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JHEP06(2019)017

Published for SISSA by Springer

Received: March 19, 2019

Accepted: May 25, 2019

Published: June 6, 2019

Assembling integrable σ-models as aﬃne Gaudin

models

F. Delduc,

S. Lacroix,

b,c

M. Magro

and B. Vicedo

Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique,

F-69342 Lyon, France

II. Institut f¨ur Theoretische Physik, Universit¨at Hamburg,

Luruper Chaussee 149, 22761 Hamburg, Germany

Zentrum f¨ur Mathematische Physik, Universit¨at Hamburg,

Bundesstrasse 55, 20146 Hamburg, Germany

Department of Mathematics, University of York,

York YO10 5DD, U.K.

E-mail: francois.delduc@ens-lyon.fr, sylvain.lacroix@desy.de,

marc.magro@ens-lyon.fr, benoit.vicedo@gmail.com

Abstract: We explain how to obtain new classical integrable ﬁeld theories by assembling

two aﬃne Gaudin models into a single one. We show that the resulting aﬃne Gaudin model

depends on a parameter γ in such a way that the limit γ → 0 corresponds to the decoupling

limit. Simple conditions ensuring Lorentz invariance are also presented. A ﬁrst application

of this method for σ-models leads to the action announced in [1] and which couples an

arbitrary number N of principal chiral model ﬁelds on the same Lie group, each with a

Wess-Zumino term. The aﬃne Gaudin model descriptions of various integrable σ-models

that can be used as elementary building blocks in the assembling construction are then

given. This is in particular used in a second application of the method which consists in

assembling N − 1 copies of the principal chiral model each with a Wess-Zumino term and

one homogeneous Yang-Baxter deformation of the principal chiral model.

Keywords: Integrable Field Theories, Sigma Models

ArXiv ePrint: 1903.00368

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP06(2019)017

JHEP06(2019)017

1 Introduction

Constructing classical ﬁeld theories which are integrable is diﬃcult. Indeed, the property

of integrability is established at the Lagrangian level if one is able to ﬁnd a Lax connection

whose zero curvature equation is equivalent to the equations of motion derived from the

given action. This ensures, in particular, the existence of inﬁnitely many integrals of motion

in the theory. Yet ﬁnding such a Lax connection is somewhat of an art. Furthermore, in

order to establish integrability at the Hamiltonian level one also needs to prove the existence

of an inﬁnite number of integrals of motion in involution. This is guaranteed if the Poisson

bracket of the Lax matrix can be brought to a certain form. But doing so is also, in and

of itself, an arduous task.

Faced with the diﬃculty of ﬁnding new integrable ﬁeld theories from scratch, one

approach to exploring the landscape of all possible Hamiltonian integrable ﬁeld theories is

to build new ones from old ones. One way of doing so is by deforming. For a very broad

class of integrable ﬁeld theories, the Poisson bracket of the Lax matrix is controlled by a

certain rational function ϕ(z) of the spectral parameter z, called the twist function. As

shown in [2, 3], this twist function can be used at the Hamiltonian level to deform the

theory in question, all the while preserving integrability. We may then perform an inverse

Legendre transform to construct the actions of the resulting deformed ﬁeld theories, which

by construction will automatically be integrable.

It was recently shown by one of us in [4] that classical integrable ﬁeld theories admitting

a twist function can be seen as realisations of classical Gaudin models associated with

untwisted aﬃne Kac-Moody algebras. In fact, all of the examples considered in [4], which

includes many integrable σ-models and their deformations, essentially correspond to aﬃne

Gaudin models with a single site. However, since the number of sites in an aﬃne Gaudin

model can be arbitrary, this opens up another avenue for constructing new integrable ﬁeld

theories from existing ones.

Indeed, the procedure detailed in the present article consists, roughly speaking, in

taking existing integrable ﬁeld theories which admit an aﬃne Gaudin model description

and ‘placing’ them at the individual sites of an N-site aﬃne Gaudin model. This results

in a new integrable ﬁeld theory which couples these individual theories together. In fact,

the strength of the coupling will be determined by the relative location of each site in

the complex plane. Finally, since aﬃne Gaudin models are intrinsically formulated at

the Hamiltonian level, to complete the procedure one has to perform an inverse Legendre

transform in order to construct the action for this new model. In [1] we presented the

result of applying such a procedure to couple together N principal chiral ﬁelds on the same

Lie group, each with a Wess-Zumino term.

To motivate why building an N-site aﬃne Gaudin model in this way corresponds to

coupling together the individual ﬁeld theories, let us consider here the simpler situation

of a Gaudin model associated with a ﬁnite-dimensional complex semisimple Lie algebra

g [5]. Let u

for i = 1, . . . , N be a collection of N distinct complex numbers. Let I

a(i)

for

a = 1, . . . , dim g denote the copy of a basis of the Lie algebra g that we think of as being

formally ‘attached’ to the point u

, for i = 1, . . . , N. One may regard the basis elements

– 1 –

JHEP06(2019)017

a(i)

, i = 1, . . . , N of the N copies of g as the degrees of freedom of the model. They satisfy

the Kostant-Kirillov Poisson bracket

a(i)

, I

b(j)

} = δ

c(i)

. (1.1)

