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Analytic Bootstrap for Boundary CFT

Agnese Bissi, Tobias Hansen, Alexander Söderberg

TL;DR

This work develops an analytic bootstrap framework for BCFT two-point functions by examining the Lorentzian analytic structure of bulk and boundary conformal blocks and exploiting an image-symmetry to simplify the crossing equations. The authors extend the Wilson-Fisher BCFT analysis to $O(\epsilon^2)$, obtaining bulk OPE coefficients, boundary OPE (BOE) data, and the full correlator through a discontinuity-based inversion that circumvents infinite sums. They unify consistency checks with known anomalous dimensions and derive explicit results for Neumann and Dirichlet boundary conditions, including new BOE coefficients and a complete expression for the two-point function up to $O(\epsilon^2)$. The approach generalizes Caron-Huot-type inversion to BCFTs, offering a robust route to analyze other defects and higher-order corrections in a controlled perturbative expansion.

Abstract

We propose a method to analytically solve the bootstrap equation for two point functions in boundary CFT. We consider the analytic structure of the correlator in Lorentzian signature and in particular the discontinuity of bulk and boundary conformal blocks to extract CFT data. As an application, the correlator $\langle φφ\rangle$ in $φ^4$ theory at the Wilson-Fisher fixed point is computed to order $ε^2$ in the $ε$ expansion.

Analytic Bootstrap for Boundary CFT

TL;DR

This work develops an analytic bootstrap framework for BCFT two-point functions by examining the Lorentzian analytic structure of bulk and boundary conformal blocks and exploiting an image-symmetry to simplify the crossing equations. The authors extend the Wilson-Fisher BCFT analysis to , obtaining bulk OPE coefficients, boundary OPE (BOE) data, and the full correlator through a discontinuity-based inversion that circumvents infinite sums. They unify consistency checks with known anomalous dimensions and derive explicit results for Neumann and Dirichlet boundary conditions, including new BOE coefficients and a complete expression for the two-point function up to . The approach generalizes Caron-Huot-type inversion to BCFTs, offering a robust route to analyze other defects and higher-order corrections in a controlled perturbative expansion.

Abstract

We propose a method to analytically solve the bootstrap equation for two point functions in boundary CFT. We consider the analytic structure of the correlator in Lorentzian signature and in particular the discontinuity of bulk and boundary conformal blocks to extract CFT data. As an application, the correlator in theory at the Wilson-Fisher fixed point is computed to order in the expansion.

Paper Structure

This paper contains 12 sections, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Analytic structure of BCFT correlators
  3. Image symmetry
  4. Review: BCFT bootstrap up to $O(\epsilon)$
  5. BCFT bootstrap at $O(\epsilon^2)$
  6. Bulk OPE coefficients
  7. Using image symmetry
  8. Boundary OE coefficients
  9. Full correlator
  10. Outlook: $O(\epsilon^3)$
  11. Conclusions and outlook
  12. $G_b$ and $G_i$ at order $\epsilon^2$

Figures (4)

  • Figure 1: Coordinates $x$, $y$ and boundary at $x_\perp = 0$. Also pictured are the mirror images $\overline{x}$ and $\overline{y}$.
  • Figure 2: Analytic structure of $g_i(n,z)$ (left) and $g_b(2n,z)$ (right) for positive integer $n$.
  • Figure 3: Analytic structure of $F(z)$ and paths of analytic continuation to negative $z$.
  • Figure 4: Causal structure for different values of $z$. For simplicity of illustration $\vec{x}$ and $\vec{y}$ are chosen to point into the direction of time. Dotted lines indicate lightcones.