This is a repository copy of On integrable field theories as dihedral affine Gaudin models.
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Vicedo, Benoit orcid.org/0000-0003-3003-559X (2020) On integrable field theories as
dihedral affine Gaudin models. International Mathematics Research Notices. pp. 4513-
4601. ISSN 1687-0247
https://doi.org/10.1093/imrn/rny128
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ON INTEGRABLE FIELD THEORIES AS
DIHEDRAL AFFINE GAUDIN MODELS
BENOÎT VICEDO
Abstract. We introduce the notion of a classical dihedral affine Gaudin model,
associated with an untwisted affine Kac-Moody algebra
e
g equipped with an action of
the dihedral group D
2T
, T ≥ 1 through (anti-)linear automorphisms. We show that
a very broad family of classical integrable field theories can be recast as examples of
such classical dihedral affine Gaudin models. Among these are the principal chiral
model on an arbitrary real Lie group G
0
and the Z
T
-graded coset σ-mo del on any
coset of G
0
defined in terms of an order T automorphism of its complexification.
Most of the multi-parameter integrable deformations of these σ-models recently
constructed in the literature provide further examples. The common feature shared
by all these integrable field theories, which makes it possible to reformulate them as
classical dihedral affine Gaudin models, is the fact that they are non-ultralocal. In
particular, we also obtain affine Toda field theory in its lesser-known non-ultralocal
formulation as another example of this construction.
We propose that the interpretation of a given classical non-ultralocal integrable
field theory as a classical dihedral affine Gaudin model provides a natural setting
within which to address its quantisation. At the same time, it may also furnish a
general framework for understanding the massive ODE/IM correspondence since
the known examples of integrable field theories for which such a correspondence
has been formulated can all be viewed as dihedral affine Gaudin models.
1. Motivation and introduction
The ODE/IM correspondence describes a striking and rather unexpected relation
between the theory of linear Ordinary Differential Equations in the complex plane on
the one hand, and that of quantum Integrable Models on the other. Concretely, the
first instance of such a correspondence was formulated by V. Bazhanov, S. Lukyanov
and A. Zamolodchikov for quantum KdV theory in the series of seminal papers [8] –
[12], building on from the original insight of P. Dorey and R. Tateo in [25]. These
works culminated in the remarkable conjecture of [12] stating that the joint spec-
trum of the quantum KdV Hamiltonians on the level L ∈ Z
≥0
subspace of an ir-
reducible module over the Virasoro algebra is in bijection with the set of certain
one-dimensional Schrödinger operators −∂
2
z
+ V
L
(z) with ‘monster’ potentials V
L
(z)
of a given form. The justification for this conjecture comes from the central obser-
vation that the functional relations and analytic properties characterising the eigen-
values of the Q-operators of quantum KdV theory [9, 10] on a given joint eigenvector
coincide with those satisfied by certain connection coefficients of the associated one-
dimensional Schrödinger equation. These ideas were soon extended to other massless
1
2 BENOÎT VICEDO
integrable field theories asso ciated with higher rank Lie algebras of classical type, see
e.g. [26, 22, 5, 21] and the review [23].
Despite the variety of examples of the ODE/IM correspondence, its mathematical
underpinning remained elusive for a number of years. This problem was addressed by
B. Feigin and E. Frenkel in [37] where they argued that the ODE/IM correspondence
for quantum
b
g-KdV theory could be understood as originating from an affine analogue
of the geometric Langlands correspondence. To explain this connection we make a
brief digression on Gaudin models, which provide a realisation of the global geometric
Langlands correspondence for rational curves over the complex numbers.
Let g be a finite-dimensional complex semisimple Lie algebra. The Gaudin model,
or g-Gaudin model, is a quantum integrable spin-chain with long-range interactions
[47]. If we let N ∈ Z
≥1
denote the number of sites then the algebra of observables of
the model is the N-fold tensor product U (g)
⊗N
of the universal enveloping algebra
U(g) of g. The quadratic Gaudin Hamiltonians are elements of U(g)
⊗N
given by
H
i
:
=
N
X
j=1
j6=i
I
a(i)
I
(j)
a
z
i
− z
j
(1)
where the z
i
, i = 1, . . . , N are arbitrary distinct complex numbers, {I
a
} and {I
a
} are
dual bases of g with respect to a fixed non-degenerate invariant bilinear form h·, ·i on
g, and x
(i)
is the element of U(g)
⊗N
with x ∈ g in the i
th
tensor factor and 1’s in every
other factor. The quantum integrability of the model is characterised by the existence
of a large commutative subalgebra Z
z
(g) ⊂ U (g)
⊗N
with z
:
= {z
i
}
N
i=1
∪{∞}, known
as the Gaudin algebra, containing in particular the quadratic Gaudin Hamiltonians.
