(glM,glN)-Dualities in Gaudin Models with Irregular Singularities
We establish (glM,glN)-dualities between quantum Gaudin models with irregular singularities. Specifically, for any M,N ∈ Z≥1, we consider two Gaudin models: the one associated with the Lie algebra glM, which has a double pole at infinity and N poles, counting multiplicities, in the complex plane, an...
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| Cite this: | (glM,glN)-Dualities in Gaudin Models with Irregular Singularities / B. Vicedo, C. Young // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 36 назв. — англ. |
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| citation_txt | (glM,glN)-Dualities in Gaudin Models with Irregular Singularities / B. Vicedo, C. Young // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 36 назв. — англ. |
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| description | We establish (glM,glN)-dualities between quantum Gaudin models with irregular singularities. Specifically, for any M,N ∈ Z≥1, we consider two Gaudin models: the one associated with the Lie algebra glM, which has a double pole at infinity and N poles, counting multiplicities, in the complex plane, and the same model but with the roles of M and N interchanged. Both models can be realized in terms of Weyl algebras, i.e., free bosons; we establish that, in this realization, the algebras of integrals of motion of the two models coincide. At the classical level, we establish two further generalizations of the duality. First, we show that there is also a duality for realizations in terms of free fermions. Second, in the bosonic realization, we consider the classical cyclotomic Gaudin model associated with the Lie algebra glM and its diagram automorphism, with a double pole at infinity and 2N poles, counting multiplicities, in the complex plane. We prove that it is dual to a non-cyclotomic Gaudin model associated with the Lie algebra sp2N, with a double pole at infinity and M simple poles in the complex plane. In the special case N=1, we recover the well-known self-duality in the Neumann model.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 040, 28 pages
(glM , glN)-Dualities in Gaudin Models
with Irregular Singularities
Benôıt VICEDO † and Charles YOUNG ‡
† Department of Mathematics, University of York, York YO10 5DD, UK
E-mail: benoit.vicedo@gmail.com
‡ School of Physics, Astronomy and Mathematics, University of Hertfordshire,
College Lane, Hatfield AL10 9AB, UK
E-mail: c.a.s.young@gmail.com
Received November 06, 2017, in final form April 27, 2018; Published online May 03, 2018
https://doi.org/10.3842/SIGMA.2018.040
Abstract. We establish (glM , glN )-dualities between quantum Gaudin models with irre-
gular singularities. Specifically, for any M,N ∈ Z≥1 we consider two Gaudin models: the
one associated with the Lie algebra glM which has a double pole at infinity and N poles,
counting multiplicities, in the complex plane, and the same model but with the roles of M
and N interchanged. Both models can be realized in terms of Weyl algebras, i.e., free bosons;
we establish that, in this realization, the algebras of integrals of motion of the two models
coincide. At the classical level we establish two further generalizations of the duality. First,
we show that there is also a duality for realizations in terms of free fermions. Second, in the
bosonic realization we consider the classical cyclotomic Gaudin model associated with the
Lie algebra glM and its diagram automorphism, with a double pole at infinity and 2N poles,
counting multiplicities, in the complex plane. We prove that it is dual to a non-cyclotomic
Gaudin model associated with the Lie algebra sp2N , with a double pole at infinity and M
simple poles in the complex plane. In the special case N = 1 we recover the well-known
self-duality in the Neumann model.
Key words: Gaudin models; dualities; irregular singularities
2010 Mathematics Subject Classification: 17B80; 81R12; 82B23
1 Introduction
Fix a set of N distinct complex numbers {zi}Ni=1 ⊂ C, and an element λ ∈ gl∗M . The quadratic
Hamiltonians of the quantum Gaudin model [12, 13] associated to glM are the following elements
of U(glM )⊗N :
Hi =
∑
j 6=i
N∑
a,b=1
E
(i)
abE
(j)
ba
zi − zj
+
N∑
a,b=1
λ(Eab)E
(i)
ba ,
where {Eab}Ma,b=1 denote the standard basis of glM and E
(i)
ab means Eab in the ith tensor factor.
The Hi belong to a large commutative subalgebra Z ⊂ U(glM )⊗N called the Gaudin [11] or
Bethe [19] subalgebra, for which an explicit set of generators is known [6, 19, 31].
If the element λ ∈ gl∗M is regular semisimple, i.e., if we can choose bases such that λ(Eab) =
λaδab for some distinct numbers {λa}Ma=1 ⊂ C, then one can also consider the following elements
of U(glN )⊗M :
H̃a =
∑
b6=a
M∑
i,j=1
Ẽ
(a)
ij Ẽ
(b)
ji
λa − λb
+
M∑
i=1
ziẼ
(a)
ii ,
mailto:benoit.vicedo@gmail.com
mailto:c.a.s.young@gmail.com
https://doi.org/10.3842/SIGMA.2018.040
2 B. Vicedo and C. Young
where
{
Ẽij
}N
i,j=1
denote the standard basis of glN . They belong to a large commutative subal-
gebra Z̃ ⊂ U(glN )⊗M .
Let CM denote the defining representation of glM . Then Z can be represented as a subalgebra
of
End
((
CM
)⊗N) ∼= End
(
CNM
) ∼= End
((
CN
)⊗M)
.
So can Z̃. In fact their images in End
(
CNM
)
coincide. This is the (glM , glN )-duality for
quantum Gaudin models first observed between the quadratic Gaudin Hamiltonians and the
dynamical Hamiltonians in [33], see also [32]. It was later proved in [21], see also [4]. (Under
this realization the Hamiltonians H̃a ∈ Z̃ of the dual model coincide with suitably defined
dynamical Hamiltonians [10] of the original glM Gaudin model. See [20, 21].) The classical
counterpart of this duality goes back to the works of J. Harnad [1, 14].
In this paper we generalize this (glM , glN )-duality in a number of ways, for both the quantum
and classical Gaudin models. Let us describe first the main result. Two natural generalizations
of the Gaudin model above are to
(a) models in which the quadratic Hamiltonians (and the Lax matrix, see below) have higher
order singularities at the marked points zi ∈ C, i = 1, . . . , N . Such models are called Gaudin
models with irregular singularities.1
(b) models in which λ ∈ gl∗M is not semisimple, i.e., has non-trivial Jordan blocks.2
We show that these two generalizations are natural (glM , glN )-duals to one another. Namely, we
show that there is a correspondence among models generalized in both directions, (a) and (b),
and that under this correspondence the sizes of the Jordan blocks get exchanged with the degrees
of the irregular singularities at the marked points in the complex plane. See Theorem 4.8 below.
The heart of the proof is the observation that the generating functions for the generators of
both algebras Z and Z̃ can be obtained by evaluating, in two different ways, the column-ordered
determinant of a certain Manin matrix. (A similar trick was also used in [4, Proposition 8].)
Given that observation, the duality between (a) and (b) above is essentially a consequence of
the simple fact that the inverse of a Jordan block matrix
x 0 . . . 0
−1 x . . . 0
...
. . .
. . .
...
0 . . . −1 x
is of the form
x−1 0 . . . 0
x−2 x−1 . . . 0
...
. . .
. . .
...
x−k . . . x−2 x−1
;
here the higher-order poles in x will give rise to the irregular singularities of the dual Gaudin
model.
Now let us give an overview of the results of the paper in more detail. Consider the direct
sum of Lie algebras
gl
(N)
M :=
N⊕
i=1
glM ⊕ glcom
M , (1.1)
1The reason for this terminology is that the spectrum of such models is described in terms of opers with
irregular singularities; see [9] and also [35]. Strictly speaking, the term λ(Eab)Eab in Hi is already an irregular
singularity of order 2 at∞ in the same sense: namely, the opers describing the spectrum have a double pole at∞.
For that reason we refer to a Gaudin model with such terms in the Hamiltonians Hi as having a double pole at
infinity.
2Let us note in passing that the case of λ semisimple but not regular is very rich; see for example [8, 23, 24].
(glM , glN )-Dualities in Gaudin Models with Irregular Singularities 3
where the Lie algebra glcom
M in the last summand is isomorphic to glM as a vector space but
endowed with the trivial Lie bracket. Henceforth we denote the copy of Eab in the ith direct
summand of gl
(N)
M by E
(zi)
ab and the copy in the last abelian summand glcom
M by E
(∞)
ab . In terms
of these data, the formal Lax matrix of the Gaudin model associated with glM , with a double
pole at infinity and simple poles at each zi, i = 1, . . . , N , is given by
L(z)dz :=
M∑
a,b=1
Eba ⊗
(
E
(∞)
ab +
N∑
i=1
E
(zi)
ab
z − zi
)
dz. (1.2)
Here Eab := ρ(Eab) where ρ : glM → MatM×M (C) is the defining representation.
Regarding L(z) as an M ×M matrix with entries in the symmetric algebra S
(
gl
(N)
M
)
, the
coefficients of its characteristic polynomial
det
(
λ1M×M − L(z)
)
span a large Poisson commutative subalgebra Z cl
(zi)
(
gl
(N)
M
)
of S
(
gl
(N)
M
)
. Given a classical model
described by a Poisson algebra P and Hamiltonian H ∈ P, the latter becomes of particular
interest if we have a homomorphism of Poisson algebras π : S
(
gl
(N)
M
)
→ P such that H lies in the
image of Z cl
(zi)
(
gl
(N)
M
)
. Indeed, π
(
Z cl
(zi)
(
gl
(N)
M
))
⊂ P then consists of Poisson commuting integrals
of motion of the model.
The Lax matrix (1.2) can also be used to describe quantum models by regarding it instead
as an M ×M matrix with entries in the universal enveloping algebra U
(
gl
(N)
M
)
. In this case,
a large commutative subalgebra Z(zi)
(
gl
(N)
M
)
⊂ U
(
gl
(N)
M
)
, called the Gaudin algebra, is spanned
by the coefficients in the partial fraction decomposition of the rational functions obtained as the
coefficients of the differential operator
cdet
(
∂z1M×M − tL(z)
)
,
where cdet is the column ordered determinant. Given a unital associative algebra U and a homo-
morphism π̂ : U
(
gl
(N)
M
)
→ U, the image of Z(zi)
(
gl
(N)
M
)
provides a large commutative subalgebra
of U.
Let U be the Weyl algebra generated by the commuting variables xai for i = 1, . . . , N and
a = 1, . . . ,M together with their partial derivatives ∂ai := ∂/∂xai . We introduce another set
{λa}Ma=1 ⊂ C of M distinct complex numbers. It is well known that
π̂
(
E
(∞)
ab
)
= λaδab, π̂
(
E
(zi)
ab
)
= xai ∂
b
i (1.3)
defines a homomorphism π̂ : U
(
gl
(N)
M
)
→ U. Therefore, in particular, π̂
(
Z(zi)
(
gl
(N)
M
))
is a com-
mutative subalgebra of U. On the other hand, given the new set of complex numbers λa,
a = 1, . . . ,M , we may now equally consider the Gaudin model associated with glN , with a dou-
ble pole at infinity and simple poles at each λa for a = 1, . . . ,M . Its formal Lax matrix is
defined as in (1.2), explicitly we let
L̃(λ)dλ :=
N∑
i,j=1
Ẽji ⊗
(
Ẽ
(∞)
ij +
M∑
a=1
Ẽ
(λa)
ij
λ− λa
)
dλ.
