Branes: from free fields to general backgrounds

TL;DR

The paper develops a comprehensive, nonperturbative framework for open conformal field theories and D-branes in arbitrary backgrounds by introducing two key boundary data: a consistent set of reflection coefficients and a fusion-rule automorphism $\omega$ that preserves conformal weights. Boundary conditions are shown to be classified by irreducible representations of a commutative associative classifying algebra $\mathfrak C_{\omega}$, with each representation encoding the worldvolume and flat-connection moduli of a D-brane. The authors formulate the open theory through oriented covers, chiral blocks, and factorization constraints, extending WZW-type structures (block algebras, co-invariants) to general CFTs via the automorphism framework. They also connect these CFT constructions to string-theoretic amplitudes, deriving disk and annulus amplitudes and clarifying how Chan-Paton data arise from boundary conditions. Overall, the work provides a principled, algebraic route to D-brane spectra in generic CFT backgrounds and suggests avenues to generalize Verlinde-like structures to open theories.

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 $ω$ of the fusion rules that preserves conformal weights. Non-trivial automorphisms $ω$ correspond to D-brane configurations for arbitrary conformal field theories. The choice of the fusion rule automorphism $ω$ 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 $ω$ 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.