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 ϵ2.
Highlights
The bootstrap philosophy [1, 2] was first applied to boundary conformal field theories (BCFTs) in [3] building on general results of [4, 5] and the methods introduced in [6]
As an application of the functional bootstrap equations, we study them in perturbation theory around free theory in 4 − dimensions to recover the Wilson-Fisher BCFT Operator Product Expansion (OPE) and Boundary Operator Expansion (BOE) data to O( 2)
In this work we have constructed two functional bases for the BCFT crossing equation, which are associated to generalized free field solutions with Neumann and Dirichlet boundary conditions
Summary
The bootstrap philosophy [1, 2] was first applied to boundary conformal field theories (BCFTs) in [3] building on general results of [4, 5] and the methods introduced in [6]. These sum rules are a complete reformulation of the original crossing equation, and have many nice properties Among them they provide upper and lower bounds on the Operator Product Expansion (OPE) data, diagonalize perturbation theory around generalized free solutions, and allow a rigorous derivation of the (Mellin-)Polyakov bootstrap [2, 22,23,24,25]. In this paper we will explain how to construct suitable bases of linear functionals that act on the crossing equation for a scalar CFT correlator φφ on a halfspace These functionals are associated to the generalized free boson CFT in the presence of a boundary with Neumann or Dirichlet boundary conditions. While this work was being completed we became aware of the work [29] which overlaps with ours
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