We can define another homomorphism ˆ̃π : U
(
gl
(M)
N
)
→ U as
ˆ̃π
(
Ẽ
(∞)
ij
)
= ziδij , ˆ̃π
(
Ẽ
(λa)
ij
)
= ∂aj x
a
i .
4 B. Vicedo and C. Young
(Note here the order between ∂aj and xai as compared, for instance, to [20, Section 5.1] where Ẽ
(λa)
ij
is realised as xai ∂
a
j .) The (glM , glN )-duality between the above two Gaudin models associated
with glM and glN can be formulated, in the present conventions, as the equality of differential
polynomials
π̂
(
N∏
i=1
(z − zi) cdet
(
∂z1M×M − tL(z)
))
= ˆ̃π
(
M∏
a=1
(∂z − λa) cdet
(
z1N×N − L̃(∂z)
))
,
whose coefficients are U-valued polynomials in z. (See Section 4.2 for the precise definition of the
expression appearing on the right hand side.) In the classical setting discussed above the same
identity holds with ∂z replaced everywhere by the spectral parameter λ, the Weyl algebra U is
replaced by the Poisson algebra P defined as the polynomial algebra in the canonically conjugate
variables (pai , x
a
i ) and column ordered determinants replaced by ordinary determinants.
We generalise this statement in a number of directions. Firstly, in both the classical and
quantum cases, we consider Gaudin models with irregular singularities. Specifically, fix a positive
integer n ∈ Z≥1 and let {τi}ni=1 ⊂ Z≥1 be such that
n∑
i=1
τi = N . We consider a glM -Gaudin model
with a double pole at infinity and an irregular singularity of order τi at each zi for i = 1, . . . , n.
The direct sum of Lie algebras (1.1) is replaced in this case by a direct sum of Takiff Lie algebras3
glDM :=
n⊕
i=1
(
glM [ε]/ετiglM [ε]
)
⊕ glcom
M , (1.4)
where D is a divisor encoding the collection of points zi for i = 1, . . . , n weighted by the integers τi
for i = 1, . . . , n. The formal Lax matrix L(z) of this Gaudin model is an M ×M matrix with
entries in the Lie algebra glDM , and the Gaudin algebra Z(zi)(gl
D
M ) is spanned by the coefficients
in the partial fraction decomposition of the rational functions obtained as the coefficients of the
differential operator
cdet
(
∂z1M×M − tL(z)
)
. (1.5)
Let U be the same unital associative algebra as above. In order to define a suitable homomor-
phism π̂ : U
(
glDM
)
→ U we combine representations of the Takiff Lie algebras glM [ε]/ετiglM [ε]→
U for each i = 1, . . . , n, naturally generalising the representation glM → U, Eab 7→ xai ∂
b
i in the
above regular singularity case, together with a constant homomorphism glcom
M → C1 ⊂ U. As be-
fore, the choice of the latter is what determines the position of the poles of the dual glN -Gaudin
model. In fact, if instead of choosing a diagonal matrix as in (1.3) we let
(
π̂
(
E
(∞)
ab
))M
a,b=1
=
λ1
1 λ1 0
. . .
. . .
1 λ1
. . .
λm
1 λm
0
. . .
. . .
1 λm
3These were introduced in the mathematics literature in [29] but have also been widely used in the mathematical
physics literature though not by this name, see for instance [22].
(glM , glN )-Dualities in Gaudin Models with Irregular Singularities 5
be a direct sum of m Jordan blocks of size τ̃a ∈ Z≥1 with λa ∈ C along the diagonal for
a = 1, . . . ,m, such that
m∑
a=1
τ̃a = M , then the dual Gaudin model associated with glN will have
a double pole at infinity and an irregular singularity at each λa of order τ̃a for a = 1, . . . ,m.
Let D̃ be the divisor corresponding to these data and glD̃N the associated direct sum of Takiff
algebras, cf. (1.4). After defining a corresponding homomorphism ˆ̃π : U
(
glD̃N
)
→ U for this
Gaudin model, we prove a (glM , glN )-duality similar to the one stated above for the regular
singularity case, see Theorem 4.8. As before, a similar result also holds in the classical setting
where π and π̃ in this case are homomorphisms from the symmetric algebras S
(
glDM
)
and S
(
glD̃N
)
,
respectively, to the Poisson algebra P, see Theorem 3.2.
In the classical setup of Section 3 we also consider fermionic generalisations of (glM , glN )-
duality. Specifically, for the Poisson algebra P we take instead the even part of the Z2-graded
Poisson algebra generated by canonically conjugate Grassmann variable pairs (πai , ψ
a
i ). The
corresponding homomorphisms of Poisson algebras πf : S
(
glDM
)
→ P and π̃f : S
(
glD̃N
)
→ P are
defined in Lemma 3.3. In this case we establish a different type of (glM , glN )-duality between
the same Gaudin models with irregular singularities and associated with glM and glN as above.
Denoting by L(z) and L̃(λ) their respective Lax matrices, it takes the form
πf
(
det
(
λ1M×M − L(z)
))
π̃f
(
det
(
z1N×N − L̃(λ)
))
=
n∏
i=1
(z − zi)τi
m∏
a=1
(λ− λa)τ̃a .
See Theorem 3.4, the proof of which is completely analogous to that of Theorem 3.2 in the
bosonic setting, using basic properties of the Berezinian of an (M |N)× (M |N) supermatrix. We
leave the possible generalisation of such a fermionic (glM , glN )-duality to the quantum setting
for future work.
Finally, in Section 5 we consider extensions of these results to cyclotomic Gaudin models
also in the classical setting. Specifically, we consider a Z2-cyclotomic glM -Gaudin model with
a double pole at infinity as usual and with irregular singularities at the origin of order τ0
and at points zi ∈ C×, with disjoint orbits under z 7→ −z, of order τi for each i = 1, . . . , n.
Let N = τ0 +
n∑
i=1
τi. Using the bosonic Poisson algebra P generated by canonically conjugate
variables (pai , x
a
i ) we prove that this model is dual to a Gaudin model associated with the Lie
algebra sp2N , with a double pole at infinity and regular singularities at M points λa, a =
1, . . . ,M , see Theorem 5.2. We show that the well know self-duality in the Neumann model
is a particular example of the latter with N = 1. Generalisations of such (glM , glN )-dualities
involving cyclotomic Gaudin models to the quantum case are less obvious since it is known [34]
that in this case the cyclotomic Gaudin algebra is not generated by a cdet-type formula as
in (1.5), see Remark 5.3.
2 Gaudin models with irregular singularities
2.1 Lie algebras glDM and glD̃N
Let M,N ∈ Z≥1. Denote by Eab for a, b = 1, . . . ,M the standard basis of glM and by Ẽij for
i, j = 1, . . . , N the standard basis of glN .
Let zi ∈ C for i = 1, . . . , n and λa ∈ C for a = 1, . . . ,m be such that zi 6= zj for i 6= j and
λa 6= λb for a 6= b. Pick and fix integers τi ∈ Z≥1 for each i = 1, . . . , n and τ̃a ∈ Z≥1 for each
a = 1, . . . ,m. We call these the Takiff degrees at zi and λa, respectively. Consider the effective
6 B. Vicedo and C. Young
divisors
D =
n∑
i=1
τi · zi + 2 · ∞, D̃ =
m∑
a=1
τ̃a · λa + 2 · ∞.
(Recall that an effective divisor is a finite formal linear combination of points in some Riemann
surface, here the Riemann sphere C ∪ {∞}, with coefficients in Z≥0.)
We require that degD = N + 2 and deg D̃ = M + 2 or in other words,
n∑
i=1
τi = N and
m∑
a=1
τ̃a = M.
Note that if τi = 1 = τ̃a for all i = 1, . . . , n and a = 1, . . . ,m then in fact we have n = N and
m = M . More generally, it will be convenient to break up the list of integers from 1 to N into n
blocks of sizes τi, i = 1, . . . , n, and similarly for the list of integers from 1 to M . To that end,
let us define
νi :=
i−1∑
j=1
τj , and ν̃a :=
a−1∑
b=1
τ̃b (2.1)
for i = 1, . . . , N and a = 1, . . . ,M , so that
(1, . . . , N) = (1, . . . , τ1; ν2 + 1, . . . , ν2 + τ2; . . . ; νn + 1, . . . , νn + τn),
(1, . . . ,M) = (1, . . . , τ̃1; ν̃2 + 1, . . . , ν̃2 + τ̃2; . . . ; ν̃m + 1, . . . , ν̃m + τ̃m).
Note that ν1 = ν̃1 = 0.
Let glM [ε] := glM ⊗ C[ε] denote the Lie algebra of polynomials in a formal variable ε with
coefficients in glM . For any k ∈ Z≥1 we have the ideal εkglM [ε] := glM ⊗ εkC[ε]. The corre-
sponding quotient glM [ε]/εk := glM [ε]/εkglM [ε] is called a Takiff Lie algebra over glM . When
k ∈ Z≥2, for every n ∈ Z≥1 with n < k we have a non-trivial ideal in glM [ε]/εk given by
εnglM [ε]/εk := εnglM [ε]/εkglM [ε], which by abuse of terminology we shall also refer to as a Ta-
kiff Lie algebra. We define direct sums of Takiff Lie algebras over glM and glN , respectively,
as
glDM := ε∞glM [ε∞]/ε2
∞ ⊕
n⊕
i=1
glM [εzi ]/ε
τi
zi ,
glD̃N := ε̃∞glN [ε̃∞]/ε̃2
∞ ⊕
m⊕
a=1
glN [ε̃λa ]/ε̃τ̃aλa .
Note that ε∞glM [ε∞]/ε2
∞ and ε̃∞glN [ε̃∞]/ε̃2
∞ are respectively isomorphic to the abelian Lie
algebras glcom
M and glcom
N in the notation used in the introduction, see, e.g., (1.1).
We use the abbreviated notation Xεk for an element X ⊗ εk ∈ glM [ε] where X ∈ glM and
k ∈ Z≥0, and likewise for elements of glN [ε]. Fix a basis of glDM defined by
E
(zi)
ab[r]
:= Eabε
r
zi , E
(∞)
ab[1]
:= Eabε∞
for i = 1, . . . , N , a, b = 1, . . . ,M and r = 0, . . . , τi− 1. Let us note, in particular, that E
(zi)
ab[r] = 0
whenever r ≥ τi. Likewise, as a basis of glD̃N we take
Ẽ
(λa)
ij[s]
:= Ẽij ε̃
s
λa , Ẽ
(∞)
ij[1]
:= Ẽij ε̃∞
for a = 1, . . . ,M , i, j = 1, . . . , N and s = 0, . . . , τ̃a − 1. Here also Ẽ
(λa)
ij[s] = 0 for s ≥ τ̃a.
