Entanglement branes and factorization in conformal field theory

TL;DR

The paper addresses how to factorize the local Hilbert space of a 2D CFT by recasting factorization as cobordisms dictated by OPE data, and introduces an entanglement brane (E-brane) boundary condition realized by the vacuum Ishibashi state to close entangling holes. It develops an explicit framework in which edge modes arise from a sum over superselection sectors labeled by Cardy boundary conditions, and shows that the factorization map is governed by the CFT co-product, connecting to fusion rules and a Bogoliubov-like transformation in the free boson case. The main contributions include the E-brane axiom, a concrete construction for the free compact boson, a three-point-function–based factorization of primaries, and a co-product formulation that yields a local tensor-product structure compatible with CFT fusion. The findings offer a CFT-based perspective on entanglement with edge modes, with potential implications for tensor-network renormalization, BC-bits in holography, and extensions to RCFTs and holographic CFTs. Overall, the work proposes a principled route to define and compute factorized CFT states using OPE data and boundary-CFT structure, linking entanglement to edge degrees of freedom and symmetry co-products.

Abstract

In this work, we consider the question of local Hilbert space factorization in 2D conformal field theory. Generalizing previous work on entanglement and open-closed TQFT, we interpret the factorization of CFT states in terms of path integral processes that split and join the Hilbert spaces of circles and intervals. More abstractly, these processes are cobordisms of an extended CFT which are defined purely in terms of the OPE data. In addition to the usual sewing axioms, we impose an entanglement boundary condition that is satisfied by the vacuum Ishibashi state. This choice of entanglement boundary state leads to reduced density matrices that sum over super-selection sectors, which we identify as the CFT edge modes. Finally, we relate our factorization map to the co-product formula for the CFT symmetry algebra, which we show is equivalent to a Boguliubov transformation in the case of a free boson.

Figures (14)

• Figure 1: The basic objects and cobordisms of a 2D extended TQFT. The extension from the closed to open sector introduces boundary labelings on the open string, which has to match whenever their endpoints meet
• Figure 2: Shown are five sewing axioms which ensures the consistency of different gluings. The last one is the Cardy condition
• Figure 3: The basic CFT cobordisms can be mapped to the punctured (half) plane. Given a choice of punctures, the cobordism is a linear map between Hilbert spaces at the punctures, whose components are determined by the OPE coefficients.
• Figure 4: By viewing the annulus path integral as a closed string amplitude, the open string boundary conditions can be mapped to closed string states.
• Figure 5: We define factorization maps from the open-closed cobordisms on the left. These satisfy the an E-brane boundary condition that allows us to close up holes.