Bootstrapping boundary-localized interactions

TL;DR

This paper investigates conformal boundary conditions for a free scalar field, revealing restrictions on boundary dynamics and hinting at a potential new boundary condition through numerical bootstrap analysis.

Contribution

It demonstrates that the bulk-to-boundary operator expansion involves at most a shadow pair and identifies a possible new boundary condition via numerical bounds.

Findings

Bulk-to-boundary expansion involves at most a shadow pair.

Numerical bootstrap excludes large parameter regions.

A kink suggests a potential new boundary condition.

Abstract

We study conformal boundary conditions for the theory of a single real scalar to investigate whether the known Dirichlet and Neumann conditions are the only possibilities. For this free bulk theory there are strong restrictions on the possible boundary dynamics. In particular, we find that the bulk-to-boundary operator expansion of the bulk field involves at most a `shadow pair' of boundary fields, irrespective of the conformal boundary condition. We numerically analyze the four-point crossing equations for this shadow pair in the case of a three-dimensional boundary (so a four-dimensional scalar field) and find that large ranges of parameter space are excluded. However a `kink' in the numerical bounds obeys all our consistency checks and might be an indication of a new conformal boundary condition.