Crossing Kernels for Boundary and Crosscap CFTs

TL;DR

This paper develops a unified integral kernel framework for boundary and crosscap conformal field theories, extending the alpha space method to higher dimensions and revealing their relation to the d=1 crossing kernel.

Contribution

It introduces integral representations for scalar two-point functions in boundary and crosscap CFTs using conformal Casimir eigenfunctions, generalizing the alpha space approach.

Findings

Boundary and crosscap kernels are special limits of the d=1 crossing kernel.

Integral equations for spectral densities are derived from CFT consistency.

The method provides a new perspective on boundary and crosscap CFTs in higher dimensions.

Abstract

This paper investigates d-dimensional CFTs in the presence of a codimension-one boundary and CFTs defined on real projective space RP^d. Our analysis expands on the alpha space method recently proposed for one-dimensional CFTs in arXiv:1702.08471. In this work we establish integral representations for scalar two-point functions in boundary and crosscap CFTs using plane-wave-normalizable eigenfunctions of different conformal Casimir operators. CFT consistency conditions imply integral equations for the spectral densities appearing in these decompositions, and we study the relevant integral kernels in detail. As a corollary, we find that both the boundary and crosscap kernels can be identified with special limits of the d=1 crossing kernel.