Free Fermion Representation of a Boundary Conformal Field Theory

TL;DR

Polchinski and Thorlacius analyze a massless 2D scalar with a periodic boundary interaction and show that at a critical period the system is a boundary conformal field theory that can be reformulated in terms of free fermions, exposing an SU(2) current algebra. They construct an exact fermionic representation by introducing an auxiliary boson and cocycles, derive the exact partition function as a function of the boundary coupling, and outline the boundary S-matrix in this language. With two interacting boundaries, the spectrum forms a band structure that interpolates between free propagation and tight-binding, analogous to a periodic tachyon background. The work provides a precise solvable framework for boundary perturbations in CFT and clarifies the role of open string tachyon backgrounds in boundary scattering and spectrum.

Abstract

The theory of a massless two-dimensional scalar field with a periodic boundary interaction is considered. At a critical value of the period this system defines a conformal field theory and can be re-expressed in terms of free fermions, which provide a simple realization of a hidden $SU(2)$ symmetry of the original theory. The partition function and the boundary $S$-matrix can be computed exactly as a function of the strength of the boundary interaction. We first consider open strings with one interacting and one Dirichlet boundary, and then with two interacting boundaries. The latter corresponds to motion in a periodic tachyon background, and the spectrum exhibits an interesting band structure which interpolates between free propagation and tight binding as the interaction strength is varied.