Also let I

(i)

for a = 1, . . . , dim g be the dual basis of I

a(i)

with respect to κ(·, ·) deﬁned as

the opposite of the Killing form on g. The quadratic Poisson commuting Hamiltonians of

the Gaudin model are then given by

k=1

k6=i

a(i)

(k)

− u

(1.2)

for i = 1, . . . , N , where the sum over the repeated Lie algebra index a = 1, . . . , dim g

is implicit. The basis elements I

a(i)

may be packaged together into g-valued observables

(i)

= I

⊗ I

a(i)

attached to each site u

for i = 1, . . . , N, and in terms of which the Lax

matrix takes the form

L(z) =

i=1

(i)

z −u

. (1.3)

We may then express the quadratic Gaudin Hamiltonians (1.2) as

res

κ(L(z), L(z))dz. (1.4)

A ﬁnite-dimensional integrable system is said to be a realisation of this Gaudin model

if there is a morphism from the algebra of observables of the Gaudin model to that of

the given integrable system. Moreover, the Hamiltonian of the integrable system should

coincide, in the simplest case, with the image of some linear combination of the Hamiltoni-

ans (1.2) under this morphism. Concretely, the existence of such a morphism means that

the integrable system should admit a Lax matrix of the same form as (1.3) and satisfying

the same Poisson bracket relations as (1.3) does. In particular, the g-valued observables

(i)

and J

(j)

should commute for i 6= j.

Suppose now that we break up the collection of N sites {u

}

i=1

into two subsets of

sizes N

and N

, respectively, which without loss of generality we take to be {u

}

i=1

and

j+N

}

j=1

. Fixing a non-zero complex number γ 6= 0, we then choose to parametrise

these two subsets of points as u

= z

−

−1

, i = 1, . . . , N

and u

j+N

= ˆz

−1

j = 1, . . . , N

. For convenience we also use the notation

a(j)

= I

a(j+N

)

and

(j)

= I

(j+N

)

for j = 1, . . . , N

. The collection of Hamiltonians (1.2) then naturally split into two subsets

as well, namely

k=1

k6=i

a(i)

(k)

− z

j=1

a(i)

(j)

− ˆz

− γ

−1

, (1.5a)

= H

j+N

k=1

k6=j

a(j)

(k)

ˆz

− ˆz

i=1

a(j)

(i)

ˆz

− z

+ γ

−1

, (1.5b)

– 2 –

JHEP06(2019)017

for every i = 1, . . . , N

and j = 1, . . . , N

. We now observe that the ﬁrst sums in each of

the above expressions (1.5) is just the usual expression for the Gaudin Hamiltonians, as

in (1.2), but corresponding to one of the two subsets of points. Moreover, the second sums

on the right hand sides of (1.5) describe interaction terms between the degrees of freedom

attached to the ﬁrst N

sites and those attached to the last N

sites. It is clear from (1.5)

that one can decouple these two sets of degrees of freedom by letting the ‘coupling’ γ go

to zero. This corresponds to moving apart the ﬁrst N

sites from the last N

sites while

keeping all the relative positions within these two sets of sites ﬁxed. In this way one can

view the Gaudin model with N = N

+ N

sites as the coupling of two separate Gaudin

models, one with N

sites and the other with N

sites. It is important to note that all

three Gaudin models are associated with the same Lie algebra g.

In fact, by reversing the logic of the previous paragraphs, we see that the coupling of

ﬁnite-dimensional integrable systems which are realisations of Gaudin models associated

with the same Lie algebra g is easy to perform at the level of the Lax matrix. Starting

with the individual Lax matrices for both models, namely

(z) =

i=1

(i)

z −z

and L

(z) =

i=1

(i)

z − ˆz

, (1.6)

where J

(i)

= I

⊗ I

a(i)

for i = 1, . . . , N

and J

(i)

= I

⊗

a(i)

for i = 1, . . . , N

, the Lax

matrix of the coupled system in (1.3) is obtained simply by adding the Lax matrices (1.6)

together after suitably shifting their spectral parameters by some amount of the inverse

−1

of the coupling parameter γ. Explicitly, we have

L(z) = L



z +

−1



+ L



z −

−1



. (1.7)

The Hamiltonian of the coupled system, which includes all the interaction terms as in (1.5),

is then constructed from this Lax matrix in the usual way as a linear combination of (1.4).

We now return to the ﬁeld theory setting. Let

g be the untwisted aﬃne Kac-Moody

algebra corresponding to g. It is useful to start by brieﬂy recalling how classical integrable

ﬁeld theories which admit a twist function can be viewed as Gaudin models associated with

g, referring the reader to [4] for the details.

The Gaudin model associated with the inﬁnite dimensional Lie algebra

g is deﬁned

in exactly the same way as in the ﬁnite-dimensional case recalled above, replacing g

everywhere by

g. The only subtlety is that, in doing so, the implicit ﬁnite sums over

a = 1, . . . , dim g are now replaced by inﬁnite sums over an index ea labelling a basis of

which one has to make sense of by working in a suitable completion of the algebra of ob-

servables. See [4] for a further discussion of this technical point which we shall not consider

any further here. In particular, the Lax matrix of the aﬃne Gaudin model takes on the

exact same form as in (1.3) but where now the

g-valued observables

(i)

= I

⊗ I

ea(i)

= d ⊗ k

(i)

+ k ⊗ d

(i)

n∈Z

a,−n

⊗ I

a(i)

– 3 –

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