rjω avoid the zeros and poles of the
hyperbolic gamma function as described in Section 2. We would thus make an O
(
β0
)
error in
the asymptotics of log I(b, β) if in (4.13) we set W0(b), along with |W | and eβEsusy(b), to unity.
In other words,
I(b, β) ≈
(
2π
β
)rG
e−E
DK
0 (b,β)
∫
hcl
drGx e−V
eff(x;b,β)W (x; b, β), (4.17)
with an O
(
β0
)
error upon taking the logarithms of the two sides.
We are hence left with the asymptotic analysis of the integral
∫
hcl
e−VW . From here, standard
methods of asymptotic analysis can be employed.
Writing V eff in terms of the Rains function Lh, (4.17) simplifies to
I(b, β) ≈
(
2π
β
)rG
e−E
DK
0 (b,β)
∫
hcl
drGx e
− 4π2
β
(
b+b−1
2
)
Lh(x)
W (x; b, β). (4.18)
The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals 17
It will be useful for us to know that W (x; b, β) is a positive semi-definite function of x; this
follows from (2.4) and (2.5).
The rest of the analysis, leading to our main result in (4.28), involves: i) showing that the
integral in (4.18) localizes around the locus hqu of minima of Lh, so that the leading O(1/β) terms
in (4.28) are justified, and ii) rescaling, in the remaining localized integral, the distance ∆x⊥
from hqu as in ∆x⊥ →
( β
2π
)
∆x⊥, so that rG − dim hqu of the (2π/β) pre-factors of (4.18)
are absorbed through integration, justifying the subleading O
(
log
(
2π
β
))
term in (4.28). The
reader not interested in the details of the derivation is invited to skip the following analysis, and
continue reading from (4.28).
To analyze the integral in (4.18), first note that the integrand is not smooth over hcl. We
hence break hcl into sets on which Lh is linear. These sets can be obtained as follows. Define
Sg :=
⋃
α+
{x ∈ hcl|〈α+ · x〉 ∈ Z}, Sj :=
⋃
ρj+
{x ∈ hcl|〈ρj+ · x〉 ∈ Z},
S :=
⋃
j
Sj ∪ Sg.
It should be clear that everywhere in hcl, except on S, the function Lh is guaranteed to be
linear – and therefore smooth.
The set S consists of a union of codimension one affine hyperplanes inside the space of the xi.
These hyperplanes chop hcl into (finitely many, convex) polytopes Pn. The integral in (4.18)
then decomposes to
I(b, β) ≈ e−EDK
0 (b,β)
∑
n
(
2π
β
)rG ∫
Pn
drGx e
− 4π2
β
(
b+b−1
2
)
Lh(x)
W (x; b, β). (4.19)
Let S(β)
g denote the set of all points in hcl that are at a distance less than N0β from Sg, with
some fixed N0 > 0. We divide Pn into i) Pn ∩ S(β)
g , and ii) the rest of Pn, which we denote
by P ′n. Now, by taking N0 to be large enough, we can push P ′n away from the zeros of ψb, and
thus make wi < W (x; b, β) < ws over P ′n (with some 0 < wi and some ws < ∞). Therefore
the contribution that the nth summand in (4.19) receives from P ′n is well approximated (with
an O
(
β0
)
error upon taking the logs) by
Jn :=
(
2π
β
)rG ∫
P ′n
drGx e
− 4π2
β
(
b+b−1
2
)
Lh(x)
. (4.20)
Let’s further replace P ′n in (4.20) with Pn; we will shortly see that this replacement introduces
a negligible error. We would hence like to estimate
In :=
(
2π
β
)rG ∫
Pn
drGx e
− 4π2
β
(
b+b−1
2
)
Lh(x)
. (4.21)
Since Lh is linear on each Pn, its minimum over Pn is guaranteed to be realized on ∂Pn. Let
us assume that this minimum occurs on the kth j-face of Pn, which we denote by jn-Fkn . We
denote the value of Lh on this j-face by Lnh,min. Equipped with this notation, we can write (4.21)
as
In =
(
2π
β
)rG
e
− 4π2
β
(
b+b−1
2
)
Lnh,min
∫
Pn
drGx e
− 4π2
β
(
b+b−1
2
)
∆Lnh(x)
, (4.22)
where ∆Lnh(x) := Lh(x)−Lnh,min is a linear function on Pn. Note that ∆Lnh(x) vanishes on jn-Fkn ,
and it increases as we go away from jn-Fkn and into the interior of Pn. (The last sentence, as
18 A.A. Ardehali
well as the rest of the discussion leading to (4.28), would receive a trivial modification if jn = rG
(corresponding to constant Lh over Pn).) Therefore as β → 0, the integral in (4.22) localizes
around jn-Fkn .
To further simplify (4.22), we now adopt a set of new coordinates – affinely related to xi and
with unit Jacobian – that are convenient on Pn. We pick a point on jn-Fkn as the new origin,
and parameterize jn-Fkn with x̄1, . . . , x̄jn . We take xin to parameterize a direction perpendicular
to all the x̄s, and to increase as we go away from jn-Fkn and into the interior of Pn. Finally, we
pick x̃1, . . . , x̃rG−jn−1 to parameterize the perpendicular directions to xin and the x̄s. Note that,
because ∆Lnh is linear on Pn, it does not depend on the x̄s; they parameterize its flat directions.
