Boundary Conformal Field Theory

TL;DR

Boundary conformal field theory extends CFT to domains with boundaries, preserving conformal symmetry while introducing boundary-specific data. The paper surveys core CFT machinery—stress tensor, Virasoro algebra, and modular invariance—and develops the boundary framework via conformal boundary conditions, boundary states (Ishibashi and Cardy), and the annulus partition function to classify allowed boundary conditions and boundary operators. It then discusses boundary entropy, bulk-boundary OPE, and the role of extended algebras, plus connections to SLE, illustrating how boundary data organize and constrain rational CFTs with practical computational tools. Overall, BCFT provides a robust toolkit for understanding boundary phenomena in statistical mechanics and string theory, with concrete links between algebraic structures and geometric boundary data.

Abstract

Boundary conformal field theory (BCFT) is simply the study of conformal field theory (CFT) in domains with a boundary. It gains its significance because, in some ways, it is mathematically simpler: the algebraic and geometric structures of CFT appear in a more straightforward manner; and because it has important applications: in string theory in the physics of open strings and D-branes, and in condensed matter physics in boundary critical behavior and quantum impurity models. In this article, however, I describe the basic ideas from the point of view of quantum field theory, without regard to particular applications nor to any deeper mathematical formulations.

Figures (2)

• Figure 1: Two equivalent parametrizations of the same torus.
• Figure 2: The annulus, with boundary conditions $a$ and $b$ on either boundary.