Massless Flows I: the sine-Gordon and O(n) models

TL;DR

This paper explores the massless flow between minimal models in conformal field theory through the sine-Gordon and O(n) models, revealing roaming behavior, analytic continuations, and applications to polymer connectivity constants.

Contribution

It provides a detailed analysis of massless flows using multiple approaches, including numerical, analytical, and lattice regularization, with novel insights into supersymmetry and polymer models.

Findings

Roaming behavior with central charge oscillations between UV and IR.

Analytic continuation of Casimir energy for massive flows.

Exact results for polymer graph connectivity constants.

Abstract

The massless flow between successive minimal models of conformal field theory is related to a flow within the sine-Gordon model when the coefficient of the cosine potential is imaginary. This flow is studied, partly numerically, from three different points of view. First we work out the expansion close to the Kosterlitz-Thouless point, and obtain roaming behavior, with the central charge going up and down in between the UV and IR values of $c=1$. Next we analytically continue the Casimir energy of the massive flow (i.e. with real cosine term). Finally we consider the lattice regularization provided by the O(n) model in which massive and massless flows correspond to high- and low-temperature phases. A detailed discussion of the case $n=0$ is then given using the underlying N=2 supersymmetry, which is spontaneously broken in the low-temperature phase. The ``index'' $\tr F(-1)^F$ follows from the Painleve III differential equation, and is shown to have simple poles in this phase. These poles are interpreted as occuring from level crossing (one-dimensional phase transitions for polymers). As an application, new exact results for the connectivity constants of polymer graphs on cylinders are obtained.