By re-scaling x̄, xin, x̃ 7→ β
2π x̄, β
2πxin, β
2π x̃, we can absorb the
(
2π
β
)rG factor in (4.22) into the
integral, and write the result as
In =
∫
2π
β
Pn
djn x̄ dxin drG−jn−1x̃ e−2π
(
b+b−1
2
)
∆Lnh(xin,x̃). (4.23)
To eliminate β from the exponent, we have used the fact that ∆Lnh depends homogenously on
the new coordinates. We are also denoting the re-scaled polytope schematically by 2π
β Pn.
Instead of integrating over all of 2π
β Pn though, we can restrict to xin < ε/β with some small
ε > 0. The reason is that the integrand of (4.23) is exponentially suppressed (as β → 0) for
xin > ε/β. We take ε > 0 to be small enough such that a hyperplane at xin = ε/β, and parallel
to jn-Fkn , cuts off a prismatoid Pnε/β from 2π
β Pn. After restricting the integral in (4.23) to Pnε/β,
the integration over the x̄s is easy to perform. The only potential difficulty is that the range of
the x̄ coordinates may depend on xin and the x̃s. But since we are dealing with a prismatoid, the
dependence is linear, and by the time the range is modified significantly (compared to its O(1/β)
size on the re-scaled j-face 2π
β (jn-Fkn)), the integrand is exponentially suppressed. Therefore we
can neglect the dependence of the range of the x̄s on the other coordinates in (4.23). The integral
then simplifies to
In ≈
(
2π
β
)jn
vol
(
jn-Fkn
) ∫
P̂n
ε/β
dxin drG−jn−1x̃ e−2π
(
b+b−1
2
)
∆Lnh(xin,x̃), (4.24)
where P̂nε/β is the pyramid obtained by restricting Pnε/β to x̄1 = · · · = x̄jn = 0. The logarithms
of the two sides of (4.24) differ by O(β), with the error mainly arising from our neglect of the
possible dependence of the range of the x̄ coordinates in (4.23) on xin and the x̃s. (Recall that
the other error, arising from restricting the integral in (4.23) to Pnε/β, is exponentially small.)
We now take ε→∞ in (4.24). This introduces an exponentially small error, as the integrand
is exponentially suppressed (as β → 0) for xin > ε/β. The resulting integral is strictly positive,
because it is the integral of a strictly positive function. We denote by An the result of the
integral multiplied by vol
(
jn-Fkn
)
. Then In can be approximated as
In ≈ e−
4π2
β
(
b+b−1
2
)
Lnh,min
(
2π
β
)jn
An. (4.25)
We are now in a position to argue Jn ≈ In. If we had integrated over P ′n, then we would end up
with an expression similar to (4.25), in which Lnh,min would be replaced with the minimum of Lh
over P ′n; but since Lh is piecewise linear, the difference between the new minimum and Lnh,min
would be O(β), which translates to an O
(
β0
)
multiplicative difference between Jn and In. Other
sources of difference between Jn and In similarly introduce negligible error; more precisely, we
have log In = log Jn +O
(
β0
)
.
The dominant contribution to I(b,β) comes, of course, from the terms/polytopes whose Lnh,min
is smallest. If these polytopes are labeled by n = n1
∗, n
2
∗, . . . , we can introduce hqu and dim hqu
The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals 19
via
hqu :=
⋃
n∗
jn∗-Fkn∗ , dim hqu := max(jn∗), (4.26)
with jn∗ the dimensions of the j-faces with minimal Lnh,min.
Put colloquially, if hqu has multiple connected components, by dim hqu we mean the dimension
of the component(s) with greatest dimension, while if a connected component consists of several
intersecting flat elements inside hcl, by its dimension we mean the dimension of the flat element(s)
of maximal dimension.
Our final estimate for the contribution to I(b, β) from ∪nP ′n is thus
Be
−EDK
0 (b,β)− 4π2
β
(
b+b−1
2
)
Lh,min
(
2π
β
)dim hqu
, (4.27)
where Lh,min := Ln∗h,min, and B is some positive real number.
We are left with determining the contribution to I(b, β) coming from S(β)
g . Over Pn ∩ S(β)
g ,
the simple estimate W (x; b, β) = O(1) (which follows from the fact that W (x; b, β) is uniformly
bounded on S(β)
g ) suffices for our purposes; we thus learn that the contribution that the inte-
gral (4.19) receives from Pn ∩ S(β)
g is not only positive, but also
O
(∫
2π
β
(
Pn∩S(β)
g
) djn x̄ dxin drG−jn−1x̃ e−2π
(
b+b−1
2
)
∆Lnh(xin,x̃)
)
.
Now, the argument of the O above is nothing but the difference between In and Jn, which we
already argued to be negligible. Thus the contribution to I(b, β) coming from S(β)
g is negligible.
