Form factors in sinh- and sine-Gordon models, deformed Virasoro algebra, Macdonald polynomials and resonance identities

TL;DR

This paper advances the algebraic understanding of form factors in sinh- and sine-Gordon models by linking them to deformed Virasoro algebra, Macdonald polynomials, and resonance identities, providing new derivations and representations.

Contribution

It introduces a novel algebraic construction connecting form factors to a limit of deformed Virasoro and Heisenberg algebras, and relates singular vectors to Macdonald polynomials.

Findings

Singular vectors expressed via degenerate Macdonald polynomials

Resonance identities derived from matrix elements with singular and cosingular vectors

New derivation of sinh-Gordon equation of motion and conserved currents

Abstract

We continue the study of form factors of descendant operators in the sinh- and sine-Gordon models in the framework of the algebraic construction proposed in [arXiv:0812.4776]. We find the algebraic construction to be related to a particular limit of the tensor product of the deformed Virasoro algebra and a suitably chosen Heisenberg algebra. To analyze the space of local operators in the framework of the form factor formalism we introduce screening operators and construct singular and cosingular vectors in the Fock spaces related to the free field realization of the obtained algebra. We show that the singular vectors are expressed in terms of the degenerate Macdonald polynomials with rectangular partitions. We study the matrix elements that contain a singular vector in one chirality and a cosingular vector in the other chirality and find them to lead to the resonance identities already known in the conformal perturbation theory. Besides, we give a new derivation of the equation of motion in the sinh-Gordon theory, and a new representation for conserved currents.