A note on entanglement edge modes in Chern Simons theory

TL;DR

The paper addresses how to consistently factorize the Hilbert space in Chern-Simons theory across a spatial entangling surface by introducing edge modes as part of an extended Hilbert space. It presents two universal derivations: (i) a UV regularization of the entangling surface in the Euclidean path integral that yields a thermally populated edge sector and an explicit Ishibashi-state embedding, and (ii) the entangling-product construction that glues edge Hilbert spaces via diagonal loop-group invariance. The results show that the reduced density matrix is a thermal state of the boundary CFT and reproduce the topological entanglement entropy, with S_top = -log D + log d_r, highlighting the role of edge degrees of freedom in topological entanglement. The work provides a framework applicable to holographic TQFTs and suggests connections to gravitational and string-theoretic descriptions of entanglement, including the notion of an entanglement brane.

Abstract

We elaborate on the extended Hilbert space factorization of Chern Simons theory and show how this arises naturally from a proper regularization of the entangling surface in the Euclidean path integral. The regularization amounts to stretching the entangling surface into a co-dimension one surface which hosts edge modes of the Chern Simons theory when quantized on a spatial subregion. The factorized state is a regularized Ishibashi state and reproduces the well known topological entanglement entropies. We illustrate how the same factorization arises from the glueing of two spatial subregions via the entangling product defined by Donnelly and Freidel.

Figures (1)

• Figure 1: The wavefunctional prepared by the Euclidean path integral can be sliced in angular time $\theta$, provided we remove an $\epsilon$ neighborhood of the entangling surface. When viewed as the initial and final time slices in $\theta$, region $A$ and $B$ are assigned opposite orientations relative to the ambient space $A\cup B$. Thus they support the Hilbert space of a chiral and anti-chiral edge CFT, which transforms non-trivially under the boundary gauge group. In addition to parity, switching from $t$ to $\theta$ also involves a time reversal because they have opposite orientations on region $B$ at the $t=0$ time slice. Sliced in angular time, the Euclidean path integral prepares a Thermofield double state in which the left right entanglement of these CFT's leads to singlet state under a diagonal action of the boundary gauge group on $\partial A$ and $\partial B$