Using the explicit expression (4.7) for EDK
0 (b, β) (which can be rewritten in terms of TrR in
equation (4.14)), and noting that (4.27) is an accurate estimate for I(b, β) up to a multiplicative
factor of order β0 (but not order β), we arrive at the following proposition as our main result.
Proposition 4.3. The Romelsberger index (3.2) of a non-chiral SUSY gauge theory with U(1)
R-symmetry (as in Definition 4.2) has the following asymptotics in the hyperbolic limit:
log I(b, β) = −π
2
3β
(
b+ b−1
2
)
(TrR+ 12Lh,min) + dim hqu log
(
2π
β
)
+O
(
β0
)
, (4.28)
with dim hqu defined precisely in (4.26).
5 The minimization problem
via generalized triangle inequalities
To the order shown in (4.28), the hyperbolic asymptotics of I(b, β) is determined by the minimum
value and the dimension of the locus of minima of the Rains function. Finding these two numbers
involves solving a minimization problem for Lh. This can be done only on a case-by-case basis at
the moment. The tool allowing us to solve the minimization problem is often Lemma 3.2 of [35],
stating that for any sequence of real numbers c1, . . . , cn, d1, . . . , dn, the following inequality
holds:
∑
1≤i,j≤n
ϑ(ci − dj)−
∑
1≤i<j≤n
ϑ(ci − cj)−
∑
1≤i<j≤n
ϑ(di − dj) ≥ ϑ
∑
1≤i≤n
(ci − di)
, (5.1)
20 A.A. Ardehali
with equality iff the sequence can be permuted so that either
{c1} ≤ {d1} ≤ {c2} ≤ · · · ≤ {dn−1} ≤ {cn} ≤ {dn},
or
{d1} ≤ {c1} ≤ {d2} ≤ · · · ≤ {cn−1} ≤ {dn} ≤ {cn}.
The proof can be found in [35].
Re-scaling with ci, di 7→ vci, vdi, taking v → 0+, and using the relation ϑ(vx) = v|x| − v2x2
(which holds for small enough v), Rains obtains the following corollary of (5.1):
∑
1≤i,j≤n
|ci − dj | −
∑
1≤i<j≤n
|ci − cj | −
∑
1≤i<j≤n
|di − dj | ≥
∣∣∣∣∣∣
∑
1≤i≤n
(ci − di)
∣∣∣∣∣∣ , (5.2)
with equality iff the sequence can be permuted so that either
c1 ≤ d1 ≤ c2 ≤ · · · ≤ dn−1 ≤ cn ≤ dn,
or
d1 ≤ c1 ≤ d2 ≤ · · · ≤ cn−1 ≤ dn ≤ cn.
The fact that the inequality (5.2) arise as a corollary of (5.1) justifies the name “generalized
triangle inequality” for the latter.
We now consider four examples of non-chiral EHIs for which the minimization problem of Lh
can be fully addressed.
5.1 SU(Nc) SQCD with Nf > Nc flavors
This theory was described in Section 3. Its corresponding EHI is
INc,Nf (b, β) =
(p; p)Nc−1(q; q)Nc−1
Nc!
∫
dNc−1x
Nc∏
i=1
ΓNf
(
(pq)rf/2z±1
i
)
∏
1≤i<j≤Nc
Γ
(
(zi/zj)±1
) , (5.3)
with rf = 1 − Nc
Nf
, and
Nc∏
i=1
zi = 1. Recall that this is essentially the same EHI as I
(m)
An
of [36]
with n = Nc − 1 and m = Nf −Nc − 1.
The Rains function of the theory is
L
Nc,Nf
h (x1, . . . , xNc−1) = Nf (1− rf )
Nc∑
i=1
ϑ(xi)−
∑
1≤i<j≤Nc
ϑ(xi − xj)
= Nc
∑
i
ϑ(xi)−
∑
1≤i<j≤Nc
ϑ(xi − xj). (5.4)
The xNc in the above expression is constrained by
Nc∑
i=1
xi ∈ Z, although since ϑ(x) is periodic
with period one we can simply replace xNc → −x1 − · · · − xNc−1. For Nc = 3 the resulting
function is illustrated in Fig. 2.
We recommend that the reader convince herself that the Rains function in (5.4) can be easily
written down by examining the integrand of (5.3). Whenever the Romelsberger index (or the
The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals 21
EHI) of a SUSY gauge theory is available in the literature, a similar examination of the integrand
quickly yields the theory’s Lh and Qh functions.
Using Rains’ generalized triangle inequality (5.1), in the special case where di = 0, we find
that the above function is minimized when all xi are zero. Therefore
L
Nc,Nf
h,min = 0, dim h
Nc,Nf
qu = 0.
A similar story applies to the Sp(2N) SQCD theory discussed in Section 3. We invite the
interested reader to reproduce the plot of the Rains function of the Sp(4) SQCD for Nf > 3
shown in Fig. 3. (The Romelsberger index of the Sp(2N) SQCD theories can be found in [17].
