Universal Dynamics of Heavy Operators in Boundary CFT$_2$

TL;DR

The authors establish universal high-energy asymptotics for averaged bulk-to-boundary and boundary OPE coefficients in unitary, irrational BCFT$_2$ with $c>1$, showing that a single universal density $C_0$ governs all heavy regimes. They derive these results via crossing constraints on higher-genus surfaces with open boundaries and irrational crossing kernels, and they express the asymptotics in terms of the central charge and boundary entropies, independently of microscopic details. The work extends the Eigenstate Thermalization Hypothesis to BCFT data and connects to holographic interpretations in AdS$_3$/BCFT$_2$, including black hole and Schwarzian regimes, as well as conical-defect limits. The results provide a unified, symmetry-driven picture of heavy BCFT data and offer a framework for exploring BCFTs in gravitational contexts such as evaporating black holes and wedge holography.

Abstract

We derive a universal asymptotic formula for generic boundary conditions for the average value of the bulk-to-boundary and boundary Operator Product Expansion coefficients of any unitary, compact two-dimensional Boundary CFT (BCFT) with $c>1$. The asymptotic limit consists of taking one or more boundary primary operators -- which transform under a single copy of the Virasoro algebra -- to have parametrically large conformal dimension for fixed central charge. In particular, we find a \textit{single} universal expression that interpolates between distinct heavy regimes, exactly as in the case of bulk OPE asymptotics\cite{Collier:2019weq}. The expression depends universally on the boundary entropy and the central charge, and not on any other details of the theory. We derive these asymptotics by studying crossing symmetry of various correlation functions on higher genus Riemann surfaces with open boundaries. Essential in the derivation is the use of the irrational versions of the crossing kernels that relate holomorphic Virasoro blocks in different channels. Our results strongly suggest an extended version of the Eigenstate Thermalization Hypothesis for boundary OPE coefficients, where the hierarchy between the diagonal and non-diagonal term in the ansatz is further controlled by the boundary entropy. We finally comment on the applications of our results in the context of $\text{AdS}_3/\text{BCFT}_2$, as well as on the recent relation of BCFTs with lower dimensional models of evaporating black holes.

Figures (11)

• Figure 1: The three elementary "legos" out of which one can construct a BCFT$_2$ correlation function on a Riemann surface with open boundaries: (i) bulk OPE structure constant , (ii) bulk-to-boundary structure constant and (iii) boundary structure constant. The solid orange lines represent points on the bulk surface, whereas solid black lines represent a conformal boundary. Local bulk operator insertions are depicted as orange circles, while local boundary operator insertions as dashed blue lines.
• Figure 2: The open-closed duality of the cylinder one-point function with an external (boundary) operator labelled by $\Psi_0$. The blocks on the two channels are related with a holomorphic copy of the modular kernel.
• Figure 3: The bulk OPE channel (left) and boundary OPE channel (right) of a bulk two-point function on the disk for identical bulk external operators. The corresponding conformal blocks are related with a holomorphic copy of the fusion kernel.
• Figure 4: The decomposition of the" tadpole" channel conformal block in terms of "OC-loop" channel conformal blocks for the genus-one surface with a conformal boundary labelled by $a$: first we interchange cycles on the torus with an S-transformation, and then we change to the dual channel of the resulting disk two-point function (of identical bulk operators, suitably summed over) with a holomorphic fusion transformation.
• Figure 5: The "boundary necklace" channel (left) and the "boundary bagel" channel (right) of the cylinder two-point function with identical boundary operators and identical boundary conditions on each boundary.