(glM , glN )-Dualities in Gaudin Models with Irregular Singularities 7
The set of non-trivial Lie brackets of these basis elements read[
E
(zi)
ab[r],E
(zj)
cd[s]
]
= δij [Eab,Ecd]
(zi)
[r+s] = δijδbcE
(zi)
ad[r+s] − δijδadE
(zi)
cb[r+s], (2.2)
for any i, j = 1, . . . , n and a, b, c, d = 1, . . . ,M , and[
Ẽ
(λa)
ij[r] , Ẽ
(λb)
kl[s]
]
= δab[Ẽij , Ẽkl]
(λa)
[r+s] = δabδjkẼ
(λa)
il[r+s] − δabδilẼ
(λa)
kj[r+s],
for any i, j, k, l = 1, . . . , N and a, b = 1, . . . ,m. Note, in particular, that E
(∞)
ab[1] and Ẽ
(∞)
ij[1] are
Casimirs of the Lie algebras glDM and glD̃N , respectively.
2.2 Lax matrices
Let ρ : glM → MatM×M (C) and ρ̃ : glN → MatN×N (C) denote the defining representations
of glM and glN , respectively. We write Eab := ρ(Eab) and Ẽij := ρ̃
(
Ẽij
)
.
The sets {Eab}Ma,b=1 and {Eba}Ma,b=1 form dual bases of glM with respect to the trace in the
representation ρ since tr(EabEcd) = δadδbc for all a, b, c, d = 1, . . . ,M . Likewise, dual bases of glN
with respect to the trace in the representation ρ̃ are given by
{
Ẽij
}N
i,j=1
and
{
Ẽji
}N
i,j=1
.
The Lax matrix of the Gaudin model associated with glDM is given by
LD(z)dz :=
M∑
a,b=1
Eba ⊗
E
(∞)
ab[1] +
n∑
i=1
τi−1∑
r=0
E
(zi)
ab[r]
(z − zi)r+1
dz. (2.3a)
It is an M ×M matrix whose coefficients are rational functions of z valued in glDM . Likewise,
the Lax matrix of the Gaudin model associated with glD̃N reads
LD̃(λ)dλ :=
N∑
i,j=1
Ẽji ⊗
Ẽ
(∞)
ij[1] +
m∑
a=1
τ̃a−1∑
s=0
Ẽ
(λa)
ij[s]
(λ− λa)s+1
dλ, (2.3b)
and is an N ×N matrix with entries rational functions of λ valued in glD̃N .
3 Classical (glM , glN)-duality
3.1 Classical Gaudin model
The algebra of observables of the classical Gaudin model associated with glDM is the symmetric
tensor algebra S
(
glDM
)
. It is a Poisson algebra: the Poisson bracket is defined to be equal to the
Lie bracket (2.2) on the subspace glDM ↪→ S
(
glDM
)
and then extended by the Leibniz rule to the
whole of S
(
glDM
)
. Consider the quantity
n∏
i=1
(z − zi)τi det
(
λ1M×M − LD(z)
)
. (3.1)
This is a polynomial of degree M in λ whose coefficients are rational functions in z with coeffi-
cients in S
(
glDM
)
. The classical Gaudin algebra Z cl
(
glDM
)
of the glDM -Gaudin model is by defi-
nition the linear subspace of S
(
glDM
)
spanned by these coefficients. It is a Poisson-commutative
subalgebra of S
(
glDM
)
.
The classical Gaudin algebra Z cl
(
glD̃N
)
of the glD̃N -Gaudin model is defined analogously in
terms of the following polynomial of degree N in z with coefficients rational in λ,
m∏
a=1
(λ− λa)τ̃a det
(
z1N×N − LD̃(λ)
)
. (3.2)
8 B. Vicedo and C. Young
3.2 Bosonic realisation
Introduce the Poisson algebra Pb := C
[
xai , p
b
j
]N M
i,j=1 a,b=1
with Poisson brackets{
xai , x
b
j
}
= 0,
{
pai , x
b
j
}
= δijδab,
{
pai , p
b
j
}
= 0, (3.3)
for a, b = 1, . . . ,M and i, j = 1, . . . , N . In the following we shall regard Pb as a Lie algebra
under the Poisson bracket.
For any x ∈ C and k ∈ Z≥1 we denote by Jk(x) the Jordan block of size k × k with x along
the diagonal and −1’s below the diagonal, namely
Jk(x) =
x 0 . . . 0
−1 x . . . 0
...
. . .
. . .
...
0 . . . −1 x
.
We note for later that if x 6= 0 then this is invertible and its inverse is given by
Jk(x)−1 =
x−1 0 . . . 0
x−2 x−1 . . . 0
...
. . .
. . .
...
x−k . . . x−2 x−1
. (3.4)
Lemma 3.1. The linear maps πb : glDM → Pb and π̃b : glD̃N → Pb defined by
πb
(
E
(zi)
ab[r]
)
=
νi+τi−r∑
u=νi+1
xau+rp
b
u, πb
(
E
(∞)
ab[1]
)
= −
(
m⊕
c=1
Jτ̃c(−λc)
)
ba
,
for every r = 0, . . . , τi − 1, i = 1, . . . , n and a, b = 1, . . . ,M , and
π̃b
(
Ẽ
(λa)
ij[s]
)
=
ν̃a+τ̃a−s∑
u=ν̃a+1
puj x
u+s
i , π̃b
(
Ẽ
(∞)
ij[1]
)
= −
(
n⊕
k=1
Jτk(−zk)
)
ji
,
for every s = 0, . . . , τ̃a − 1, i, j = 1, . . . , N and a = 1, . . . ,m, are homomorphisms of Lie
algebras. They extend uniquely to homomorphisms of Poisson algebras πb : S
(
glDM
)
→ Pb and
π̃b : S
(
glD̃N
)
→ Pb.
Proof. We will prove the corresponding result in the quantum case in detail below. See
Lemma 4.7. That proof applies line-by-line here, with ∂ replaced by p. �
Let C(z)[λ] denote the algebra of polynomials in λ with coefficients rational in z. Given any
Poisson algebra P we introduce the Poisson algebra P(z)[λ] := P⊗C(z)[λ] with Poisson bracket
defined using multiplication in the second tensor factor. Extend the homomorphisms πb and π̃b
from Lemma 3.1 to homomorphisms of Poisson algebras
πb : S
(
glDM
)
(z)[λ] −→ Pb(z)[λ], π̃b : S
(
glD̃M
)
(λ)[z] −→ Pb(λ)[z],
by letting them act trivially on the tensor factors C(z)[λ] and C(λ)[z], respectively. In particular,
we may apply these homomorphisms respectively to the expressions (3.1) and (3.2). It follows
from Theorem 3.2 below that the resulting expressions in fact live in the common subalgebra
Pb[z, λ] := Pb ⊗ C[z, λ] of both Pb(z)[λ] and Pb(λ)[z], where C[z, λ] denotes the algebra of
(glM , glN )-Dualities in Gaudin Models with Irregular Singularities 9
polynomials in the variables z and λ. The coefficients of these polynomials in Pb[z, λ] span the
images of the classical Gaudin algebras in Pb, namely
πb
(
Z cl
(
glDM
))
⊂ Pb and π̃b
(
Z cl
(
glD̃N
))
⊂ Pb,
respectively. The following theorem establishes that these Poisson-commutative subalgebras
of Pb coincide.
Theorem 3.2. We have the following relation
πb
(
n∏
i=1
(z − zi)τi det
(
λ1M×M − LD(z)
))
= π̃b
(
m∏
a=1
(λ− λa)τ̃a det
(
z1N×N − LD̃(λ)
))
,
as an equality in Pb[z, λ].
Proof. Introduce the M ×M and N ×N block diagonal matrices
Λ :=
m⊕
a=1
tJτ̃a(λ− λa), Z :=
n⊕
i=1
Jτi(z − zi).
Also introduce the M ×N matrices
P := (pai )
M N
a=1 i=1, X := (xai )
M N
a=1 i=1.
Consider the block matrix
M :=
(
Λ X
tP Z
)
, (3.5)
with entries in the commutative algebra Pb[λ, z]. We may evaluate its determinant in two ways.
On the one hand, we have
detM = det
(
M
(
1 −Λ−1X
0 1
))
= det
(
Λ 0
tP Z − tPΛ−1X
)
= det Λ det
(
Z − tPΛ−1X
)
.
On the other hand,
detM = det
(
Z tP
X Λ
)
= det
((
Z tP
X Λ
)(
1 −Z−1 tP
0 1
))
= det
(
Z 0
X Λ−XZ−1 tP
)
= detZ det
(
Λ−XZ−1 tP
)
.
Hence we obtain the relation
detZ det
(
Λ−XZ−1 tP
)
= det Λ det
(
Z − tPΛ−1X
)
. (3.6)
It remains to note that the square matrices Z and Λ can be written as
Z =
N∑
i,j=1
Ẽij
(
zδij − πb
(
Ẽ
(∞)
ij[1]
))
, Λ =
M∑
a,b=1
Eab
(
λδab − πb
(
E
(∞)
ba[1]
))
10 B. Vicedo and C. Young
with πb and π̃b as defined in Lemma 3.1, and that their inverses are given by
Z−1 =
n⊕
i=1
Jτi(z − zi)−1, Λ−1 =
m⊕
a=1
tJτ̃a(λ− λa)−1.
Thus we have
Λ−XZ−1 tP =
M∑
a,b=1
Eab
(
Λ−XZ−1 tP
)
ab
= λ1−
M∑
a,b=1
Eab
πb(E(∞)
ab[1]
)
+
n∑
i=1
νi+τi∑
j,k=νi+1
xaj
(
Jτi(z − zi)−1
)
jk
pbk
,
which is nothing but λ1 − πb
(
tLD(z)
)
using Lemma 4.7, the expression (2.3a) for the Lax
matrix LD(z) and (3.4) for the inverse of a Jordan block. Likewise
Z − tPΛ−1X =
N∑
i,j=1
Ẽij
(
Z − tPΛ−1X
)
ij
= z1−
N∑
i,j=1
Ẽij
π̃b(Ẽ(∞)
ji[1]
)
+
m∑
a=1
ν̃a+τ̃a∑
b,c=ν̃a+1
pbi
(
Jτ̃a(λ− λa)−1
)
cb
xcj
,
which coincides with z1− π̃b
(
L̃(λ)
)
, as required. Since det tA = detA for any square matrix A
and noting that detZ =
n∏
i=1
(z − zi)τi and det Λ =
m∏
a=1
(λ− λa)τ̃a , the result follows. �
3.3 Fermionic realisation
Let V := spanC
{
ψai , π
b
j
}N M
i,j=1 a,b=1
and define the exterior algebra Pf :=
∧
V =
⊕2MN
k=0
∧k V ,
whose skew-symmetric product we denote simply by juxtaposition. We refer to an element
u ∈
∧k V as being homogeneous of degree k and write |u| = k. In particular, |ψai | = |πai | = 1 for
any a = 1, . . . ,M and i = 1, . . . , N . We endow Pf with a Z2-graded Poisson structure defined
by {
πai , ψ
b
j
}
+
=
{
ψbj , π
a
i
}
+
= δijδab,
for any a, b = 1, . . . ,M and i, j = 1, . . . , N , and extended to the whole of Pf by the Z2-graded
skew-symmetry property and the Z2-graded Leibniz rule, i.e.,
{u, v}+ = −(−1)|u||v|{v, u}+,
{u, vw}+ = {u, v}+w + (−1)|u||v|v{u,w}+
for any homogeneous elements u, v, w ∈ Pf .