Lemma 3.3 of Rains [35] establishes that L
Sp(2N)
h (x) is minimized only at x = 0, for any
Nf > N + 1.)
All-orders asymptotics
A more careful study shows [2, 35]
log INc,Nf (b, β) ∼ −π
2
3β
(
b+ b−1
2
)
TrR+ logZ
Nc,Nf
S3 (b) + βEsusy(b), as β → 0
with
Z
Nc,Nf
S3 (b) =
1
Nc!
∫
dNc−1x
Nc∏
i=1
Γ
Nf
h (rfω ± xi)∏
1≤i<j≤Nc
Γh(±(xi − xj))
,
where ω := (ω1 + ω2)/2, and the integral is over x1, . . . , xNc−1 ∈ ]−∞,∞[.
The content of this subsection is essentially due to Rains [35].
5.2 SO(2N + 1) SQCD with Nf > 2N − 1
For G =SO(n), and nχ = Nf chiral multiplets of R-charge r = 1− n−2
Nf
in the vector representa-
tion of SO(n), we get the SO(n) SQCD theory with Nf flavors. For the R-charges to be greater
than zero, and the gauge group to be semi-simple, we require 0 < n− 2 < Nf .
We consider odd n. The analysis for even n is similar. The EHI of the SO(2N + 1) SQCD
with Nf flavors reads
ISO(2N+1)(b, β) =
(p; p)N (q; q)N
2NN !
ΓNf
(
(pq)r/2
)
×
∫
dNx
N∏
j=1
ΓNf
(
(pq)r/2z±1
j
)
N∏
j=1
Γ
(
z±1
j
) ∏
i<j
(
Γ
(
(zizj)±1
)
Γ
(
(zi/zj)±1
)) . (5.5)
The Rains function of the theory is
L
SO(2N+1)
h (x) = (2N − 2)
N∑
j=1
ϑ(xj)−
∑
1≤i<j≤N
ϑ(xi + xj)−
∑
1≤i<j≤N
ϑ(xi − xj). (5.6)
For the case N = 2, corresponding to the SO(5) theory, this function is illustrated in Fig. 4.
22 A.A. Ardehali
Figure 4. The Rains function of the SO(5) SQCD theory with Nf > 3.
To find the minima of the above function, we need the following result. For −1/2 ≤ xi ≤ 1/2,
(2N − 2)
∑
1≤j≤N
ϑ(xj)−
∑
1≤i<j≤N
ϑ(xi + xj)−
∑
1≤i<j≤N
ϑ(xi − xj)
= 2
∑
1≤i<j≤N
min(|xi|, |xj |)
= 2(N − 1) min(|xi|) + 2(N − 2) min
2
(|xi|) + · · ·+ 2 min
N−1
(|xi|), (5.7)
where min(|xi|) stands for the smallest of |x1|, . . . , |xN |, while min2(|xi|) stands for the next to
smallest element, and so on. To prove (5.7), one can first verify it for N = 2, and then use
induction for N > 2.
Applying (5.7) we find that the Rains function in (5.6) is minimized to zero when one (and
only one) of the xj is nonzero, and the rest are zero. This follows from the fact that max(|xi|)
does not show up on the r.h.s. of (5.7). Therefore we have
L
SO(2N+1)
h,min = 0, dim hSO(2N+1)
qu = 1.
More precise asymptotics
A careful study shows [2]
log ISO(2N+1)(b, β) = −EDK
0 (b, β) + log
(
2π
β
)
+ log Y3d(b) + o(1) as β → 0 (5.8)
where
Y3d(b) =
Γ
Nf
h (ωr)
2N (N − 1)!
×
∫
dN−1x
N−1∏
j=1
Γ
Nf
h (ωr ± xj)
N−1∏
j=1
Γh(±xj)
∏
1≤i<j≤N−1
(Γh(±(xi + xj))Γh(±(xi − xj)))
, (5.9)
with the integral over x1, . . . , xN−1 ∈ ]−∞,∞[.
The hyperbolic reduction (5.9) of the EHI (5.5) is unusual, compared with the cases studied
by Rains [35] for which the hyperbolic reduction has essentially the same integrand as the elliptic
integral but with hyperbolic gamma functions replacing elliptic gamma functions.
The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals 23
Figure 5. The building block of the puncture-less SU(2) class-S theories.
All-orders asymptotics for the SO(3) theory with Nf = 2 when b = 1
Interestingly, for this case an indirect approach (through the machinery of [7]) can be used to
complete the asymptotic expansion in (5.8) to all orders, with the result reading [2]
log ISO(3)(β) ∼ log
(
π
2β
− 1
2π
)
+
3
8
β as β → 0.
Note that for the SO(3) theory with Nf = 2 we have TrR = 0, so that the generically-
leading O(1/β) term (i.e., −EDK
0 (b, β)) vanishes.
The content of this subsection first appeared in [2].
