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.
Highlights
In this paper, we consider the following question: how does one define entanglement entropy for spatial subregions of the target space in string theory? In ordinary quantum field theory, the entanglement entropy of a subregion inside a spatial slice is defined as the von Neumann entropy of the reduced density matrix on the subregion
We will show that BRST exact modes in string field theory become dynamical at the entanglement cut, much like in conventional gauge field theories where edge modes appear in the computation of entanglement entropy
We demonstrated that in the free limit, this entropy is a sum over the one-loop entropies of particle excitations of the string, and that the inclusion of the tower of stringy d.o.f. softens the divergences in the entanglement entropy
Summary
We consider the following question: how does one define entanglement entropy for spatial subregions of the target space in string theory? In ordinary quantum field theory, the entanglement entropy of a subregion inside a spatial slice is defined as the von Neumann entropy of the reduced density matrix on the subregion. In order to study the entanglement properties of the vacuum in this theory, we can partition the surface S into subregions Note that this is very different from the situation in ordinary field theory—here we are required to partition not a Cauchy surface Σ in the target spacetime M, but instead an infinite-dimensional surface S in the space of open string configurations Mopen. R such that the entire string is contained inside R, but we will not attempt this here; we will briefly return to this point in the Discussion.) In any case, once we choose the subregion R, the reduced density matrix over R, and subsequently the entanglement entropy, can be defined and computed in the standard way, much like in conventional quantum field theory. We will show that BRST exact modes in string field theory become dynamical at the entanglement cut, much like in conventional gauge field theories where edge modes appear in the computation of entanglement entropy
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