Let P0̄
f :=
⊕MN
k=0
∧2k V denote the even subspace of Pf . The restriction of the Z2-graded
Poisson bracket {·, ·}+ to P0̄
f defines a Lie algebra structure on P0̄
f .
Lemma 3.3. The linear maps πf : glDM → P0̄
f and π̃f : glD̃N → P0̄
f defined by
πf
(
E
(zi)
ab[r]
)
=
νi+τi−r∑
u=νi+1
πau+rψ
b
u, πf
(
E
(∞)
ab[1]
)
= −
(
m⊕
c=1
Jτ̃c(−λc)
)
ab
,
(glM , glN )-Dualities in Gaudin Models with Irregular Singularities 11
for every i = 1, . . . , n and a, b = 1, . . . ,M , and
π̃f
(
Ẽ
(λa)
ij[s]
)
=
ν̃a+τ̃a−s∑
u=ν̃a+1
ψui π
u+s
j , π̃f
(
Ẽ
(∞)
ij[1]
)
= −
(
n⊕
k=1
Jτk(−zk)
)
ij
,
for every i, j = 1, . . . , N and a = 1, . . . ,m, are homomorphisms of Lie algebras.
Proof. For each i, j = 1, . . . , n and a, b = 1, . . . ,M we have
{
πf
(
E
(zi)
ab[r]
)
, πf
(
E
(zj)
cd[s]
)}
+
=
νi+τi−r∑
u=νi+1
νj+τj−s∑
v=νj+1
{πau+rψ
b
u, π
c
v+sψ
d
v}+
=
νi+τi−r∑
u=νi+1
νi+τi−s∑
v=νi+1
(
πau+r
{
ψbu, π
c
v+s
}
+
ψdv − πcv+s
{
πau+r, ψ
d
v
}
+
ψbu
)
δij
=
νi+τi−r−s∑
u=νi+1
(
δcbπ
a
u+r+sψ
d
u − δadπcu+r+sψ
b
u
)
δij
=
(
δbcπf
(
E
(zi)
ad[r+s]
)
− δadπf
(
E
(zi)
cb[r+s]
))
δij = πf
({
E
(zi)
ab[r],E
(zj)
cd[s]
})
.
Likewise, for each i, j = 1, . . . , N and a, b = 1, . . . ,m one shows that{
π̃f
(
Ẽ
(λa)
ij[r]
)
, π̃f
(
Ẽ
(λb)
kl[s]
)}
+
= π̃f
({
Ẽ
(λa)
ij[r] , Ẽ
(λb)
kl[s]
})
,
and all Poisson brackets involving the generators at infinity are also easily seen to be preserved
by the linear maps πf and π̃f since zi ∈ C and λa ∈ C are central in P0̄
f . �
Theorem 3.4. We have the following relation
πf
(
det
(
λ1M×M − LD(z)
))
π̃f
(
det
(
z1N×N − LD̃(λ)
))
=
n∏
i=1
(z − zi)τi
m∏
a=1
(λ− λa)τ̃a .
Proof. Consider the same M ×M and N ×N block diagonal matrices Z and Λ as in the proof
of Theorem 3.2. Introduce the M ×N and N ×M matrices
Π := (πai )M N
a=1 i=1, Ψ := (ψai )N M
i=1 a=1,
and consider the following even supermatrix
M :=
(
Λ Π
Ψ Z
)
.
Since Λ and Z are both invertible, we can define the Berezinian, or superdeterminant, of M
which is given by BerM = det Λ
(
det
(
Z −ΨΛ−1Π
))−1
. Alternatively, the Berezinian of M can
equally be expressed as BerM = det
(
Λ−ΠZ−1Ψ
)
(detZ)−1, see for instance [2]. Equating these
two expressions of BerM we obtain the relation
det
(
Λ−ΠZ−1Ψ
)
det
(
Z −ΨΛ−1Π
)
= detZ det Λ.
Recalling the expressions for the square matrices Z and Λ and their inverses given in the
proof of Theorem 3.2, we can write
Λ−ΠZ−1Ψ =
M∑
a,b=1
Eab
(
Λ−ΠZ−1Ψ
)
ab
12 B. Vicedo and C. Young
= λ1−
M∑
a,b=1
Eab
πf(E(∞)
ba[1]
)
+
n∑
i=1
νi+τi∑
j,k=νi+1
πaj
(
Jτi(z − zi)−1
)
jk
ψbk
,
which is nothing but λ1− πf
(
LD(z)
)
. Likewise
Z −ΨΛ−1Π =
N∑
i,j=1
Eij
(
Z −ΨΛ−1Π
)
ij
= z1−
N∑
i,j=1
Eij
π̃f(E(∞)
ij[1]
)
+
m∑
a=1
ν̃a+τ̃a∑
b,c=ν̃a+1
ψbi
(
Jτ̃a(λ− λa)−1
)
cb
πcj
,
which is z1− π̃f
(
tLD̃(λ)
)
. The result now follows as in the proof of Theorem 3.2. �
4 Quantum (glM , glN)-duality
There is a natural quantum version of Theorem 3.2. In order to state it, we first need a short
digression on Manin matrices. In this section we do not consider the fermionic counterpart of
Theorem 3.2, namely Theorem 3.4, but leave this for future work.
4.1 Manin matrices
Let A be an associative (but possibly noncommutative) algebra over C. Suppose M = (Mij) is
a matrix with entries in A.
Definition 4.1. The matrix M is a Manin matrix if
(i) [Mij ,Mkj ] = 0 for all i, j, k, and
(ii) [Mij ,Mkl] = [Mkj ,Mil] for all i, j, k, l.
That is, elements of the same column must commute amongst themselves, and commutators of
cross terms of 2×2 submatrices must be equal (for example [M11,M22] = [M21,M12]). Actually
the second of these conditions implies the first (set j = l) but it is convenient to think of them
separately.
In the literature Manin matrices have been also called right quantum matrices [15, 16, 17, 18]
or row-pseudo-commutative matrices [3]. For a review of their properties, and further references,
see [5].
Definition 4.2. The column(-ordered) determinant of an N ×N matrix M is
cdetM :=
∑
σ∈SN
(−1)|σ|Mσ(1)1Mσ(2)2 · · ·Mσ(N)N .
Lemma 4.3. The column determinant cdetM changes only by a sign under the exchange of
any two rows of M . If M is Manin, then cdetM also changes only by a sign under the exchange
of any two columns of M .
Proof. The first part is manifest. See [5, Section 3.4] for the second. �
Proposition 4.4. Let M be an N×N Manin matrix with coefficients in A. Let X be a k×(N−k)
matrix with coefficients in A, for some 0 ≤ k ≤ N . Then
cdetM = cdet
(
M
(
1 X
0 1
))
.
(glM , glN )-Dualities in Gaudin Models with Irregular Singularities 13
Proof. See [5, Section 5.1]. �
This has the following corollary which will be important for us.
Proposition 4.5. Let M =
(
A B
C D
)
be the block form of an N × N Manin matrix with
coefficients in A.
(i) Suppose A is a subalgebra of a (possibly larger) algebra A′ over which A has a right inverse,
i.e., AA−1 = 1 for some matrix A−1 with coefficients in A′. Then
cdetM = cdetA cdet
(
D − CA−1B
)
as an equality in A.
(ii) Suppose A is a subalgebra of a (possibly larger) algebra A′′ over which D has a right inverse,
i.e., DD−1 = 1 for some matrix D−1 with coefficients in A′′. Then
cdetM = cdetD cdet
(
A−BD−1C
)
as an equality in A.
Proof. We work initially over A′. Suppose A has a right inverse. By Proposition 4.4 we have
cdet
(
A B
C D
)
= cdet
((
A B
C D
)(
1 −A−1B
0 1
))
= cdet
(
A 0
C D − CA−1B
)
= cdetA cdet
(
D − CA−1B
)
as an equality in A′. But cdetM belongs to A, so in fact this is an equality in A. This establishes
part (i).
For part (ii) note that, by Lemma 4.3, cdetM is invariant under the exchange of any pair of
rows followed by the exchange of the corresponding pair of columns. So we can rearrange the
blocks to find
cdetM = cdet
(
D C
B A
)
and then argue as for part (i). �
Remark 4.6. The proposition above is the first half of [5, Proposition 10], specifically li-
nes (5.17) and (5.18). The subsequent lines (5.19) and (5.20) appear to contain misprints. For
example, if M =
(
a b
c d
)
is a 2 × 2 Manin matrix with d invertible then cdetM = ad − cb =(
a− cbd−1
)
d =
(
a− cd−1b
)
d whereas [5, line (5.20)] gives cdetM =
(
a− bd−1c
)
d, which is not
in general the same.
4.2 Quantum Gaudin model
The algebra of observables of the quantum Gaudin model associated with glDM is the enveloping
algebra U
(
glDM
)
, equipped with its usual associative product. Let ∂z := ∂
∂z and consider the
same Lax matrix given by (2.3a), as in the classical model we considered above but now regarded
as taking values in glDM ↪→ U
(
glDM
)
. Its transpose is
tLD(z)dz =
M∑
a,b=1
Eab ⊗
E
(∞)
ab[1] +
n∑
i=1
τi−1∑
r=0
E
(zi)
ab[r]
(z − zi)r+1
dz.
14 B. Vicedo and C. Young
Recall the definition of the column-ordered determinant, Definition 4.2, and consider the
quantity
n∏
i=1
(z − zi)τi cdet
(
∂z1M×M − tLD(z)
)
=:
M∑
k=0
Sk(z)∂
k
z . (4.1)
This is a differential operator in z of order M . For each 0 ≤ k ≤M , the coefficient Sk(z) of ∂kz
is a rational function in z valued in U
(
glDM
)
.
