SymTFT Entanglement and Holographic (Non)-Factorization

Abstract

Given two otherwise decoupled $D$-dimensional CFTs which possess a common (finite) symmetry subcategory, one can consider entangled boundary states of their $(D+1)$-dimensional SymTFTs. This roughly corresponds to performing a gauging of the tensor product of two CFTs, and we call this phenomena ``SymTFT entanglement" (or ``S-entanglement" for short). In the case when these CFTs have semiclassical holographic duals, the S-entanglement relates the bulk gauge charges between two otherwise disconnected AdS spacetimes as we highlight in several top-down examples. We show that taking partial traces of such S-entangled states leads to a streamlined approach to preparing ensemble-averaged CFTs in string theory. This ensemble averaging coincides with that generated by $\alpha$-states in the baby universe Hilbert space, and we propose a symmetry-enriched generalization of this Hilbert space via generalized global symmetries. We quantify how this symmetry-governed averaging violates holographic factorization and leads to the emergence of bulk global symmetries. We also consider the eternal (two-sided) AdS black hole geometries, where our SymTFT entanglement considerations imply that there exist refinements of the usual theromofield double state preparation of the system. We show that one may prepare the system in such a way that the total CFT data does not factorize into left and right copies. As anticipated by Marolf and Wall \cite{Marolf:2012xe}, we highlight that such considerations are necessary to define the gauge charges of eternal black holes, and in certain cases, can imply that there exist extended bulk objects stretching across the wormhole which cannot be expressed in terms of a product of left and right CFT operators.