Branes: from free fields to general backgrounds

TL;DR

This paper explores the mathematical structure of boundary conditions in conformal field theories, linking automorphisms and algebraic classifications to D-brane configurations in string theory.

Contribution

It introduces a framework connecting boundary conditions, fusion rule automorphisms, and algebraic classifications for D-branes in arbitrary conformal field theories.

Findings

Boundary conditions classified by irreducible representations of a generalized fusion algebra.

Automorphisms of fusion rules correspond to D-brane configurations.

Moduli of D-branes linked to algebraic representations and flat connections.

Abstract

Motivated by recent developments in string theory, we study the structure of boundary conditions in arbitrary conformal field theories. A boundary condition is specified by two types of data: first, a consistent collection of reflection coefficients for bulk fields on the disk; and second, a choice of an automorphism $\omega$ of the fusion rules that preserves conformal weights. Non-trivial automorphisms $\omega$ correspond to D-brane configurations for arbitrary conformal field theories. The choice of the fusion rule automorphism $\omega$ amounts to fixing the dimension and certain global topological features of the D-brane world volume and the background gauge field on it. We present evidence that for fixed choice of $\omega$ the boundary conditions are classified as the irreducible representations of some commutative associative algebra, a generalization of the fusion rule algebra. Each of these irreducible representations corresponds to a choice of the moduli for the world volume of the D-brane and the moduli of the flat connection on it.