5.3 Puncture-less SU(2) class-S quivers
An interesting class of SUSY gauge theories with U(1) R-symmetry arises from quiver gauge
theories associated to Riemann surfaces of genus g ≥ 2, without punctures. They are referred
to as class-S theories, and they bridge EHIs to topological QFT on Riemann surfaces [21]. (See
also [49] for a discussion of these theories (as A1 theories of class S) in the context of the
celebrated AGT correspondence, and [6] for a discussion of their Romelsberger index.)
These quivers can be constructed from fundamental blocks of the kind shown in Fig. 5. The
triangle in Fig. 5 represents eight chiral multiplets of R-charge 2/3 transforming in the tri-
fundamental representation of the three SU(2) gauge groups represented by the (semi-circular)
nodes; more precisely, when two semi-circular nodes are connected together to form a circle,
they represent an SU(2) vector multiplet along with a chiral multiplet with R-charge 2/3 in the
adjoint of that SU(2). A class-S theory of genus g arises when 2g − 2 of these blocks are glued
back-to-back (and forth-to-forth) along a straight line, with the leftmost and the rightmost blocks
having two of their half-circular nodes glued together; the gauge group G is then SU(2)3(g−1).
Fig. 6 shows the genus three example, which has six nodes and thus G = SU(2)6.
The Romelsberger index of the genus g theory takes the form:
ISg(b, β) =
(p; p)3(g−1)(q; q)3(g−1)
23(g−1)
∫
d3(g−1)x∆block(x1, x1, x2)∆block(x2, x3, x4)
×∆block(x3, x4, x5) · · ·∆block(x3g−4, x3(g−1), x3(g−1)),
where
∆block(x1, x2, x3) := Γ
(
(pq)1/3z±1
1 z±1
2 z±1
3
)Γ
(
(pq)1/3
)3∏
i
Γ
(
(pq)1/3z±2
i
)
∏
i
Γ(z±2
i )
,
with zi = e2πixi .
24 A.A. Ardehali
Figure 6. The quiver diagram of the g = 3 class-S theory.
An SU(2) vector multiplet along with its accompanying adjoint chiral multiplet contribute
to the Rains function of the theory as
Lnode
h (x) = −2
3
ϑ(2x).
A semi-circular node contributes half as much, and thus the three semi-circular nodes in Fig. 5
contribute together as
Lthree semi-nodes
h (x, y, z) = −1
3
(ϑ(2x) + ϑ(2y) + ϑ(2z)) . (5.10)
The eight chiral multiplets represented by the triangle in Fig. 5 contribute to the Rains
function of the theory as
Ltriangle
h (x, y, z) =
1
3
(ϑ(x+ y + z) + ϑ(x+ y − z) + ϑ(x− y + z) + ϑ(−x+ y + z)) . (5.11)
Adding up (5.10) and (5.11) we obtain the contribution of a single block to the Rains function:
Lblock
h (x, y, z) =
1
3
[ϑ(x+ y + z) + ϑ(x+ y − z) + ϑ(x− y + z) + ϑ(−x+ y + z)
− ϑ(2x)− ϑ(2y)− ϑ(2z)]. (5.12)
With the Rains function of the block (5.12) at hand, we can now write down the Rains
function of genus g class-S theories. For example, the Rains function of the g = 2 theory is
given by
L
Sg=2
h (x1, x2, x3) = Lblock
h (x1, x1, x2) + Lblock
h (x2, x3, x3),
and the Rains function of the g = 3 theory (illustrated in Fig. 6) is obtained as
L
Sg=3
h (x1, x2, x3, x4, x5, x6) = Lblock
h (x1, x1, x2) + Lblock
h (x2, x3, x4)
+ Lblock
h (x3, x4, x5) + Lblock
h (x5, x6, x6).
Importantly, Rains’s GTI (5.1), with c1 = x+ y, c2 = x− y, d1 = z, d2 = −z, implies that
Lblock
h (x, y, z) ≥ 0.
It is not difficult to show that the equality holds in a finite-volume subspace of the x, y, z space;
take for instance x, y, z ≈ .1 within .01 of each other, and use the fact that for small argument Lh
reduces to L̃S3 to show that Lh vanishes in the domain just described.
Since the Rains function of a g ≥ 2 class-S theory is the sum of several block Rains functions,
the positive semi-definiteness of Lblock
h guarantees the positive semi-definiteness of L
Sg≥2
h (xi);
hence
L
Sg≥2
h,min = 0.
Moreover, taking all xi to be around 0.1, and within 0.01 of each other, we can easily conclude
(as in the previous paragraph) that for the genus g theory
dim h
Sg≥2
qu = 3(g − 1).
These results first appeared in [3].
The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals 25
Figure 7. The Rains function of the SU(2) ISS theory.
5.4 The SU(2) ISS model
The Intriligator–Seiberg–Shenker (ISS) model is an SU(2) gauge theory with a single chiral
multiplet of R-charge 3/5 in the four-dimensional (or spin-3/2) representation of the gauge
group.
The Romelsberger index of this theory is (cf. [50])
IISS(b, β) =
(p; p)(q; q)
2
∫
dx
Γ
(
(pq)3/10z±1
)
Γ
(
(pq)3/10z±3
)
Γ
(
z±2
) . (5.13)
The Rains function of the theory is
LISS
h (x) =
2
5
ϑ(x) +
2
5
ϑ(3x)− ϑ(2x).
