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The Functional Bootstrap for Boundary CFT

Apratim Kaviraj, Miguel F. Paulos

TL;DR

This work develops a functional bootstrap for boundary conformal field theories by constructing a dual basis of linear functionals to generalized free field BCFT solutions. Acting on the BCFT crossing equation, these functionals yield sum rules that constrain BCFT data and fix contact-term ambiguities, connecting to a Polyakov-block formulation via Witten-diagram decompositions in AdS with a brane. The authors show how the functional basis diagonalizes perturbation theory around the generalized free solution, enabling a controlled ε-expansion to recover Wilson-Fisher BCFT data to $O(\epsilon^2)$. They provide explicit Witten-diagram computations for BCFTs under Neumann/Dirichlet boundary conditions and establish a consistent bridge between the functional bootstrap and Polyakov-type bootstrap approaches, with potential extensions to higher dimensions, spinning operators, and supersymmetric settings.

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 $ε$-expansion to order $ε^2$.

The Functional Bootstrap for Boundary CFT

TL;DR

This work develops a functional bootstrap for boundary conformal field theories by constructing a dual basis of linear functionals to generalized free field BCFT solutions. Acting on the BCFT crossing equation, these functionals yield sum rules that constrain BCFT data and fix contact-term ambiguities, connecting to a Polyakov-block formulation via Witten-diagram decompositions in AdS with a brane. The authors show how the functional basis diagonalizes perturbation theory around the generalized free solution, enabling a controlled ε-expansion to recover Wilson-Fisher BCFT data to . They provide explicit Witten-diagram computations for BCFTs under Neumann/Dirichlet boundary conditions and establish a consistent bridge between the functional bootstrap and Polyakov-type bootstrap approaches, with potential extensions to higher dimensions, spinning operators, and supersymmetric settings.

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 -expansion to order .

Paper Structure

This paper contains 38 sections, 199 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Review of the boundary bootstrap
  3. Functionals and a simple basis
  4. Functional ansatz and constraints
  5. Warm-up: a simple basis
  6. Special values of ${\Delta_\phi}$ and orthonormal functionals
  7. Shifted functionals and general ${\Delta_\phi}$
  8. Functional bootstrap equations
  9. Bootstrap equations and BCFT basis
  10. Witten diagrams and functional equations
  11. Constraining the kernels
  12. Boundary functionals
  13. General solution
  14. Neumann contact term ambiguity
  15. Shifted functionals
  16. ...and 23 more sections

Figures (4)

  • Figure 1: The bulk exchange, boundary exchange and contact Witten diagrams respectively. The semicircular area represents $AdS_{d+1}$ bulk in the presence of a brane, which is represented by the dashed line.
  • Figure 2: The boundary exchange Witten diagram.
  • Figure 3: The bulk exchange Witten diagrams can be written as a sum over two diagrams on full $AdS_{d+1}$. In the figure we show the Neumann case, Dirichlet would correspond to a minus sign on the RHS.
  • Figure 4: The contact Witten diagram.