Bootstrapping Boundaries and Branes

TL;DR

This paper develops a bootstrap approach to classify and constrain conformal boundary conditions in 2D CFTs, deriving universal bounds and identifying solutions in rational theories, with implications for string theory and quantum gravity.

Contribution

It introduces a systematic bootstrap framework for conformal boundaries in 2D CFTs, revealing strong constraints and solutions, and connecting to sphere packing and gravity.

Findings

Universal bounds on boundary entropy for stable boundary conditions

Unique solutions for stable branes in rational CFTs

Connections between bootstrap constraints and sphere packing problem

Abstract

The study of conformal boundary conditions for two-dimensional conformal field theories (CFTs) has a long history, ranging from the description of impurities in one-dimensional quantum chains to the formulation of D-branes in string theory. Nevertheless, the landscape of conformal boundaries is largely unknown, including in rational CFTs, where the local operator data is completely determined. We initiate a systematic bootstrap study of conformal boundaries in 2d CFTs by investigating the bootstrap equation that arises from the open-closed consistency condition of the annulus partition function with identical boundaries. We find that this deceivingly simple bootstrap equation, when combined with unitarity, leads to surprisingly strong constraints on admissible boundary states. In particular, we derive universal bounds on the tension (boundary entropy) of stable boundary conditions, which provide a rigorous diagnostic for potential D-brane decays. We also find unique solutions to the bootstrap problem of stable branes in a number of rational CFTs. Along the way, we observe a curious connection between the annulus bootstrap and the sphere packing problem, which is a natural extension of previous work on the modular bootstrap. We also derive bounds on the boundary entropy at large central charge. These potentially have implications for end-of-the-world branes in pure gravity on AdS$_3$.