arXiv:2207.03624

Quantum Entanglement in String Theory

Atish Dabholkar A

Published Jul 8, 2022• 2 citations

Abstract

We define entanglement entropy in string perturbation theory using the orbifold method -- a stringy analog of the replica method in field theory. To this end, we use the Newton series to analytically continue in $N$ the partition functions for string orbifolds on $\mathbb{C}/\mathbb{Z}_N$ conical spaces, known for all odd integer $N$. In the concrete example of ten-dimensional Type-IIB strings, the one-loop partition function can be computed explicitly and the one-loop entropy can be expressed as a manifestly modular invariant series in terms of the Weierstrass $\wp$ function. The convergence of the series is not evident but, from physical arguments based on holography, it is expected to yield a finite answer together with the tree level contribution. This method has a natural generalization to other string compactifications and to higher genus Riemann surfaces; it can provide a modular invariant definition of generalized entropy in a given string vacuum to all orders, of potential interest for the generalized second law of thermodynamics.

DOI: 10.48550/arxiv.2207.03624

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arXiv:2207.03624

Quantum Entanglement in String Theory

Atish Dabholkar A

Published Jul 8, 2022• 2 citations

arXiv HTML PDF

Abstract

N

the partition functions for string orbifolds on

\mathbb{C}/\mathbb{Z}_N

conical spaces, known for all odd integer

N

. In the concrete example of ten-dimensional Type-IIB strings, the one-loop partition function can be computed explicitly and the one-loop entropy can be expressed as a manifestly modular invariant series in terms of the Weierstrass

\wp

function. The convergence of the series is not evident but, from physical arguments based on holography, it is expected to yield a finite answer together with the tree level contribution. This method has a natural generalization to other string compactifications and to higher genus Riemann surfaces; it can provide a modular invariant definition of generalized entropy in a given string vacuum to all orders, of potential interest for the generalized second law of thermodynamics.

DOI: 10.48550/arxiv.2207.03624

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arXiv:1905.06708May 16, 20191 citation

N

the partition functions for string orbifolds on

\mathbb{C}/\mathbb{Z}_N

conical spaces, known for all odd integer

N

\wp

D

-dimensional bosonic string theory, the entanglement entropy of spin

1

subsystem and the system of all the spin values is given by

D{\log{λ_0}}+{D\over{λ_0}}\sum_{N=3}^\infty{{|β(N)|}\over{λ(N)}} {\log{({{λ(N)}\over{|β(N)|}})}}

, where

λ(N)

is the weighted number of partitions of

N

β(N)={{(N-1)ζ(3)-ζ(2)}\over{(N-1)^4}}

and

λ_0=\sum_{N=1}^{\infty}{{β(N)}\over{λ(N)}}

(

ζ

is Riemann's zeta-function). The first term,

D{\log{λ_0}}

, represents the entanglement swapping between string vacuum and string excitations. We generalize this result to obtain the entanglement for a subsystem of a given spin

s

in a given space-time dimension. We also discuss how open string field theory may be used to study the entanglement of systems other than higher spin excitations in string theory.