Let M
i
, i = 1, . . . , N be g-modules. One is interested in finding the joint spectrum of
Z
z
(g) on the spin-chain
N
N
i=1
M
i
. Note that a joint eigenvalue of the Gaudin algebra
defines a homomorphism Z
z
(g) → C sending each element of Z
z
(g) to its eigenvalue.
The joint spectrum can therefore be described as a subset of the maximal spectrum
of the commutative algebra Z
z
(g), i.e. the set of all homomorphisms Z
z
(g) → C. It
was shown by E. Frenkel in [43, Theorem 2.7(1)] that the maximal spectrum of the
Gaudin algebra Z
z
(g) is isomorphic to a certain subquotient of the space of
L
g-valued
connections on P
1
, known as
L
g-opers, with regular singularities in the set z, where
L
g
denotes the Langlands dual of the Lie algebra g. In other words, each joint eigenvalue
of the Gaudin algebra Z
z
(g) on the given spin-chain
N
N
i=1
M
i
will be described by
such an
L
g-oper. In fact, when all the g-modules M
i
are finite-dimensional irreducibles
V
λ
i
of highest weights λ
i
∈ h
∗
, [43, Conjecture 1] states that for each integral dominant
weight λ
∞
∈ h
∗
, the joint spectrum of Z
z
(g) on the subspace of weight λ
∞
singular
vectors in
N
N
i=1
V
λ
i
is in bijection with the subspace of such
L
g-opers with residue at
the points z
i
and infinity given by the shifted Weyl orbits of the weights λ
i
and λ
∞
respectively, and with trivial monodromy representation.
The description of the maximal spectrum of the Gaudin algebra Z
z
(g) in terms
of
L
g-opers also generalises to the case of Gaudin models with irregular singularities;
see [40, 39]. Another possible generalisation of Gaudin models is given by cyclotomic
Gaudin models, introduced in [86, 87] and more recently [88] for the case with irregular
ON INTEGRABLE FIELD THEORIES AS DIHEDRAL AFFINE GAUDIN MODELS 3
singularities. A similar description of the corresponding cyclotomic Gaudin algebra
of [86] was recently conjectured in [62] in terms of cyclotomic
L
g-opers, i.e.
L
g-opers
equivariant under an action of the cyclic group. In fact, these descriptions of the
various Gaudin algebras in terms of global
L
g-opers on P
1
follow (conjecturally in the
cyclotomic case) from the ‘local’ version proved by B. Feigin and E. Frenkel in their
seminal paper [36] (see also [44, 45]) which states that the space of singular vectors
in the vacuum Verma module V
crit
0
(g) at the critical level over the untwisted affine
Kac-Moody algebra
b
g, which naturally forms a commutative algebra, is isomorphic to
the algebra of functions on the space of
L
g-opers on the formal disc.
The apparent similarity between the description of the joint spectrum of the Gaudin
algebra on any given spin-chain in terms of certain
L
g-opers and the statement of the
ODE/IM correspondence for quantum KdV theory is more than just a coincidence.
Indeed, as argued in [37], quantum
b
g-KdV theory can be regarded as a generalised
Gaudin model associated with the untwisted affine Kac-Moody algebra
b
g, or
b
g-Gaudin
model for short, with a regular singularity at the origin and an irregular singularity of
the mildest possible form at infinity. Unfortunately, much less is know at present ab out
Gaudin models associated with general Kac-Moody algebras; see however [72, 42]. In
particular, there is currently no known analogue of the Feigin-Frenkel isomorphism
for describing the space of singular vectors in the suitably completed vacuum Verma
module over the double affine, or toroidal, Lie algebra
b
b
g. It is not even clear what
the critical level should be in this setting. Nevertheless, the notion of an affine oper,
or
b
g-oper, on P
1
can certainly be defined [42] and so it is tempting to speculate that
the description of the spectrum of the g-Gaudin Hamiltonians in terms of
L
g-opers
persists when g is replaced by an affine Kac-Moody algebra.
