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arXiv:1802.00169 · Physics Letters B

Entanglement entropy for open bosonic strings on Dp-branes

Taejin Lee

Published Jun 4, 2018• 3 citations
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Abstract

We study the entanglement entropy for open bosonic strings on multiple Dp -branes by using the covariant open string field theory. Choosing one of the spatial coordinates which are tangential to the hyperplane on which Dp -branes are located, we divide the hyperplane into two halves. By using the string wavefunction in the Fock space representation, we evaluate the entanglement entropy. The entanglement entropy is found to be proportional to the area of ( p − 1 ) -dimensional boundary of the bipartite hyperplanes and divergent in the ultraviolet (UV) region as well as in the infrared (IR) region. However, the leading divergences are mainly due to tachyon contributions to the entanglement entropy, which may be absent in supersymmetric string theories . Apart from the divergences thanks to tachyons, the entanglement entropy for open bosonic strings on Dp -branes is finite for 2 ≤ p ≤ d critical − 2 and logarithmically divergent for p = 1 , d critical − 1 .

DOI
10.1016/j.physletb.2018.05.080

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We study the entanglement entropy for open bosonic strings on multiple Dp -branes by using the covariant open string field theory. Choosing one of the spatial coordinates which are tangential to the hyperplane on which Dp -branes are located, we divide the hyperplane into two halves. By using the string wavefunction in the Fock space representation, we evaluate the entanglement entropy. The entanglement entropy is found to be proportional to the area of ( p − 1 ) -dimensional boundary of the bipartite hyperplanes and divergent in the ultraviolet (UV) region as well as in the infrared (IR) region. However, the leading divergences are mainly due to tachyon contributions to the entanglement entropy, which may be absent in supersymmetric string theories . Apart from the divergences thanks to tachyons, the entanglement entropy for open bosonic strings on Dp -branes is finite for 2 ≤ p ≤ d critical − 2 and logarithmically divergent for p = 1 , d critical − 1 .
We compute the left-right entanglement entropy for Dp-branes in string theory. We employ the CFT approach to string theory Dp-branes, in particular, its presentation as coherent states of the closed string sector. The entanglement entropy is computed as the von Neumann entropy for a density matrix resulting from integration over the left-moving degrees of freedom. We discuss various crucial ambiguities related to sums over spin structures and argue that different choices capture different physics; however, we advance a themodynamic argument that seems to favor a particular choice of replica. We also consider Dp branes on compact dimensions and verify that the effects of T-duality act covariantly on the Dp brane entanglement entropy. We find that generically the left-right entanglement entropy provides a suitable generalization of boundary entropy and of the D-brane tension.
Abstract We investigate the left-right entanglement entropy of a boundary state, corresponding to a dynamical Dp-brane with the internal and background fields. We assume that the brane has a tangential linear motion and a rotation, and is dressed with an internal U(1) gauge potential and the Kalb–Ramond tensor field BμνB_{\mu \nu }Bμν​ Bμν . We derive the entanglement entropy via the Rényi entropy by applying the replica trick. Our calculations will be in the context of the bosonic string theory.
We compute the left-right entanglement entropy for Dp-branes in string theory. We employ the CFT approach to string theory Dp-branes, in particular, its presentation as coherent states of the closed string sector. The entanglement entropy is computed as the von Neumann entropy for a density matrix resulting from integration over the left-moving degrees of freedom. We discuss various crucial ambiguities related to sums over spin structures and argue that different choices capture different physics; however, we advance a themodynamic argument that seems to favor a particular choice of replica. We also consider Dp branes on compact dimensions and verify that the effects of T-duality act covariantly on the Dp brane entanglement entropy. We find that generically the left-right entanglement entropy provides a suitable generalization of boundary entropy and of the D-brane tension.
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 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.
In this work we explain the construction of the thermal vacuum for the bosonic string, as well that of the thermal boundary state interpreted as a Dp-brane at finite temperature. In both case we calculate the respective entropy using the entropy operator of the Thermo Field Dynamics theory. We show that the contribution of the thermal string entropy is explicitly present in the Dp-brane entropy. Furthermore, we show that the Thermo Field approach is suitable to introduce temperature in boundary states.
What is the meaning of entanglement in a theory of extended objects such as strings? To address this question we consider the spatial entanglement between two intervals in the Gross-Taylor model, the string theory dual to two-dimensional Yang-Mills theory at large N . The string diagrams that contribute to the entanglement entropy describe open strings with endpoints anchored to the entangling surface, as first argued by Susskind. We develop a canonical theory of these open strings, and describe how closed strings are divided into open strings at the level of the Hilbert space. We derive the modular Hamiltonian for the Hartle-Hawking state and show that the corresponding reduced density matrix describes a thermal ensemble of open strings ending on an object at the entangling surface that we call an entanglement brane, or E-brane.