This function is plotted in Fig. 7.
A direct examination reveals that LISS
h (x) is minimized at x = ±1/3, and LISS
h (±1/3) =
−2/15. Therefore
LISS
h,min = −2/15, dim h
Nc,Nf
qu = 0.
All-orders asymptotics
A careful study shows [2]
log IISS(b, β) ∼ −π
2
3β
(
b+ b−1
2
)(
TrR+ 12LISS
h,min
)
+ log Y ISS
S3 (b) + βEsusy(b) as β → 0
with
Y ISS
S3 (b) =
∫ ∞
−∞
dx′ e−
4π
5
(b+b−1)x′Γh(3x′ + (3/5)ω)Γh(−3x′ + (3/5)ω), (5.14)
and TrR = 7/5. A numerical evaluation using
log Γh(ix; i, i) = (x− 1) log
(
1− e−2πix
)
− 1
2πi
Li2
(
e−2πix
)
+
iπ
2
(x− 1)2 − iπ
12
,
yields Y ISS
S3 (b = 1) ≈ 0.423.
The EHI in (5.13) gives the simplest example known to the author where Lh is minimized
away from the origin. The hyperbolic reduction (5.14) is unusual, compared with the cases
studied by Rains [35] for which the hyperbolic reduction has essentially the same integrand as
the elliptic integral but with hyperbolic gamma functions replacing elliptic gamma functions.
The content of this subsection first appeared in [2].
26 A.A. Ardehali
6 Open problems of physical interest
We take the relation (3.2) as our definition of an EHI. Let us first restrict attention to non-chiral
EHIs: those in which the non-zero weights ρj come in pairs with opposite signs. (All the EHIs
studied by Rains in [35] are non-chiral.)
As mentioned below (4.6), the range of integration in an EHI can be interpreted in the
gauge theory picture as the classical moduli space hcl of holonomies around the circle S1
β of
the background spacetime. The function V eff , and hence also the Rains function Lh (defined
in (4.10), and related to V eff as in (4.8)), can then be interpreted as a quantum effective potential
for the holonomies. The locus of minima of Lh gives the “quantum moduli space” hqu of the
holonomies.
According to our main result (4.28), when the quantum moduli space hqu consists only of the
origin x = 0 (where Lh always vanishes) the hyperbolic asymptotics of the EHI is given by the
Cardy-like formula of [13]:
log I(b, β) = −π
2
3β
(
b+ b−1
2
)
TrR+O
(
β0
)
,
with TrR defined in (4.14). The hyperbolic reduction of such EHIs works as in the cases studied
by Rains [35].5 An example of this relatively simple scenario is provided by the EHI I
(m)
An
of the
SU(Nc) SQCD theory, analyzed in Section 5.1.
For EHIs whose quantum moduli space hqu is extended, but still contains the origin x = 0,
relation (4.28) gives the hyperbolic asymptotics as
log I(b, β) = −π
2
3β
(
b+ b−1
2
)
TrR+ dim hqu log
(
2π
β
)
+O
(
β0
)
.
For such EHIs the hyperbolic reduction is more subtle than in the cases studied by Rains [35].
The EHI (5.5) of the SO(2N + 1) SQCD theory analyzed in Section 5.2 is an example realizing
this scenario. More examples of this kind can be found in [2, Section 3.2]. The EHIs of the
class-S theories, discussed in Section 5.3 above, are also examples of this kind.
Now, the most physically interesting scenario is when Lh is minimized away from the origin,
in which case the hyperbolic asymptotics looks like
log I(b, β) = −π
2
3β
(
b+ b−1
2
)
(TrR+ 12Lh,min) + dim hqu log
(
2π
β
)
+O
(
β0
)
,
and the hyperbolic reduction of the EHI is again more subtle than in the cases studied in [35].
In this last scenario, we can make an analogy with the Higgs mechanism in the standard
model of particle physics. The ISS model of Section 5.4 gives a clear example. Its Rains func-
tion resembles a Mexican-hat potential familiar from the Higgs mechanism. We might roughly
say that in this example an “infinite-temperature Higgs mechanism” moves the quantum moduli
space hqu away from the origin (“infinite-temperature” because the hyperbolic limit is roughly
like the high-temperature limit in the gauge theory picture). In particle physics, Higgs mecha-
nism describes the spontaneous breaking of gauge groups. Analogously we see an SU(2)→ U(1)
breaking of the gauge group as we go from the ISS model’s EHI (5.13) in the form of an SU(2)
matrix-integral, to its hyperbolic reduction (5.14) which is roughly in the form of a U(1) matrix-
integral.
Besides the ISS model, one other SUSY gauge theory with Lh minimized away from the origin
was studied in [2] (more examples have since been studied in [12, 23]); that is the BCI model
5Since Qh is stationary at x = 0, it seems like when hqu = {x = 0} the analysis of the hyperbolic asymptotics
should proceed similarly for chiral theories with non-zero Qh as well.
