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Crossing Kernels for Boundary and Crosscap CFTs

Matthijs Hogervorst

TL;DR

This work extends the alpha-space bootstrap framework to two-point functions in BCFTs and crosscap CFTs in arbitrary dimensions, formulating integral representations via Casimir-eigenfunction bases and deriving spectral-density equations governed by crossing kernels. It shows that both boundary and crosscap kernels can be obtained as special limits of the d=1 crossing kernel, revealing a unifying structure behind disparate backgrounds. The analysis provides explicit Wilson-function-based kernels, unitary crossing maps between bulk/boundary and crosscap spaces, and mean-field alpha-space solutions, suggesting analytic pathways for solving crossing in these settings and potential links to holography and Liouville theory. Overall, the paper deepens the analytical understanding of conformal constraints in nontrivial backgrounds and lays groundwork for further exploration of defect and nontrivial topology 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.

Crossing Kernels for Boundary and Crosscap CFTs

TL;DR

This work extends the alpha-space bootstrap framework to two-point functions in BCFTs and crosscap CFTs in arbitrary dimensions, formulating integral representations via Casimir-eigenfunction bases and deriving spectral-density equations governed by crossing kernels. It shows that both boundary and crosscap kernels can be obtained as special limits of the d=1 crossing kernel, revealing a unifying structure behind disparate backgrounds. The analysis provides explicit Wilson-function-based kernels, unitary crossing maps between bulk/boundary and crosscap spaces, and mean-field alpha-space solutions, suggesting analytic pathways for solving crossing in these settings and potential links to holography and Liouville theory. Overall, the paper deepens the analytical understanding of conformal constraints in nontrivial backgrounds and lays groundwork for further exploration of defect and nontrivial topology 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.

Paper Structure

This paper contains 23 sections, 3 theorems, 151 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Warm-up: toy bootstrap in alpha space
  3. Review of the Jacobi transform
  4. Boundary CFTs
  5. Boundary bootstrap equation
  6. Sturm-Liouville theory in the bulk…
  7. … and on the boundary
  8. Intermezzo: conformal Regge theory
  9. Crossing kernel
  10. Crosscap CFTs
  11. Bootstrap condition
  12. Sturm-Liouville theory for the crosscap
  13. Bootstrap in alpha space
  14. Analyzing the crossing kernels
  15. Recap: $d=1$ crossing kernel
  16. ...and 8 more sections

Key Result

Theorem 1

$\mathsf{K}$ is a unitary operator between $\mathcal{H}[\mathsf{h}_1,\mathsf{h}_2,\mathsf{h}_3,\mathsf{h}_4]$ and $\mathcal{H}[\mathsf{h}_3,\mathsf{h}_2,\mathsf{h}_1,\mathsf{h}_4]$.

Figures (2)

  • Figure 1: Schematic representation of the boundary bootstrap equation in alpha space.
  • Figure 2: Boundary (resp. bulk) partial waves can be expressed in terms of bulk (boundary) ones by means of the kernels $E_{\mathrm{bdy} \to \mathrm{bulk}}$ and ${E}_{\mathrm{bulk}\to \mathrm{bdy}}$.

Theorems & Definitions (3)

  • Theorem : 1.1 of Hogervorst:2017sfd
  • Theorem : 1
  • Theorem : 2