^{1}

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^{2}

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We consider the AdS_{5} × S^{5} integrable model. As it turns out, relying on well known arguments, we claim that the conformally invariant fermionic model is solvable, the resulting solution given in terms of two current algebras realizations.

Integrable models have a long and successful history [

There is also at least one model where no mass gap exists, but comprising non trivial conservation laws. It is the case of the chiral Gross-Neveu model [

This means that an integrable model can also contain a conformally invariant solution. This is a quite non trivial fact that we wish to explore in case of integrable models relevant for string theory, where conformal invariance is a very desirable property.

In the framework of string theory, it is possible to gather information about the Yang-Mills theory at intermediate coupling. Obtaining a strongly coupled field theory underlying the QCD string actually provides an integrable model in the world sheet, and the low dimensionality of the problem may imply exact solvability [

In that case, the symmetry of the integrable model is

The bosonic part of such a coset is AdS_{5} × S^{5}, which will be our main concern. Most of the literature is related, in this case, to integrable models and their nonlocal conservation laws [_{5} × S^{5} have subsequently been constructed [_{5} is largely discussed in relation to string solutions [

Later, the non local charges have also been related to a BRST cohomology [

On the other hand, in string theory, a lot has been done concerning integrability of the underlying symmetry of strings in certain backgrounds. In Maldacena’s conjecture, four-dimensional N = 4 super Yang-Mills theory is dual to super strings in AdS_{5} × S^{5} background [

This means that the model is defined on a symmetric space, thus implying a non trivial (and non local) conservation law [

We shall consider a fermionic model defined upon the space (1). Following old and well established arguments we see that at a well defined value of the coupling constant the theory is conformally invariant.

The above mentioned fermionic model is defined by the lagrangian density

where we define the currents are given by _{1} and g_{2} are, up to now, arbitrary coupling constants.

The field equation for

while

is the field equation for

The Noether currents related to the symmetries SO(5,1) and SO(6), respectively, obey the conservation equations

Let us now consider the axial currents (non) conservation laws. Using the relations for the γ_{μ} matrices we have

We can compute the divergence of the axial current,

Taking into account the field equations we get

Here we note that the terms with the g_{1} coefficient are products of two currents while the terms with g_{2} coefficient are cancelled, that is,

Using the identity

the final result is

Therefore, the axial current

A similar result follows for the axial current

We consider now the axial anomaly contribution to the field equations. We introduce the gauge field

to be added to the divergence equation for

where N is the number of species, in this case equal to 6. We are thus led to

Therefore, the choice

This means conformal invariance in the coset SO(5,1)/O(4,1). Notice that, mutatis mutandis we get similar a result for the SO(6)/O(5) factor, as well as conformal invariance for all spaces of the kind

and the fermionic theory in the coset SO(6)/(O(5) is conformally invariant.

An alternative and equivalent proof of conformal invariance at a given coupling can be obtained by arguments already known in [

We can write the equal-time commutation rules

where C_{1} and C_{2} (= 0 or − C_{1}) are c-number Schwinger terms. In addition, we also have

where D_{1} and D_{2} (= 0 or − D_{1}) are also c-number Schwinger terms.

Here we note the structure constants

Using

we can deduce from the equal-time commutation relations the commutation rules for any space-time point,

We can now decompose also the currents

where (+) is the creation part and (‒) the annihilation part. Note that here two creation or two annihilation operators of different SO(5,1)/O(4,1) indices do not commute.

One finds also

where, due to Jacobi identity

Correlation functions are now immediately obtained from the methods of two-dimensional conformally invariant Quantum Field Theory [

The by now rather expected results displayed above mean that integrable models can have a conformally invariant counterpart. The fact that in string theory one needs conformal invariance as a building block forces us into the above solution at least for the fermionic models in question.

The rather important unanswered question is about what happens in case of a purely bosonic theory, or also, maybe even more important, to the model defined on a graded manifold. In the last case, in view of the unbroken supersymmetry, we are led to a conjecture concerning such sigma models, namely we conjecture that such models have a conformal fix point where the correlators are exactly solvable and present the previous symmetry.

This work has been supported by FAPESP and CNPq, Brazil.