An Analytic Approach to BCFT$_d$

TL;DR

The paper develops an analytic BCFT bootstrap framework for the two-point function of scalar primaries in the presence of a boundary, recasting crossing as a vector problem in a space of analytic functions and enforcing Regge boundedness. It constructs a MF-adapted basis of bulk and boundary blocks, and defines Polyakov blocks tied to improved Witten diagrams, establishing a BCFT version of the Polyakov bootstrap together with a Lorentzian inversion perspective. The authors derive explicit actions of dual functionals on conformal blocks, relate these to Witten-diagram decompositions in AdS with Neumann boundary conditions, and present Euclidean and Lorentzian inversion formulae that yield Polyakov expansions and sum rules. They further develop a deformation of MF theory interpolating Neumann and Dirichlet boundary conditions, providing holographic bounds and analytic control over boundary spectra, and showing how their results illuminate the Polyakov bootstrap in BCFT. The work offers a versatile analytic toolkit for perturbations around mean-field BCFT, with potential applications to holographic BCFTs and boundary conformal manifolds.

Abstract

We develop an analytic approach to Boundary Conformal Field Theory (BCFT), focussing on the two-point function of a general pair of scalar primary operators. The resulting crossing equation can be thought of as a vector equation in an infinite-dimensional space ${\cal V}$ of analytic functions of a single complex variable. We argue that in a unitary theory, functions in ${\cal V}$ satisfy a boundedness condition in the Regge limit. We identify a useful basis for ${\cal V}$, consisting of bulk and boundary conformal blocks with scaling dimensions which appear in OPEs of the mean field theory correlator. Our main achievement is an explicit expression for the action of the dual basis (the basis of liner functionals on ${\cal V}$) on an arbitrary conformal block. The practical merit of our basis is that it trivializes the study of perturbations around mean field theory. Our results are equivalent to a BCFT version of the Polyakov bootstrap. Our derivation of the expressions for the functionals relies on the identification of the Polyakov blocks with (suitably improved) boundary and bulk Witten exchange diagrams in $AdS_{d+1}$. We also provide another conceptual perspective on the Polyakov block expansion and the associated functionals, by deriving a new Lorentzian OPE inversion formula for BCFT.

Figures (8)

• Figure 1: Left: The Regge limit of a two-point function in the presence of a conformal boundary condition. The thick vertical line represents the (timelike) boundary. In the Regge limit, $\mathcal{O}_2$ approaches the lightcone of the mirror reflection of $\mathcal{O}_1$. Right: To reach the Regge limit, we should start with $\mathcal{G}(\xi)$ in the Euclidean regime $\xi>0$ and analytically continue to $\xi\rightarrow -1$. In doing so, we need to go around the branch point $\xi = 0$.
• Figure 2: Tree level Witten diagrams in $hAdS_{d+1}^N$. The semi-disk represents the $hAdS_{d+1}^N$ space which terminates at an $AdS_d$, represented by the horizontal line.
• Figure 3: A contact Witten diagram $W^{contact}$ in the probe brane setup. The disk represent $AdS_{d+1}$ space and the horizontal line represent the $AdS_d$ interface.
• Figure 4: A bulk channel exchange Witten diagram $W^{bulk}$ in the probe brane setup.
• Figure 5: A boundary channel exchange Witten diagram $W^{boundary}$ in the probe brane setup.