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arXiv:2407.17553 · Journal of High Energy Physics

Heterotic strings and quantum entanglement

Atish Dabholkar A, Upamanyu Moitra

Published Dec 3, 2024• 3 citations
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Abstract

A bstract We construct ℤ N orbifolds of the ten-dimensional heterotic string theories appropriate for implementing the stringy replica method for the calculation of quantum entanglement entropy. A novel feature for the heterotic string is that the gauge symmetry must be broken by a Wilson line to ensure modular invariance. We completely classify the patterns of symmetry breaking. We show that the tachyonic contributions in all cases can be analytically continued, with a finite answer in the domain 0 < N ≤ 1, relevant for calculating entanglement entropy across the Rindler horizon. We discuss the physical implications of our results.

DOI
10.1007/jhep12(2024)012

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A bstract We construct ℤ N orbifolds of the ten-dimensional heterotic string theories appropriate for implementing the stringy replica method for the calculation of quantum entanglement entropy. A novel feature for the heterotic string is that the gauge symmetry must be broken by a Wilson line to ensure modular invariance. We completely classify the patterns of symmetry breaking. We show that the tachyonic contributions in all cases can be analytically continued, with a finite answer in the domain 0 < N ≤ 1, relevant for calculating entanglement entropy across the Rindler horizon. We discuss the physical implications of our results.
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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 NNN the partition functions for string orbifolds on C/ZN\mathbb{C}/\mathbb{Z}_NC/ZN​ conical spaces, known for all odd integer NNN. 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.
A bstract We define a notion of ‘quantum entanglement index’ with the aim to compute it for black hole horizons in string theory at one-loop order using the stringy replica method. We consider the horizon of BTZ black holes to construct the relevant conical orbifolds, labeled by an odd integer N , and compute the partition function as a function of N , corresponding to the fractional indexed Rényi entropy. We show that it is free of tachyons and naturally finite both in the ultraviolet and the infrared, even though it is generically ultraviolet divergent in the field theory limit. Thus, the index provides a useful diagnostic of the entanglement structure of string theory without the need for analytic continuation in N .
A bstract We consider ℤ N orbifolds of Type-II compactifications to four and six dimensions on several Calabi-Yau manifolds in the orbifold limit with the aim to compute the entanglement entropy. The spectrum can contain tachyons in the doubly-twisted sectors which can lead to new infrared divergences for the partition function that are not present in the orbifolds of the uncompactified ten-dimensional theory. We show that all tachyonic contributions in these models admit a resummation and analytic continuation that yields finite entropy in the physical region 0 < N ≤ 1 just as in ten dimensions.
A bstract We study entanglement entropy in the low-energy effective field theory of two-dimensional string theory as well as in the singlet sector of the dual c = 1 matrix quantum mechanics. From the target space perspective, we argue that a generic bulk subregion is expected to have an associated generalized entanglement entropy combining a dilaton-dependent gravitational term and a matter contribution coming from the tachyon, extending the construction from black hole horizons to arbitrary subregions. Motivated by the absence of the gravitational area-like term in previous analyses of entanglement entropy in the c = 1 model, we compute the effect on the entanglement entropy of the nonlocal transformation induced by the leg-pole factor that relates the target space tachyon and the matrix model collective excitations, finding that it cannot account for the area-like term. We offer preliminary comments on the possible origin of the area term in the non-singlet sectors of the theory.
The standard, gapped entanglement boundary condition in Chern Simons theory breaks the topological invariance of the theory by introducing a complex structure on the entangling surface. This produces an infinite dimensional subregion Hilbert space, a non-trivial modular Hamiltonion, and a UV-divergent entanglement entropy that is a universal feature of local quantum field theories. In this work, we appeal to the combinatorial quantization of Chern Simons theory to define a purely topological notion of a subregion. The subregion operator algebras are spaces of functions on a quantum group. We develop a diagrammatic calculus for the associated qqq-deformed entanglement entropy, which arise from the entanglement of anyonic edge modes. The qqq-deformation regulates the divergences of the QFT, producing a finite entanglement entropy associated to a qqq-tracial state. We explain how these ideas provide an operator algebraic framework for the entanglement entropy computations in topological string theory \cite{Donnelly:2020teo,Jiang:2020cqo, wongtopstring}, where a large- NNN limit of the qqq-deformed subregion algebra plays a key role in the stringy description of spacetime.