The quantum Gaudin algebra Z
(
glDM
)
of the glDM -Gaudin model is by definition the unital
subalgebra of U
(
glDM
)
generated by the coefficients in the partial fraction decomposition of these
rational functions Sk(z). It is a commutative subalgebra of U
(
glDM
)
, [19, 30].4
The quantum Gaudin algebra Z
(
glD̃N
)
of the glD̃N -Gaudin model is defined in exactly the
same way in terms of the N th order differential operator in λ,
m∏
a=1
(λ− λa)τ̃a cdet
(
∂λ1N×N − tLD̃(λ)
)
,
where, cf. (2.3b),
tLD̃(λ)dλ =
N∑
i,j=1
Ẽij ⊗
Ẽ
(∞)
ij[1] +
m∑
a=1
τ̃a−1∑
s=0
Ẽ
(λa)
ij[s]
(λ− λa)s+1
dλ.
There is an automorphism of glDN defined by LD̃(λ) 7→ − tLD̃(λ). The Gaudin algebra is stabilized
by this automorphism. (This statement follows from applying a tensor product of evaluation
homomorphisms of Takiff algebras to the statement of [19, Proposition 8.4]). Therefore we may
equivalently consider the N th order differential operator
m∏
a=1
(λ− λa)τ̃a cdet
(
∂λ1N×N + LD̃(λ)
)
=:
N∑
k=0
S̃k(λ)∂kλ (4.2)
and define the quantum Gaudin algebra Z
(
glD̃N
)
to be the unital subalgebra of U
(
glD̃N
)
generated
by the coefficients in the partial fraction decomposition of the rational functions S̃k(λ) in λ. It
is a commutative subalgebra of U
(
glD̃N
)
.
To state our result on quantum (glM , glN )-duality, it will be convenient to write (4.2) in the
equivalent form
m∏
a=1
(∂z − λa)τ̃a cdet
(
− z1N×N + LD̃(∂z)
)
=
N∑
k=0
S̃k(∂z)(−z)k.
Let us explain the meaning of the expression
cdet
(
− z1N×N + LD̃(∂z)
)
.
The quantity
cdet
(
∂λ1N×N + LD̃(λ)
)
,
4It is shown in [19] that cdet
(
∂z1M×M − Eab ⊗
∞∑
n=0
(Eab ⊗ tn)z−n−1
)
generates a commutative subalgebra
of U(glM [t]). The algebra Z
(
glDM
)
is a homomorphic image of this algebra in U
(
glDM
)
.
(glM , glN )-Dualities in Gaudin Models with Irregular Singularities 15
which appears in (4.2), belongs to the algebra U
(
glD̃N
)
(λ)[∂λ] of differential operators in λ whose
coefficients are rational functions of λ with coefficients in U
(
glD̃N
)
. Here λ and ∂λ can be re-
garded as formal generators obeying the commutation relation [∂λ, λ] = 1. We can relabel
these generators as we wish, provided we preserve this relation. In particular, we may send
(∂λ, λ) 7→ (−z, ∂z), since [−z, ∂z] = 1. Thus cdet
(
− z1N×N + LD̃(∂z)
)
is an element of the
algebra U
(
glD̃N
)
(∂z)[z].
More precisely, we shall be concerned in what follows with the quantity
m∏
a=1
(∂z − λa)τ̃a cdet
(
z1N×N − LD̃(∂z)
)
=
N∑
k=0
(−1)N−kS̃k(∂z)z
k. (4.3)
4.3 Bosonic realisation
We consider realisations of U
(
glDM
)
and U
(
glD̃N
)
acting by differential operators on the polynomial
algebra C[xai ]
N M
i=1 a=1. Namely, let ∂ai := ∂
∂xai
and let us denote by Ub the unital associative algebra
generated by {xai }N M
i=1 a=1 and {∂ai }N M
i=1 a=1 subject to the commutation relations
[xai , x
b
j ] = 0, [∂ai , x
b
j ] = δijδab, [∂ai , ∂
b
j ] = 0,
for a, b = 1, . . . ,M and i, j = 1, . . . , N .
Ub is in particular a Lie algebra, with the Lie bracket given by the commutator.
Lemma 4.7. The linear maps π̂b : glDM → Ub and ˆ̃πb : glD̃N → Ub defined by
π̂b
(
E
(zi)
ab[r]
)
=
νi+τi−r∑
u=νi+1
xau+r∂
b
u, π̂b
(
E
(∞)
ab[1]
)
= −
(
m⊕
c=1
Jτ̃c(−λc)
)
ba
,
for every r = 0, . . . , τi − 1, i = 1, . . . , n and a, b = 1, . . . ,M , and
ˆ̃πb
(
Ẽ
(λa)
ij[s]
)
=
ν̃a+τ̃a−s∑
u=ν̃a+1
∂uj x
u+s
i , ˆ̃πb
(
Ẽ
(∞)
ij[1]
)
= −
(
n⊕
k=1
Jτk(−zk)
)
ji
,
for every s = 0, . . . , τ̃a − 1, i, j = 1, . . . , N and a = 1, . . . ,m, are homomorphisms of Lie
algebras. They extend uniquely to homomorphisms of associative algebras π̂b : U
(
glDM
)
→ Ub
and ˆ̃πb : U
(
glD̃N
)
→ Ub.
Proof. For each i, j = 1, . . . , n and a, b = 1, . . . ,M we have
[
π̂b
(
E
(zi)
ab[r]
)
, π̂b
(
E
(zj)
cd[s]
)]
=
νi+τi−r∑
u=νi+1
νj+τj−s∑
v=νj+1
[
xau+r∂
b
u, x
c
v+s∂
d
v
]
=
νi+τi−r∑
u=νi+1
νi+τi−s∑
v=νi+1
(
xau+r
[
∂bu, x
c
v+s
]
∂dv + xcv+s
[
xau+r, ∂
d
v
]
∂bu
)
δij
=
νi+τi−r−s∑
u=νi+1
(
δbcx
a
u+r+s∂
d
u − δadxcu+r+s∂
b
u
)
δij
=
(
δbcπ̂b
(
E
(zi)
ad[r+s]
)
− δadπ̂b
(
E
(zi)
cb[r+s]
))
δij = π̂b
([
E
(zi)
ab[r],E
(zj)
cd[s]
])
.
In the second equality we have used the fact that if i 6= j then all commutators vanish due to
the restriction in the range of values in the sums over u and v.
16 B. Vicedo and C. Young
Likewise, for all i, j = 1, . . . , N and a, b = 1, . . . ,m we find
[
ˆ̃πb
(
Ẽ
(λa)
ij[r]
)
, ˆ̃πb
(
Ẽ
(λb)
kl[s]
)]
=
ν̃a+τ̃a−r∑
u=ν̃a+1
ν̃b+τ̃b−s∑
v=ν̃b+1
[
∂uj x
u+r
i , ∂vl x
v+s
k
]
=
ν̃a+τ̃a−r∑
u=ν̃a+1
ν̃a+τ̃a−s∑
v=ν̃a+1
(
∂uj
[
xu+r
i , ∂vl
]
xv+s
k + ∂vl
[
∂uj , x
v+s
k
]
xu+r
i
)
δab
=
ν̃a+τ̃a−r−s∑
u=ν̃a+1
(
−δil∂uj xu+r+s
k + δjk∂
u
l x
u+r+s
i
)
δab
=
(
δjk ˆ̃πb
(
Ẽ
(λa)
il[r+s]
)
− δil ˆ̃πb
(
Ẽ
(λa)
kj[r+s]
))
δab = ˆ̃πb
([
Ẽ
(λa)
ij[r] , Ẽ
(λb)
kl[s]
])
,
as required. Moreover, all the commutators involving the generators at infinity are also easily
seen to be preserved by the linear maps π̂b and ˆ̃πb since zi ∈ C and λa ∈ C are central in Ub. �
Given any unital associative algebra U we denote by U[z, ∂z] the tensor product of unital
associative algebras U ⊗ C[z, ∂z]. As in the classical setting of Section 3.2, consider also the
unital associative algebras U(z)[∂z] := U ⊗ C(z)[∂z] and U(∂z)[z] := U ⊗ C(∂z)[z], both con-
taining U[z, ∂z] as a subalgebra. We extend the homomorphisms π̂b and ˆ̃πb from Lemma 4.7 to
homomorphisms of tensor product algebras,
π̂b : U
(
glDM
)
(z)[∂z]→ Ub(z)[∂z], ˆ̃πb : U
(
glD̃M
)
(∂z)[z]→ Ub(∂z)[z],
respectively. Applying these homomorphisms respectively to the expressions given by (4.1)
and (4.3), Theorem 4.8 below shows that the resulting expressions in fact live in the common
subalgebra Ub[z, ∂z]. The coefficients of the resulting differential operators in z span the respec-
tive images of the quantum Gaudin algebras in Ub, namely
π̂b
(
Z
(
glDM
))
⊂ Ub and ˆ̃πb
(
Z
(
glD̃N
))
⊂ Ub.
The following theorem establishes that these commutative subalgebras of Ub coincide.
Theorem 4.8. We have
π̂b
(
n∏
i=1
(z − zi)τi cdet
(
∂z1M×M − tLD(z)
))
= ˆ̃πb
(
m∏
a=1
(∂z − λa)τ̃a cdet
(
z1N×N − LD̃(∂z)
))
,
as an equality of polynomial differential operators in z.
Proof. Introduce the M ×M and N ×N block diagonal matrices
Λ :=
m⊕
a=1
tJτ̃a(∂z − λa), Z :=
n⊕
i=1
Jτi(z − zi).
Also introduce the M ×N matrices
D := (∂ai )M N
a=1 i=1, X := (xai )
M N
a=1 i=1.
Consider the block matrix
M :=
(
Λ X
tD Z
)
,
(glM , glN )-Dualities in Gaudin Models with Irregular Singularities 17
with entries in the noncommutative algebra A := Ub[z, ∂z]. The key observation is that this is
a Manin matrix. Indeed, the only non-trivial check is for the 2× 2 submatrices of the form(
∂z − λa xai
∂ai z − zi
)
and for these we have [∂z − λa, z − zi] = 1 = [∂ai , x
a
i ] as required. This fact means that we can
follow the proof of Theorem 3.2, with suitable modifications, as follows.
The square matrices Z and Λ with entries in C[z, ∂z] ⊂ Ub[z, ∂z] have (two-sided) inverses in
the enlarged algebras A′′ := Ub(z)[∂z] and A′ := Ub(∂z)[z], respectively, both of which contain A
as a subalgebra. These inverses are given explicitly by
Z−1 =
n⊕
i=1
Jτi(z − zi)−1, Λ−1 =
m⊕
a=1
tJτ̃a(∂z − λa)−1.
We are therefore in the setup of Proposition 4.5. We may apply it to evaluate cdetM in two
different ways. We obtain
cdet Λ cdet
(
Z − tDΛ−1X
)
= cdetZ cdet
(
Λ−XZ−1 tD
)
, (4.4)
as an equality in A = Ub[z, ∂z], namely this is an equality of polynomial differential operators
in z with coefficients in Ub.