The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals 27
(cf. [50]), which has G = SO(n) with n > 1, and a single chiral multiplet of R-charge 4/(n+ 2)
in the two-index symmetric traceless tensor representation of the gauge group. The result of [2]
on the hyperbolic reduction for n = 5 shows an SO(5)→ U(1)× SO(3) breaking in that case.
For both the ISS and the BCI model, we have TrR > 0: specifically TrRISS = 7/5 and
TrRBCI = n− 1. Therefore the following problem arises [2, 12].
Problem 6.1. Prove (or disprove) that, in all non-chiral EHIs, Lh,min < 0 only if TrR > 0.
Note that in the statement of the above problem we are writing “only if”; writing “if and only
if” would be incorrect because the class-S theories of Section 5.3 have TrR = 2(g − 1)/3 > 0
but Lh,min = 0.
Our case-by-case study also shows that in SUSY gauge theories whose Lh is positive in a punc-
tured neighborhood of x = 0, Lh is positive semi-definite everywhere, implying (in combination
with Lh(x = 0) = 0) that Lh,min = 0. This brings up another problem.
Problem 6.2. Prove (or disprove) that, for non-chiral EHIs, if the function Lh (and thus L̃S3
as defined in (4.12)) is strictly positive in some punctured neighborhood of the origin, then Lh
is positive semi-definite.
The significance of the above problem arises from the fact that if L̃S3 is strictly positive
in some punctured neighborhood of the origin, then the squashed three-sphere partition func-
tion ZS3(b) of the dimensionally reduced theory is finite [2]. Therefore in such cases no infinity
obviously threatens the simplistic physical intuition (spelled out in the introduction) for the
hyperbolic reduction of the Romelsberger index. On the other hand, in the ISS and BCI models,
the function ZS3(b) diverges, signalling the breakdown of the simplistic physical intuition and
the need for an infinite-temperature Higgs mechanism to save the day. Whether an infinite-
temperature Higgs mechanism can happen even when ZS3(b) is finite, is the question formalized
by the above problem.
Finally, the obvious problem of finding the hyperbolic asymptotics of chiral EHIs remains
open.
Problem 6.3. Find the hyperbolic asymptotics of the master EHI in (3.2), without assuming it
to be non-chiral. In particular, prove (or disprove) Conjecture 4.1.
A Continuous non-R symmetries: flavor fugacities
and R-charge deformations
In this appendix we explain how additional parameters, known as flavor fugacities, can be
incorporated into the EHIs of SUSY gauge theories that have (compact, Lie) symmetries besides
their U(1)R. Moreover, we show that when these extra symmetries include U(1) factors, the R-
charge assignment of the chiral multiplets can be continuously deformed. These are well-known
matters in the community of physicists working on EHIs.
A.1 U(1) flavor symmetries and R-charge deformation
We say a SUSY gauge theory with U(1) R-symmetry (as in Definition 3.1) has a U(1)a flavor
symmetry, if to each of its chiral multiplets {Rj , rj} we can assign a U(1)a flavor charge qj ∈ R,
such that the following anomaly cancellation condition holds:∑
j
qj
∑
ρj∈∆j
ρjl ρ
j
m = 0 for all l, m.
28 A.A. Ardehali
(A special case of such U(1) flavor symmetries is what in the physics literature is called a “bary-
onic” U(1) symmetry.)
Then the following two results are easy to establish.
• A flavor fugacity ua = e2πiβma [wherein we take ma ∈ R] can be incorporated into the
expression (3.2), by simply modifying its numerator gamma functions to Γ
(
(pq)rj/2zρ
j
u
qj
a
)
;
the resulting function I(b, β,ma) is continuous over b, β,ma ∈ ]0,∞[, but not necessarily
real for ma 6= 0. As an example, see the EHI in [44, equation (9.2)], in the special case
where si = (pq)
1
2
− N
(K+1)Nf ua and ti = (pq)
− 1
2
+ N
(K+1)Nf ua.
• The deformed R-charges r′j = rj + λqj , for λ(∈ R) small enough such that r′j ∈ ]0, 2[, can
replace rj and lead to new SUSY gauge theories and new EHIs.
Generalization to several U(1) flavor symmetries is straightforward.
The “non-uniqueness” of R-charges in the presence of U(1) flavor symmetries raises the fol-
lowing question regarding Problem 1 in Section 6: should TrR be strictly positive for the whole
family labeled by λ(s)? Reference [12] conjectures that Lh,min < 0 only if TrR∗ > 0, with ∗ refer-
ring to the specific value of λ(s) for which a := 3 TrR3 −TrR is maximized. (The R-symmetry
chosen through this “a-maximization” procedure can potentially serve as the preferred U(1)R of
the superconformal field theory that the SUSY gauge theory flows to in the infrared [25].)
A.2 Semi-simple flavor symmetries
Let F be a semi-simple matrix Lie group of rank rF . We say a SUSY gauge theory with U(1)
R-symmetry (as in Definition 3.1) has flavor symmetry group F if its chiral multiplets come in
irreducible finite dimensional representations RF1 ,RF2 , . . . of F , such that the following anomaly
cancellation condition holds:∑
j
ρF,jl
∑
ρj∈∆j
ρjmρ
j
n = 0 for all l, m, n.
