Degeneration of the Elliptic Algebra $A_{q,p}(\widehat{sl_2})$ and Form Factors in the sine-Gordon Theory

TL;DR

This paper introduces a degeneration of the elliptic algebra related to $ ext{sl}_2$, explores its rational limit as a Yangian double, and reformulates form factor calculations in sine-Gordon theory using these algebraic structures.

Contribution

It presents a new degeneration of the elliptic algebra $A_{q,p}( ext{sl}_2)$, its boson realization, and applies these to reformulate form factor bootstrap in sine-Gordon theory.

Findings

Derived a new boson realization of the degenerated algebra.

Reformulated form factor bootstrap approach for sine-Gordon theory.

Proposed a conjectural integral formula for sine-Gordon form factors.

Abstract

Following the work with Jimbo and Miwa, we introduce a certain degeneration of the elliptic algebra $A_{q,p}(\widehat{sl_2})$ and its boson realization. We investigate its rational limit. The limit is the central extension of the Yangian double DY(sl_2) at level one. We give a new boson realization of it. Based on these algebras, we reformulate the Smirnov's form factor bootstrap approach to the sine-Gordon theory and the SU(2) invariant Thirring model. A conjectural integral formula for form factor in the sine-Gordon theory is derived.