^{1}

^{2}

^{1}

^{3}

^{3}.

A systematic procedure for constructing classical integrable field theories with arbitrarily many free parameters is outlined. It is based on the recent interpretation of integrable field theories as realizations of affine Gaudin models. In this language, one can associate integrable field theories with affine Gaudin models having arbitrarily many sites. We present the result of applying this general procedure to couple together an arbitrary number of principal chiral model fields on the same Lie group, each with a Wess-Zumino term.

The very scarceness of the property of integrability in classical and quantum systems makes its ubiquity in high energy physics as well as its rich history in condensed matter physics seem even more remarkable. For instance, integrability has played a pivotal role in recent years (see, e.g., the review

One particular aspect of the subject of classical integrable field theories which has proved extremely fruitful in some of these recent developments is the concept

Recently, it was shown by one of us in

Viewing the above class of integrable field theories in this new light opens up the possibility of constructing new classical integrable field theories by considering more general affine Gaudin models. Indeed, all the examples of integrable field theories discussed in

To illustrate the main idea behind the construction, it is useful to recall that, when regarded as an integrable spin chain, a (classical) Gaudin model has the following two salient features: (i) the degrees of freedom, i.e., “spins,” at different sites mutually (Poisson) commute, and (ii) the Hamiltonian describes interactions between the spins at every pair of sites.

Consider an affine Gaudin model with an arbitrary number of sites

When the

The purpose of the present Letter is to report on the result of applying this procedure to

Action: Let us fix a set of

In terms of the above data, the action obtained in

We note that there is some redundancy in the choice of the

The action

Field equations and Lax connection: The field equations obtained by determining the extrema of the action

The

Twist function and integrability: The integrability of the field theory defined by the action

The twist function is known to play a fundamental role in many aspects of the classical integrability of field theories which can be realized as affine Gaudin models. For instance, in these theories, an infinite family of Poisson commuting local charges can be constructed as the integrals of certain invariant polynomials of

Decoupling limit: In the model just described, a given field

Global symmetries: Since the action

Because the currents at the various sites are all coupled, the symmetry under right multiplication is limited to the diagonal action of

Examples: As a simple illustration of the general action in

It is also possible to couple two principal chiral fields without introducing WZ terms and with the same decoupling limit

The general procedure illustrated in this Letter opens up the possibility of constructing infinite families of relativistic integrable field theories with a given underlying Lie algebra. The basic building blocks for the construction are relativistic integrable field theories which can be realized as affine Gaudin models. An arbitrary number of these building blocks can then be assembled together to form a new relativistic integrable field theory. In turn, since the latter is an affine Gaudin model by definition, it could be used as a new building block for subsequent constructions. Determining the scope of all possibilities certainly requires a thorough investigation.

The fact that the twist function of the model we have considered is a generic rational function with

To illustrate the method, we have focused on the class of noncyclotomic affine Gaudin models. Working within this class, it is possible to couple together any integrable

Another natural direction for further study would be to extend the procedure to the class of cyclotomic affine Gaudin models, which includes

A possible next step would be to generalize the whole procedure to the supersymmetric case. This would require introducing the notion of classical affine Gaudin model associated with a Kac-Moody superalgebra first. An obvious question in this setting is to look for type IIB superstring

Having constructed classical integrable field theories with arbitrarily many parameters, a very important and natural question concerns the study of these models at the quantum level. It would be very interesting to determine the renormalization group flow in these multiparameter field theories and identify their fixed points. The recent results obtained for the models mentioned above in

We thank H. Samtleben for useful discussions and G. Arutyunov and A. Tseytlin for useful discussions and comments on the draft. This work is partially supported by the French Agence Nationale de la Recherche (ANR) under Grant No. ANR-15-CE31-0006 DefIS.