Russian Doll Renormalization Group, Kosterlitz-Thouless Flows, and the Cyclic sine-Gordon model

TL;DR

This paper explores the cyclic behavior of Kosterlitz-Thouless RG flows, connecting it to quantum affine symmetry, S-matrix formulations, and potential string theory implications, supported by analytical and symmetry-based arguments.

Contribution

It introduces a novel cyclic RG regime in the Kosterlitz-Thouless flow, links it to quantum affine symmetry, and proposes specific S-matrices that describe this cyclic behavior, including a string theory-like spectrum.

Findings

The RG cycle period is computed via RG invariants.

A quantum affine symmetry $U_q(\,hat{sl(2)})$ is established in the theory.

A conjectured S-matrix describes the cyclic RG regime and exhibits a Russian doll scaling of resonances.

Abstract

We investigate the previously proposed cyclic regime of the Kosterlitz-Thouless renormalization group (RG) flows. The period of one cycle is computed in terms of the RG invariant. Using bosonization, we show that the theory has $U_q (\hat{sl(2)})$ quantum affine symmetry, with $q$ {\it real}. Based on this symmetry, we study two possible S-matrices for the theory, differing only by overall scalar factors. We argue that one S-matrix corresponds to a continuum limit of the XXZ spin chain in the anti-ferromagnetic domain $\Delta < -1$. The latter S-matrix has a periodicity in energy consistent with the cyclicity of the RG. We conjecture that this S-matrix describes the cyclic regime of the Kosterlitz-Thouless flows. The other S-matrix we investigate is an analytic continuation of the usual sine-Gordon one. It has an infinite number of resonances with masses that have a Russian doll scaling behavior that is also consistent with the period of the RG cycles computed from the beta-function. Closure of the bootstrap for this S-matrix leads to an infinite number of particles of higher spin with a mass formula suggestive of a string theory.