It remains to evaluate both sides of (4.4) more explicitly. We have
cdetZ =
n∏
i=1
(z − zi)τi , cdet Λ =
m∏
a=1
(∂z − λa)τ̃a ,
where the order of the products on the right of these equalities does not matter. Now Z and Λ
can be written explicitly as follows
Z =
N∑
i,j=1
Ẽij
(
zδij − ˆ̃πb
(
Ẽ
(∞)
ji[1]
))
, Λ =
M∑
a,b=1
Eab
(
∂zδab − π̂b
(
E
(∞)
ab[1]
))
with π̂b and ˆ̃πb as defined in Lemma 4.7. In terms of these expressions we can write
Λ−XZ−1 tD =
M∑
a,b=1
Eab
(
Λ−XZ−1 tD
)
ab
= ∂z1−
M∑
a,b=1
Eab
π̂b(E(∞)
ab[1]
)
+
n∑
i=1
νi+τi∑
j,k=νi+1
xaj
(
Jτi(z − zi)−1
)
jk
∂bk
.
The latter expression is exactly ∂z1− π̂b
(
tLD(z)
)
by virtue of Lemma 4.7, the expression (2.3a)
for the Lax matrix LD(z) and the expression (3.4) for the inverse of a Jordan block. Likewise
Z − tDΛ−1X =
N∑
i,j=1
Ẽij
(
Z − tDΛ−1X
)
ij
= z1−
N∑
i,j=1
Ẽij
ˆ̃πb
(
Ẽ
(∞)
ji[1]
)
+
m∑
a=1
ν̃a+τ̃a∑
b,c=ν̃a+1
∂bi
(
Jτ̃a(∂z − λa)−1
)
cb
xcj
,
which coincides with z1− ˆ̃πb
(
LD̃(∂z)
)
. The result now follows. �
In the special case of no Jordan blocks and no non-trivial Takiff algebras, Theorem 4.8 can
be found in [21]. See also [4, Proposition 8], where it is noted that the relation cdetM =
detZ cdet
(
Λ−XZ−1 tD
)
leads to a relation between the classical spectral curve and the “quan-
tum spectral curve”.
18 B. Vicedo and C. Young
5 Z2-cyclotomic Gaudin models with irregular singularities
Another possible class of generalisations of Gaudin models are those whose Lax matrix is equiv-
ariant under an action of the cyclic group, determined by a choice of automorphism of the Lie
algebra (here glM ). Such models were considered in [25, 26, 27] and in [7] for automorphisms of
order 2, and for automorphisms of arbitrary finite order in [34, 35].
It is natural to ask whether (glM , glN )-dualities also exist, in the sense of Section 3, be-
tween cyclotomic Gaudin models. Theorem 5.2, which can be deduced from the results of [1],
establishes a duality between a cyclotomic glM -Gaudin model associated with the diagram au-
tomorphism of glM and a non-cyclotomic spN -Gaudin model.
5.1 Z2-cyclotomic Lax matrix for the diagram automorphism
Let zi ∈ C for i = 1, . . . , n be such that 0 6= zi 6= ±zj for i 6= j. Pick and fix integers τi ∈ Z≥1
for i = 0 and for each i = 1, . . . , n. Consider the effective divisor
C = 2τ0 · 0 +
n∑
i=1
τi · zi +
n∑
i=1
τi · (−zi) + 2 · ∞.
Note, in particular, that the Takiff degree at the origin is always even. Let N ∈ Z≥1. We require
that deg C = 2N + 2 or in other words,
τ0 +
n∑
i=1
τi = N.
Let M ∈ Z≥1. As before, cf. Section 2.1, denote by Eab for a, b = 1, . . . ,M the standard basis
of glM . There is an automorphism σ of glM defined by
σ(Eab) := −Eba.
We call this the diagram automorphism of glM . The Lie algebra glM decomposes into the direct
sum of the ±1 eigenspaces of σ,
glM = soM ⊕ pM .
Here the subalgebra of invariants, i.e., the (+1)-eigenspace, is a copy of the Lie algebra soM .
The (−1)-eigenspace pM is a copy of the symmetric second rank tensor representation of soM .
We shall write
E±ab := Eab ± Eba,
so that E+
ab ∈ soM and E−ab ∈ pM , for all a, b = 1, . . . ,M . We introduce the pair of maps
Π(0) : glM → soM , Eab 7→ E−ab and Π(1) : glM → pM , Eab 7→ E+
ab. More generally, for r ∈ Z≥0 we
define Π(r) := Π(rmod 2) : glM → glM , so that Π(r)Eab = Eab − (−1)rEba.
There is an extension of the automorphism σ to an automorphism of the polynomial algebra
glM [ε] defined by
Xεk 7→ σ(X)(−ε)k.
Let glM [ε]σ denote the subalgebra of invariants. As vector spaces, we have
glM [ε]σ ∼= soM
[
ε2
]
⊕ εpM
[
ε2
]
.
(glM , glN )-Dualities in Gaudin Models with Irregular Singularities 19
Define glCM to be the direct sum of Takiff Lie algebras
glCM := (ε∞glM [ε∞])σ/ε2
∞ ⊕
n⊕
i=1
glM [εzi ]/ε
τi
zi ⊕ glM [ε0]σ/ε2τ0
0 .
Note that as a vector space the Takiff algebra attached to the point at infinity is simply
(ε∞glM [ε∞])σ/ε2
∞
∼= pMε∞.
As before we let ρ : glM → MatM×M (C) denote the defining representation of glM and write
Eab := ρ(Eab). The formal Lax matrix of the Z2-cyclotomic Gaudin model associated with glCM
is the M ×M matrix with entries consisting of glCM -valued rational functions of z, given by
L̃C(z)dz :=
M∑
a,b=1
Eba ⊗
E
+(∞)
ab[1] +
2τ0−1∑
r=0
(Π(r)Eab)
(0)
[r]
zr+1
+
n∑
i=1
τi−1∑
r=0
E
(zi)
ab[r]
(z − zi)r+1
+
n∑
i=1
τi−1∑
r=0
(−1)r+1E
(zi)
ba[r]
(z + zi)r+1
dz. (5.1)
It obeys the following Lax algebra[
L̃C
1 (z), L̃C
2 (w)
]
=
[
r12(z, w), L̃C
1 (z)
]
−
[
r21(w, z), L̃
C
2 (w)
]
(5.2)
where r12(z, w) denotes the (non-skew-symmetric) classical r-matrix
r12(z, w) :=
M∑
a,b=1
(
Eba ⊗ Eab
w − z
− Eba ⊗ Eba
w + z
)
.
Consider the quantity(
z2τ0
n∏
i=1
(z − zi)τi(z + zi)
τi
)
det
(
λ1M×M − L̃C(z)
)
This is a polyomial in λ of order M . For each 0 ≤ k ≤ M , the coefficient of λk is a rational
function in z valued in S
(
glCM
)
. The classical cyclotomic Gaudin algebra Z
(
glCM
)
associated with
the divisor C and the diagram automorphism σ is by definition the Poisson subalgebra of S
(
glCM
)
generated by the coefficients of these rational functions. It follows from (5.2) that Z
(
glCM
)
is
a Poisson-commutative subalgebra of S
(
glCM
)
.
5.2 Lax matrix of sp2N -Gaudin model with regular singularities
Denote by ẼIJ the standard basis of gl2N , where, for convenience, we shall let I, J run over
the index set I := {−N, . . . ,−1, 1, . . . , N}. There is a subalgebra of gl2N , isomorphic to the Lie
algebra sp2N , spanned by
ĒIJ := ẼIJ − σIσJ Ẽ−J,−I ,
for all I, J ∈ I. Here we denote by σI the sign of I, equal to 1 if I > 0 and to −1 if I < 0. We
have the relation Ē−J,−I = −σIσJ ĒIJ for every I, J ∈ I. Let
I2 :=
{
(I, J) ∈ I× I
∣∣ I, J > 0 or σIσJ = −1 with |I| ≤ |J |
}
.
20 B. Vicedo and C. Young
Then
{
ĒIJ
}
(I,J)∈I2 is a basis of the subalgebra sp2N . A dual basis with respect to half the trace
in the fundamental representation is given by
{
ĒIJ
}
(I,J)∈I2 where
ĒIJ := ẼJI − σIσJ Ẽ−I,−J , ĒI,−I := Ẽ−I,I ,
for any I, J ∈ I with J 6= −I. Indeed, if we let ĒIJ := ρ
(
ĒIJ
)
and ĒIJ := ρ
(
ĒIJ
)
for all I, J ∈ I
then we have 1
2 tr
(
ĒIJ Ē
KL
)
= δILδJK for all (I, J), (K,L) ∈ I2.
Let D̄ denote the special case of the effective divisor D̃ of Section 2.1 obtained by setting
τ̃a = 1 for each a = 1, . . . ,m, and hence m = M . That is,
D̄ =
M∑
a=1
λa + 2 · ∞. (5.3)
Introduce the direct sum of Lie algebras
spD̄2N := ε̃∞sp2N [ε̃∞]/ε̃2
∞ ⊕
M⊕
a=1
sp2N .
The Lax matrix of the classical Gaudin model associated with the divisor D̄ is the 2N × 2N
matrix of spD̄2N -valued rational functions of λ given by
LD̄(λ)dλ :=
∑
(I,J)∈I2
ĒIJ ⊗
(
Ē
(∞)
IJ +
M∑
a=1
Ē
(λa)
IJ
λ− λa
)
dλ, (5.4)
where by abuse of notation we drop the subscript on the Takiff generators, namely we define
Ē
(λa)
IJ := Ē
(λa)
IJ [0] for all a = 1, . . . ,M and Ē
(∞)
IJ := Ē
(∞)
IJ [1]. It obeys the Lax algebra
[
LD̄
1 (λ),LD̄
2 (µ)
]
=
[
r̄12(λ, µ),LD̄
1 (λ) + LD̄
2 (µ)
]
(5.5)
where r̄12(λ, µ) is the standard skew-symmetric classical r-matrix with spectral parameter for
the Lie algebra sp2N , namely
r̄12(λ, µ) :=
∑
(I,J)∈I2
ĒIJ ⊗ ĒIJ
µ− λ
.
Just as in Section 3.1 we may consider the subalgebra Z
(
spD̄2N
)
of the Poisson algebra S
(
spD̄2N
)
generated by the coefficients rational functions in λ obtained as the coefficients of the polynomial
in z defined by
M∏
a=1
(λ− λa) det
(
z1N×N − LD̄(λ)
)
,
which is Poisson-commutative by virtue of the relation (5.5).
5.3 Bosonic realisation
Consider the Poisson algebra Pb := C[xai , p
b
j ]
N M
i,j=1 a,b=1, as in Section 3.2, with Poisson brackets
given by (3.3).
