A Monoidal Category for Perturbed Defects in Conformal Field Theory

TL;DR

This paper constructs a monoidal category framework to model perturbed conformal defects in 2D conformal field theory, enabling the derivation of functional relations that reveal integrable structures.

Contribution

It introduces a new abelian rigid monoidal category C_F capturing perturbed defects, linking algebraic structures to conformal field theory operators.

Findings

Defines the category C_F from C for perturbed defects

Associates operators to objects in C_F on CFT state space

Derives functional relations among perturbed defect operators

Abstract

Starting from an abelian rigid braided monoidal category C we define an abelian rigid monoidal category C_F which captures some aspects of perturbed conformal defects in two-dimensional conformal field theory. Namely, for V a rational vertex operator algebra we consider the charge-conjugation CFT constructed from V (the Cardy case). Then C = Rep(V) and an object in C_F corresponds to a conformal defect condition together with a direction of perturbation. We assign to each object in C_F an operator on the space of states of the CFT, the perturbed defect operator, and show that the assignment factors through the Grothendieck ring of C_F. This allows one to find functional relations between perturbed defect operators. Such relations are interesting because they contain information about the integrable structure of the CFT.