In this spirit, the explicit form of the
L
b
g-opers which ought to describe the joint
spectrum of the quantum
b
g-KdV Hamiltonians on certain irreducible modules over the
W -algebra associated with g was conjectured in [37], by using as a finite-dimensional
analogy a certain description of the finite W -algebra for a regular nilpotent element
in terms of
L
g-opers. Remarkably, when
b
g =
b
sl
2
so that also
L
b
g =
b
sl
2
, these
b
sl
2
-opers
were shown to coincide exactly, after a simple change of coordinate on P
1
, with the
one-dimensional Schrödinger operators written down in [12]. This result not only
confirms the idea that the ODE/IM correspondence can be thought of as a particular
instance of the geometric Langlands correspondence but also provides strong evidence
in support of the general claim that the joint spectrum of the higher Hamiltonians of
an affine Gaudin model can be described in terms of affine opers for the Langlands
dual affine Kac-Moody algebra.
Another approach to testing the proposed link between the joint spectrum of the
quantum
b
g-KdV Hamiltonians and
L
b
g-opers of the prescribed form is to follow the
same strategy originally used to establish the ODE/IM correspondence for quantum
KdV theory. Specifically, one should compare the functional relations and analytic
properties of the joint eigenvalues of the Q-operators of quantum
b
g-KdV theory on
joint eigenvectors in the irreducibles over the W -algebra associated with g, with those
satisfied by the connection coefficients of the associated
L
b
g-opers. This programme
was initiated in [81] and was further developed very recently in [68, 69] where some
4 BENOÎT VICEDO
remarkable functional relations, referred to as the Q
e
Q-system, were obtained for cer-
tain generalised spectral determinants of the ODE associated with the
L
b
g-opers of
[37] corresponding to highest weight states in representations of the W -algebra. Even
more recently in [46], the very same Q
e
Q-system was shown to arise as relations in the
Grothendieck ring K
0
(O) of the category O of representations of the Borel subalge-
bra of the quantum affine algebra U
q
(
b
g) for an untwisted affine Kac-Moody algebra
b
g. Analogous relations were also conjectured to hold when
b
g is a twisted affine Kac-
Moody algebra. Since non-local quantum
b
g-KdV Hamiltonians can be associated with
elements of K
0
(O) by the construction of [9, 10, 5], the joint spectrum of these Hamil-
tonians also satisfy the Q
e
Q-system, thereby providing further evidence in favour of
the ODE/IM correspondence for quantum
b
g-KdV theory.
The recent developments towards formulating and ultimately proving the ODE/IM
correspondence for quantum
b
g-KdV theory, which we briefly recalled above, can be
summarised in the following commutative diagram
b
g-Gaudin
model
quantum
b
g-KdV theory
L
b
g-opers
Q
e
Q-system
[37] [37]
[46]
[68, 69]
(2)
The top line of this diagram, corresponding to the work [37], consisted of two steps.
The first was to reinterpret quantum
b
g-KdV theory as a particular affine
b
g-Gaudin
model. The second, which we represent by a dashed arrow to emphasise its conjectural
status, was to make use of the existing description of the spectrum of g-Gaudin models
in terms of
L
g-opers as an analogy. The big open problem here is to establish the affine
counterpart of the latter statement to put the second step on a firm mathematical
footing. Indeed, this would promote the top line in the above diagram to a proof of
the ODE/IM correspondence for quantum
b
g-KdV theory. While the top line is still
partly conjectural, the bottom part of the diagram provides a solid link between both
sides of the ‘KdV-oper’ correspondence of [37] through the common Q
e
Q-system.
Until relatively recently, the study of the ODE/IM correspondence had been limited
to describing integrable structures in conformal field theories only. This left open the
important question of whether similar ideas could be used to describe the spectrum of
massive quantum integrable field theories as well. The first example of such a massive
ODE/IM correspondence was put forward by S. Lukyanov and A. Zamolodchikov for
quantum sine-Gordon and sinh-Gordon theories in their pioneering paper [64]. Specif-
ically, they showed that the functional relations and analytic properties characterising
the vacuum eigenvalues of the Q-operators of quantum sine/sinh-Gordon theory were
the same as those satisfied by certain connection coefficients of the auxiliary linear
problem of the classical modified sinh-Gordon equation for a suitably chosen classical
solution. Subsequently, various higher rank generalisations of this massive ODE/IM
correspondence for quantum affine
e
g-Toda field theories were also conjectured, when
e
g is of type A for rank 3 in [24] and for general rank n in [1], and more recently for
