Null-Vectors in Integrable Field Theory

Abstract

The form factor bootstrap approach allows to construct the space of local fields in the massive restricted sine-Gordon model. This space has to be isomorphic to that of the corresponding minimal model of conformal field theory. We describe the subspaces which correspond to the Verma modules of primary fields in terms of the commutative algebra of local integrals of motion and of a fermion (Neveu-Schwarz or Ramond depending on the particular primary field). The description of null-vectors relies on the relation between form factors and deformed hyper-elliptic integrals. The null-vectors correspond to the deformed exact forms and to the deformed Riemann bilinear identity. In the operator language, the null-vectors are created by the action of two operators Q (linear in the fermion) and C (quadratic in the fermion). We show that by factorizing out the null-vectors one gets the space of operators with the correct character. In the classical limit, using the operators Q and C we obtain a new, very compact, description of the KdV hierarchy. We also discuss a beautiful relation with the method of Whitham. 0 Membre du CNRS

Cohomologies of Affine Hyperelliptic Jacobi Varieties and Integrable Systems

“…Finally we would like to say that this relation to cohomology groups became clear analyzing the results of papers [2] and [3] which deal with quantum integrable models. Very briefly the reason for that is as follows.…”

On form factors and Macdonald polynomials

“…Some evidence for the one-to-one correspondence for a more general case was found in [14]. Rigorous proofs for some rather general cases were found in [15][16][17][18]. In the particular case of the Lee-Yang model a simple proof was proposed in [19,20].…”

Hidden local, quasi-local and non-local symmetries in integrable systems

“…those objects which at the quantum level constitute the spectrum of fields. 3 The spectrum of fields in the A (1) 1 framework.…”

“…Besides, the extension of this method in the scaling limit out of criticality is up to now unknown. More recenlty, it has been conjectured in [3] that one could add to LIMI I 2m+1 additional non-commuting charges J 2m in such a way that the resulting algebra (actually it is not clear from [3] if these objects close an algebra) would be sufficient to create all the states of a particular class of perturbed theory (Restricted Sine-Gordon Theory) of type (1.1). Therein it was also discovered that a sort of null-vector condition appears in the above procedure leading to certain equations for the form factors.…”

“…Therein it was also discovered that a sort of null-vector condition appears in the above procedure leading to certain equations for the form factors. However, what remains unclear in [3] are the field theory expressions of null vectors, the general procedure for finding them and above all the symmetry structures lying behind their arising. Besides, heavy use is made of the very specific form of the form factors of the RSG model, and it is not clear how to extend this procedure to other integrable theories of the form (1.1).…”