The weight ρF,j , assigned to the chiral multiplet {Rj , rj}, belongs to ∆F , the set of all the
weights of the representation RFk in which the chiral multiplet sits.
Then flavor fugacities uF,1 = e2πiβmF,1 , . . . , uF,rF = e2πiβmF,rF (wherein we take mF,i ∈ R)
can be incorporated into the expression (3.2), by simply modifying its numerator gamma func-
tions to Γ
(
(pq)rj/2zρ
j
uρ
F,j
F
)
, where uρ
F,j
F stands for u
ρF,j1
F,1 × · · · × u
ρF,jrF
F,rF
; the resulting function
I(b, β,mF,1, . . . ,mF,rF ) is continuous over b, β,mF,1, . . . ,mF,rF ∈ ]0,∞[, but not necessarily real
when some of mF,i are nonzero. As an example, see the EHI in equation (4.3) of [44], but use
uF = (s1, . . . , sNf−1, t1, . . . tNf−1) instead of si, ti.
Generalization to compact F , possibly containing several U(1) flavor symmetries besides
a semi-simple factor, is straightforward.
Acknowledgements
I would like to thank P. Miller and E. Rains for discussions on the mathematical side, as well as
G. Festuccia, J.T. Liu, and P. Szepietowski for discussions on the physical side of the subject.
I am also grateful to the anonymous referees, whose informative comments and constructive feed-
backs on a draft of this manuscript has contributed significantly to its subsequent improvement.
This work was supported in part by the National Elites Foundation of Iran.
The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals 29
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1 Introduction
2 The required special functions and their asymptotics
3 Elliptic hypergeometric integrals from supersymmetric gauge theory
3.1 How a SUSY gauge theory with U(1) R-symmetry gives an EHI
3.2 How SUSY dualities lead to transformation identities for EHIs
4 The rich structure in the hyperbolic limit
4.1 A conjecture for the general case
4.2 The answer for non-chiral theories
5 The minimization problem via generalized triangle inequalities
5.1 SU(Nc) SQCD with Nf>Nc flavors
5.2 SO(2N+1) SQCD with Nf>2N-1
5.3 Puncture-less SU(2) class-S quivers
5.4 The SU(2) ISS model
6 Open problems of physical interest
A Continuous non-R symmetries: flavor fugacities and R-charge deformations
A.1 U(1) flavor symmetries and R-charge deformation
A.2 Semi-simple flavor symmetries
References
|
| id | nasplib_isofts_kiev_ua-123456789-209529 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T14:44:27Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ardehali, A.A. 2025-11-24T10:42:40Z 2018 The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals Arising in Supersymmetric Gauge Theory / A.A. Ardehali // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 50 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D67; 33E05; 41A60; 81T13; 81T60 arXiv: 1712.09933 https://nasplib.isofts.kiev.ua/handle/123456789/209529 https://doi.org/10.3842/SIGMA.2018.043 The purpose of this article is to demonstrate that i) the framework of elliptic hypergeometric integrals (EHIs) can be extended by input from supersymmetric gauge theory, and ii) analyzing the hyperbolic limit of the EHIs in the extended framework leads to a rich structure containing sharp mathematical problems of interest to supersymmetric quantum field theorists. Both of the above items have already been discussed in the theoretical physics literature. Item i was demonstrated by Dolan and Osborn in 2008. Item ii was discussed in the present author's Ph.D. Thesis in 2016, wherein crucial elements were borrowed from the 2006 work of Rains on the hyperbolic limit of certain classes of EHIs. This article contains a concise review of these developments, along with minor refinements and clarifying remarks, written mainly for mathematicians interested in EHIs. In particular, we work with a representation-theoretic definition of a supersymmetric gauge theory, so that readers without any background in gauge theory - but familiar with the representation theory of semi-simple Lie algebras - can follow the discussion. I would like to thank P. Miller and E. Rains for discussions on the mathematical side, as well as G. Festuccia, J.T. Liu, and P. Szepietowski for discussions on the physical side of the subject. I am also grateful to the anonymous referees, whose informative comments and constructive feedback on a draft of this manuscript have contributed significantly to its subsequent improvement. This work was supported in part by the National Elites Foundation of Iran. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals Arising in Supersymmetric Gauge Theory Article published earlier |
| spellingShingle | The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals Arising in Supersymmetric Gauge Theory Ardehali, A.A. |
| title | The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals Arising in Supersymmetric Gauge Theory |
| title_full | The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals Arising in Supersymmetric Gauge Theory |
| title_fullStr | The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals Arising in Supersymmetric Gauge Theory |
| title_full_unstemmed | The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals Arising in Supersymmetric Gauge Theory |
| title_short | The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals Arising in Supersymmetric Gauge Theory |
| title_sort | hyperbolic asymptotics of elliptic hypergeometric integrals arising in supersymmetric gauge theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209529 |
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