Russian doll renormalization group and Kosterlitz–Thouless flows

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( sl(2)) quantum affine symmetry, with q 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 Δ<−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.