(glM , glN )-Dualities in Gaudin Models with Irregular Singularities 21
We now want to break up the list of integers from 1 to N into n+ 1 blocks of size τi for each
i = 0, 1, . . . , n. Define the integers νi by – in contrast to (2.1) –
νi :=
i−1∑
j=0
τj ,
for each i = 0, . . . , N (note in particular that now ν0 = 0), so that
(1, . . . , N) = (1, . . . , τ0; ν1 + 1, . . . , ν1 + τ1; . . . ; νn + 1, . . . , νn + τn).
Lemma 5.1. Let µ ∈ C be arbitrary and define a pair of linear maps πb : glCM → Pb and
π̄b : spD̄2N → Pb by
πb
(
E
(zi)
ab[r]
)
=
νi+τi−r∑
u=νi+1
xau+rp
b
u, πb
(
E
+(∞)
ab[1]
)
= λaδab,
πb
(
(Π(s)Eab)
(0)
[s]
)
=
τ0−s∑
u=1
(
xau+sp
b
u − (−1)sxbu+sp
a
u
)
− µ
τ0∑
u,v=1
u+v=s+1
(−1)vxaux
b
v
for every r = 0, . . . , τi − 1, s = 0, . . . , 2τ0 − 1, i = 1, . . . , n and a, b = 1, . . . ,M , and
π̄b
(
Ē
(λa)
ij
)
= pajx
a
i π̄b
(
Ē
(λa)
i,−j
)
= −xajxai , π̄b
(
Ē
(λa)
−i,j
)
= pajp
a
i ,
π̄b
(
Ē
(∞)
IJ
)
= −
(
1⊕
i=n
(
− Jτi(−zi)
)
⊕
(
− Jτ0(0)
)
⊕ Jτ0(0)⊕
n⊕
i=1
Jτi(−zi) + µẼ1,−1
)
JI
,
for every i, j = 1, . . . , N , I, J ∈ I and a = 1, . . . ,m. These maps are homomorphisms of Lie
algebras. They extend uniquely to homomorphisms of Poisson algebras πb : S
(
glCM
)
→ Pb and
π̄b : S
(
spD̄2N
)
→ Pb.
Proof. We first show that πb is a homomorphism. It follows, exactly as in the proof of
Lemma 3.1 (see Lemma 4.7) that{
πb
(
E
(zi)
ab[r]
)
, πb
(
E
(zj)
cd[s]
)}
= πb
([
E
(zi)
ab[r],E
(zj)
cd[s]
])
, (5.6)
for any r, s = 0, . . . , τi − 1, i, j = 1, . . . , n and a, b, c, d = 1, . . . ,M . We also clearly have{
πb
(
(Π(s)Eab)
(0)
[s]
)
, πb
(
E
(zi)
cd[r]
)}
= 0
for any r = 0, . . . , τi − 1 i = 1, . . . , n and a, b, c, d = 1, . . . ,M since the canonical variables
entering each argument of the Poisson brackets mutually commute.
To simplify the notation, introduce yabr :=
τ0−r∑
u=1
(
xau+rp
b
u − (−1)rxbu+rp
a
u
)
. We can then write
πb
(
(Π(s)Eab)
(0)
[s]
)
= yabs − µ
τ0∑
u,v=1
u+v=s+1
(−1)vxaux
b
v.
By a similar computation to the one leading to (5.6), we find that{
yabr , y
cd
s
}
= δbcy
ad
r+s + (−1)sδacy
db
r+s + (−1)rδbdy
ca
r+s + (−1)r+sδady
bc
r+s.
22 B. Vicedo and C. Young
Likewise, we have
−
τ0∑
v,w=1
v+w=s+1
(−1)w
{
yabr , x
c
vx
d
w
}
= −
τ0∑
u=r+1
s∑
w=1
u+w=r+s+1
(−1)w
(
δbcx
a
ux
d
w + (−1)rδbdx
c
ux
a
w + (−1)sδacx
d
ux
b
w + (−1)r+sδadx
b
ux
c
w
)
.
and also by symmetry we obtain
−
τ0∑
v,w=1
v+w=r+1
(−1)w
{
xavx
b
w, y
cd
s
}
=
τ0∑
v,w=1
v+w=r+1
(−1)w
{
ycds , x
a
vx
b
w
}
= −
r∑
u=1
τ0∑
w=s+1
u+w=r+s+1
(−1)w
(
δbcx
a
ux
d
w + (−1)rδbdx
c
ux
a
w + (−1)sδacx
d
ux
b
w + (−1)r+sδadx
b
ux
c
w
)
.
It now follows by combining all the above that{
πb
(
(Π(r)Eab)
(0)
[r]
)
, πb
(
(Π(s)Ecd)
(0)
[s]
)}
= δbcπb
(
(Π(r)Ead)
(0)
[r+s]
)
+ (−1)sδacπb
(
(Π(r)Edb)
(0)
[r+s]
)
+ (−1)rδbdπb
(
(Π(r)Eca)
(0)
[r+s]
)
+ (−1)r+sδadπb
(
(Π(r)Ebc)
(0)
[r+s]
)
= πb
([
(Π(r)Eab)
(0)
[r] , (Π(s)Ecd)
(0)
[s]
])
,
as required. And finally, since E
+(∞)
ab[1] is a Casimir and is sent to a constant under πb, all Poisson
brackets involving it are preserved by πb.
We now turn to showing that π̄b is also a homomorphism. Define qaI for each I ∈ I and
a = 1, . . . ,M by letting qai := xai and qa−i := pai for every i = 1, . . . , N . In this notation the
Poisson brackets (3.3) can be rewritten more uniformly as{
qaI , q
b
J
}
= σJδI,−Jδab,
for all I, J ∈ I and a, b = 1, . . . ,M . Moreover, we also have π̄b
(
Ē
(λa)
IJ
)
= σJq
a
I q
a
−J for all I, J ∈ I
and a = 1, . . . ,M . We then have{
π̄b
(
Ē
(λa)
IJ
)
, π̄b
(
Ē
(λb)
KL
)}
= σJσL
(
σKδI,−Kq
a
−Jq
a
−L + σ−LδI,Lq
a
Kq
a
−J + σKδJ,Kq
a
I q
a
−L + σ−LδJ,−Lq
a
I q
a
K
)
δab
= σJσK
(
π̄b
(
Ē
(λa)
IL
)
δJ,K + π̄b
(
Ē
(λa)
−J,−K
)
δI,L + π̄b
(
Ē
(λa)
I,−K
)
δ−J,L + π̄b
(
Ē
(λa)
−J,L
)
δK,−I
)
δab
= π̄b
([
Ē
(λa)
IJ , Ē
(λb)
KL
])
,
where in the second equality we have made use of the fact that σIσ−I = −1 for any I ∈ I.
Finally, the Poisson brackets involving the generators Ē
(∞)
IJ attached to infinity are all trivially
preserved by π̄b. �
We are now in a position to prove the analogue of Theorem 3.2 in the present context.
(glM , glN )-Dualities in Gaudin Models with Irregular Singularities 23
Theorem 5.2. For any µ ∈ C as in Lemma 5.1, we have the relation
πb
(
z2τ0
n∏
i=1
(z − zi)τi(z + zi)
τi det
(
λ1M×M − L̃C(z)
))
= π̄b
(
M∏
a=1
(λ− λa) det
(
z1N×N − LD̃(λ)
))
.
Proof. We follow the argument given in the proof of Theorem 3.2 very closely. Consider the
M ×M and 2N × 2N block matrices
Λ :=
(
(λ− λa)δab
)M
a,b=1
,
Z :=
1⊕
i=n
(
− Jτi(−z − zi)
)
⊕
(
−Jτ0(−z)
)
⊕ Jτ0(z)⊕
n⊕
i=1
Jτi(z − zi) + µẼ1,−1.
We use here the convention, cf. Section 5.2, that indices on components of the 2N × 2N matrix
Z run through the index set I = {−N, . . . ,−1, 1, . . . , N}. As an example of the form of the
matrix Z, if n = 2, τ0 = 2, τ1 = 1 and τ2 = 2 then we have
Z =
z + z2 0
1 z + z2 0
z + z1
z 0 0 0
1 z 0 0
0 µ z 0
0 0 −1 z
z − z1
0 z − z2 0
−1 z − z2
.
We define a pair of M × 2N matrices P and X, whose columns are also indexed by the set I,
as
tP :=
−x1
N . . . −xMN
...
. . .
...
−x1
1 . . . −xM1
p1
1 . . . pM1
...
. . .
...
p1
N . . . pMN
, X :=
p
1
N . . . p1
1 x1
1 . . . x1
N
...
. . .
...
...
. . .
...
pMN . . . pM1 xM1 . . . xMN
.
Consider now the block (M+2N)×(M+2N) square matrix (3.5) with Λ, Z, X and P defined as
above. Now the derivation leading to the equation (3.6) from the proof of Theorem 3.2 still holds
and so it just remains to compute the determinants appearing on both sides of this identity.
On the one hand, we have
Λ−XZ−1 tP =
M∑
a,b=1
Eab
(
Λ−XZ−1 tP
)
ab
= λ1−
M∑
a,b=1
Eab
πb(E+(∞)
ab[1]
)
+
n∑
i=1
νi+τi∑
j,k=νi+1
xaj
(
Z−1
)
jk
pbk +
τ0∑
j,k=1
xaj
(
Z−1
)
jk
pbk
24 B. Vicedo and C. Young
−
τ0∑
j,k=1
paj
(
Z−1
)
−j,−kx
b
k −
τ0∑
j,k=1
xaj
(
Z−1
)
j,−kx
b
k −
n∑
i=1
νi+τi∑
j,k=νi+1
paj
(
Z−1
)
−j,−kx
b
k
.
For each i = 1, . . . , n we note using the expression (3.4) for the inverse of a Jordan block together
with Lemma 5.1 that
νi+τi∑
j,k=νi+1
xaj
(
Z−1
)
jk
pbk =
τi−1∑
r=0
πb
(
E
(zi)
ab[r]
)
(z − zi)r+1
,
−
νi+τi∑
j,k=νi+1
paj
(
Z−1
)
−j,−kx
b
k =
τi−1∑
r=0
(−1)r+1πb
(
E
(zi)
ba[r]
)
(z + zi)r+1
.
Next, for the two terms in the middle line above, corresponding to the origin, we find
τ0∑
j,k=1
(
xaj
(
Z−1
)
jk
pbk − paj
(
Z−1
)
−j,−kx
b
k
)
=
τ0−1∑
s=0
1
zs+1
τ0−s∑
u=1
(
xau+sp
b
u − (−1)sxbu+sp
a
u
)
.
Finally, for the remaining term we have
−
τ0∑
j,k=1
xaj
(
Z−1
)
j,−kx
b
k = −
2τ0−1∑
s=1
µ
zs+1
τ0∑
u,v=1
u+v=s+1
(−1)vxaux
b
v.
Putting all the above together we deduce that Λ−XZ−1 tP = λ1− πb
(
tL̃C(z)
)
.
