Comments on Entanglement Entropy in String Theory

TL;DR

This work proposes a string-field-theory-based definition of entanglement entropy for spatial subregions by extending spacetime subregions into the space of open-string configurations and analyzing entanglement in the free theory. In the light-cone gauge, the EE reduces to a sum over entropies of individual string excitations, reproducing effective-field-theory expectations while benefiting from stringy UV finiteness. The paper then shows that gauge symmetry obstructs Hilbert space factorization for subregions, necessitating an extended Hilbert space with stringy edge modes, and develops this picture via the covariant phase-space formalism, including a toy Chern-Simons example. Together, these results indicate a consistent, intrinsically stringy approach to EE with potential connections to horizon entropy and holography, while outlining significant questions for closed strings and finite coupling corrections.

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.

Figures (3)

• Figure 1: (Left) From a spacetime point of view, we're interested in the degrees of freedom inside a subregion $R$ of the Cauchy Surface $S$. However, in string theory we can have strings which either lie in $R$ (blue), or straddle the cut $\partial R$ (violet), or lie outside $R$ (red). (Right) In string field theory, we get around this issue by considering a subregion $\mathcal{R}$ inside the space of open string configurations. One natural choice of the subregion is the trivial extension of $R$ along the stringy directions $X^{\mu}_{n>0}$; the open string configurations from the left panel are displayed as points.
• Figure 2: Consider the simplified setup where we discreteize the string to two points $X^{\mu} = (x^{\mu}_1, x^{\mu}_2)$. In terms of the center of mass $X_0 = \frac{1}{2}(x_1+x_2)$ and the separation $Y = \frac{1}{2}(x_1- x_2)$, we can extend the subregion (in blue) from target space to the entire configuration space in multiple ways. On the left, we count all strings with center of mass in the blue region, while on the right we count all strings which are entirely in the blue region. Note that while the subregion on the left looks non-compact, the string tension implies that the wavefunction of physical states of the string will be exponentially localized around $Y=0$.
• Figure 3: Pictorial representations of the operations $\int$ and $*$. The black dot with $\pm$ signs indicates an insertion of $e^{\pm \frac{3i}{2}\phi}$ at the midpoint.