# The Functional Bootstrap for Boundary CFT

## TL;DR

This paper develops a new functional bootstrap approach for boundary conformal field theories (BCFTs), providing sum rules, fixing ambiguities, and recovering Wilson-Fisher BCFT data in the epsilon expansion.

## Contribution

It introduces a basis for the crossing equation in BCFTs, relates it to a Polyakov-type approach, and diagonalizes perturbation theory around generalized free fields.

## Key findings

- Derives boundary bootstrap sum rules from a new basis.
- Fixes contact term ambiguities in the Polyakov approach.
- Recovers Wilson-Fisher BCFT data to order epsilon^2.

## Abstract

We introduce a new approach to the study of the crossing equation for CFTs in the presence of a boundary. We argue that there is a basis for this equation related to the generalized free field solution. The dual basis is a set of linear functionals which act on the crossing equation to give a set of sum rules on the boundary CFT data: the functional bootstrap equations. We show these equations are essentially equivalent to a Polyakov-type approach to the bootstrap of BCFTs, and show how to fix the so-called contact term ambiguity in that context. Finally, the functional bootstrap equations diagonalize perturbation theory around generalized free fields, which we use to recover the Wilson-Fisher BCFT data in the $\epsilon$-expansion to order $\epsilon^2$.