On the other hand, we have
Z − tPΛ−1X =
∑
I,J∈I
ẼIJ
(
Z − tPΛ−1X
)
IJ
= z1−
∑
(I,J)∈I2
ĒIJ
(
π̄b
(
Ē
(∞)
IJ
)
−
M∑
a=1
π̄b
(
Ē
(λa)
IJ
)
λ− λa
)
= z1− π̄b
(
LD̄(λ)
)
.
To see the second equality we note that setting z = 0 in Z− tPΛ−1X yields a 2N×2N symplectic
matrix, i.e., of the block form
M =
(
A B
C −Ã
)
with B̃ = B and C̃ = C, where for an N ×N matrix A we denote by à the transpose of A along
the minor diagonal. And for any such matrix M we have
M =
∑
I,J∈I
ẼIJMIJ =
N∑
i,j=1
((
Ẽij − Ẽ−j,−i
)
Aij + Ẽi,−jCij − Ẽ−i,jBij
)
=
N∑
i,j=1
ĒijAji +
N∑
i,j=1
i≤j
Ē−i,jCji −
N∑
i,j=1
i≤j
Ēi,−jBji =
∑
(I,J)∈I2
ĒIJMIJ .
Lastly, we clearly have det Λ =
M∏
a=1
(λ − λa) and detZ = z2τ0
n∏
i=1
(z − zi)τi(z + zi)
τi from which
the result now follows, using again the fact that det tA = detA for any square matrix A, as in
the proof of Theorem 3.2. �
(glM , glN )-Dualities in Gaudin Models with Irregular Singularities 25
Remark 5.3. Consider replacing p by ∂ in the (M + 2N)× (M + 2N) square matrix
λ− λ1 0 p1
N . . . p1
1 x1
1 . . . x1
N
. . .
...
. . .
...
...
. . .
...
0 λ− λM pMN . . . pM1 xM1 . . . xMN
−x1
N . . . −xMN
...
. . .
...
−x1
1 . . . −xM1
p1
1 . . . pM1 Z
...
. . .
...
p1
N . . . pMN
used in the proof of Theorem 5.2. The resulting square matrix with non-commutative entries
is not Manin since, for example, the entries of the first column are not mutually commuting.
Consequently, we do not immediately obtain a quantum analogue of the classical relation in
Theorem 5.2.
A related remark is that in the quantum case, higher Gaudin Hamiltonians for cyclotomic
Gaudin models do exist but they are not in general given by a simple cdet-type formula. See
[34, 36] (and especially Remark 2.5 in [34]).
Remark 5.4. Note that we did not allow irregular singularities on the sp2N side (appart from
the double pole at infinity).
From the point of view of (glM , glN )-duality, the absence of irregular singularities in the
sp2N -Gaudin model is controlled by the fact that the matrix(
πb
(
E
+(∞)
ab[1]
))M
a,b=1
, (5.7)
representing the Casimir generators attached to infinity in the cyclotomic glM -Gaudin model,
is purely diagonal and in particular has no Jordan blocks, as in Lemma 5.1. Yet this is forced
on us since the matrix (5.7) is symmetric.
Alternatively, note that if one naively attempts to run the arguments above for the divi-
sor D̃ in place of D̄, one does not obtain a homomorphism spD̃2N → Pb. For example, Poisson
brackets of the form
{
−
∑
u x
u
i x
u+1
j ,
∑
v p
v
kp
v+1
l
}
produce two sorts of terms: “good” terms like∑
u x
u
i p
u+2
l δjk, which respect the gradation of the Takiff algebra, but also “bad” terms like∑
u x
u+1
j pu+1
l δik, which do not.
5.4 Example: Neumann model
We end this section by considering the special case of Theorem 5.2 when N = 1 and µ = −1.
Specifically, for the Z2-cyclotomic Gaudin model of Section 5.1 we take n = 0 and τ0 = 1.
The formal Lax matrix (5.1) of the corresponding cyclotomic glM -Gaudin model with effective
divisor C = 2 · 0 + 2 · ∞ then reduces to
L̃C(z)dz =
M∑
a,b=1
Eba ⊗
E
+(∞)
ab[1] +
E
−(0)
ab[0]
z
+
E
+(0)
ab[1]
z2
dz. (5.8)
When N = 1 in Section 5.2 we have the canonical isomorphism sp2 ' sl2 given by Ē11 7→ −H,
Ē1,−1 7→ 2F and Ē−1,1 7→ 2E. The dual basis elements are sent under this isomorphism to
26 B. Vicedo and C. Young
Ē11 = Ē11 7→ −H, Ē1,−1 = 1
2 Ē−1,1 7→ E and Ē−1,1 = 1
2 Ē1,−1 7→ F. The formal Lax matrix (5.4) of
the sl2-Gaudin model with effective divisor (5.3) then becomes,
LD̄(λ)dλ =
(
H ⊗ H(∞) + 2E ⊗ F(∞) + 2F ⊗ E(∞)
+
M∑
a=1
H ⊗ H(λa) + 2E ⊗ F(λa) + 2F ⊗ E(λa)
λ− λa
)
dλ, (5.9)
where we have used the notation
E := ρ(E) =
(
0 1
0 0
)
, F := ρ(F) =
(
0 0
1 0
)
, H := ρ(H) =
(
1 0
0 −1
)
.
The Poisson algebra Pb in the present context is simply C[xa, pa]
M
a,b=1 where we have dropped
the subscript 1 from the canonical variables by defining xa := xa1 and pa := pa1. In terms of this
notation, the representation πb : glCM → Pb from Theorem 5.2 reads
πb
(
E
+(∞)
ab[1]
)
= λaδab, πb
(
E
−(0)
ab[0]
)
= xapb − xbpa, πb
(
E
+(0)
ab[1]
)
= −xaxb,
recalling that µ = −1. Correspondingly, the map π̄b : spC2 → Pb takes the form
π̄b
(
E(∞)
)
= 1
2 , π̄b
(
F(∞)
)
= 0, π̄b
(
H(∞)
)
= 0,
π̄b
(
E(λa)
)
= 1
2p
2
a, π̄b
(
F(λa)
)
= − 1
2x
2
a, π̄b
(
H(λa)
)
= xapa.
Applying the first representation πb to the formal Lax matrix (5.8) we find
L̃(z)dz := πb
(
L̃C(z)
)
dz
=
M∑
a=1
λaEaa − z−1
M∑
a,b=1
(xapb − xbpa)Eab − z−2
M∑
a,b=1
xaxbEab
dz.
If we introduce variables ωa, a = 1, . . . ,M such that ω2
a = λa then the above coincides with the
M ×M Lax matrix of the Neumann model, with Hamiltonian
H =
1
4
M∑
a,b=1
a6=b
(xapb − xbpa)2 +
1
2
M∑
a=1
ω2
ax
2
a,
describing the motion of a particle constrained to the sphere
M∑
a=1
x2
a = 1 in RM and subject to
harmonic forces with frequency ωa along the ath axis. On the other hand, applying π̄b to the
formal Lax matrix (5.9) yields
L(λ)dλ := π̄b
(
LD̄(λ)
)
dλ = 2
M∑
a=1
xapa
λ−λa
M∑
a=1
−x2a
λ−λa
1 +
M∑
a=1
p2a
λ−λa −
M∑
a=1
xapa
λ−λa
dλ,
which coincides with the expression for the 2× 2 Lax matrix of the same model. The statement
of Theorem 5.2 corresponds to the well known relation between the above two Lax formulations
of the Neumann model (see, e.g., [28, Section 12])
z2 det
(
λ1M×M − L̃(z)
)
=
M∏
a=1
(λ− λa) det
(
z12×2 − L(λ)
)
.
(glM , glN )-Dualities in Gaudin Models with Irregular Singularities 27
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28 B. Vicedo and C. Young
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1 Introduction
2 Gaudin models with irregular singularities
2.1 Lie algebras glDM and glN
2.2 Lax matrices
3 Classical (glM, glN)-duality
3.1 Classical Gaudin model
3.2 Bosonic realisation
3.3 Fermionic realisation
4 Quantum (glM, glN)-duality
4.1 Manin matrices
4.2 Quantum Gaudin model
4.3 Bosonic realisation
5 Z2-cyclotomic Gaudin models with irregular singularities
5.1 Z2-cyclotomic Lax matrix for the diagram automorphism
5.2 Lax matrix of sp2N-Gaudin model with regular singularities
5.3 Bosonic realisation
5.4 Example: Neumann model
References
|
| id | nasplib_isofts_kiev_ua-123456789-209532 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:01:03Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Vicedo, B. Young, C. 2025-11-24T10:43:43Z 2018 (glM,glN)-Dualities in Gaudin Models with Irregular Singularities / B. Vicedo, C. Young // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 36 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B80; 81R12; 82B23 arXiv: 1710.08672 https://nasplib.isofts.kiev.ua/handle/123456789/209532 https://doi.org/10.3842/SIGMA.2018.040 We establish (glM,glN)-dualities between quantum Gaudin models with irregular singularities. Specifically, for any M,N ∈ Z≥1, we consider two Gaudin models: the one associated with the Lie algebra glM, which has a double pole at infinity and N poles, counting multiplicities, in the complex plane, and the same model but with the roles of M and N interchanged. Both models can be realized in terms of Weyl algebras, i.e., free bosons; we establish that, in this realization, the algebras of integrals of motion of the two models coincide. At the classical level, we establish two further generalizations of the duality. First, we show that there is also a duality for realizations in terms of free fermions. Second, in the bosonic realization, we consider the classical cyclotomic Gaudin model associated with the Lie algebra glM and its diagram automorphism, with a double pole at infinity and 2N poles, counting multiplicities, in the complex plane. We prove that it is dual to a non-cyclotomic Gaudin model associated with the Lie algebra sp2N, with a double pole at infinity and M simple poles in the complex plane. In the special case N=1, we recover the well-known self-duality in the Neumann model. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications (glM,glN)-Dualities in Gaudin Models with Irregular Singularities Article published earlier |
| spellingShingle | (glM,glN)-Dualities in Gaudin Models with Irregular Singularities Vicedo, B. Young, C. |
| title | (glM,glN)-Dualities in Gaudin Models with Irregular Singularities |
| title_full | (glM,glN)-Dualities in Gaudin Models with Irregular Singularities |
| title_fullStr | (glM,glN)-Dualities in Gaudin Models with Irregular Singularities |
| title_full_unstemmed | (glM,glN)-Dualities in Gaudin Models with Irregular Singularities |
| title_short | (glM,glN)-Dualities in Gaudin Models with Irregular Singularities |
| title_sort | (glm,gln)-dualities in gaudin models with irregular singularities |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209532 |
| work_keys_str_mv | AT vicedob glmglndualitiesingaudinmodelswithirregularsingularities AT youngc glmglndualitiesingaudinmodelswithirregularsingularities |