Null-vectors in Integrable Field Theory

TL;DR

This paper explores the structure of null-vectors in integrable quantum field theories, linking form factors, algebraic operators, and classical hierarchies to deepen understanding of the sine-Gordon model and conformal field theory.

Contribution

It introduces a novel description of null-vectors using deformed hyper-elliptic integrals and operators, connecting quantum null-vectors to classical integrable hierarchies.

Findings

Null-vectors correspond to deformed exact forms and Riemann bilinear identities.

Operators $ ext{Q}$ and $ ext{C}$ generate null-vectors and help recover the operator space.

Classical limit yields a compact description of the KdV hierarchy and links to Whitham 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 $\CQ$ (linear in the fermion) and $\CC$ (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 $\CQ$ and $\CC$ we obtain a new, very compact, description of the KdV hierarchy. We also discuss a beautiful relation with the method of Whitham.