Comments on Entanglement Entropy in String Theory

TL;DR

This paper proposes a new definition of entanglement entropy in string theory using string field theory, addressing divergences and the role of edge modes at the entanglement cut.

Contribution

It introduces a string field theory framework for defining entanglement entropy, including the treatment of gauge modes and edge states, and demonstrates divergence regulation.

Findings

Reproduces effective field theory entanglement entropy as a sum over string excitations.

Shows string theory regularizes ultraviolet divergences in entanglement entropy.

Identifies dynamical gauge modes at the entanglement cut requiring an extended Hilbert space.

Abstract

Entanglement entropy for spatial subregions is difficult to define in string theory because of the extended nature of strings. Here we propose a definition for Bosonic open strings using the framework of string field theory. The key difference (compared to ordinary quantum field theory) is that the subregion is chosen inside a Cauchy surface in the "space of open string configurations". We first present a simple calculation of this entanglement entropy in free light-cone string field theory, ignoring subtleties related to the factorization of the Hilbert space. We reproduce the answer expected from an effective field theory point of view, namely a sum over the one-loop entanglement entropies corresponding to all the particle-excitations of the string, and further show that the full string theory regulates the ultraviolet divergences in the entanglement entropy. We then revisit the question of factorization of the Hilbert space by analyzing the covariant phase-space associated with a subregion in Witten's covariant string field theory. We show that the pure gauge (i.e., BRST exact) modes in the string field become dynamical at the entanglement cut. Thus, a proper definition of the entropy must involve an extended Hilbert space, with new stringy edge modes localized at the entanglement cut.