Entanglement entropy and edge modes in topological string theory. Part I. Generalized entropy for closed strings

Abstract

Progress in identifying the bulk microstate interpretation of the Ryu-Takayanagi formula requires understanding how to define entanglement entropy in the bulk closed string theory. Unfortunately, entanglement and Hilbert space factorization remains poorly understood in string theory. As a toy model for AdS/CFT, we study the entanglement entropy of closed strings in the topological A-model in the context of Gopakumar-Vafa duality. We will present our results in two separate papers. In this work, we consider the bulk closed string theory on the resolved conifold and give a self-consistent factorization of the closed string Hilbert space using extended TQFT methods. We incorporate our factorization map into a Frobenius algebra describing the fusion and splitting of Calabi-Yau manifolds, and find string edge modes transforming under a q-deformed surface symmetry group. We define a string theory analogue of the Hartle-Hawking state and give a canonical calculation of its entanglement entropy from the reduced density matrix. Our result matches with the geometrical replica trick calculation on the resolved conifold, as well as a dual Chern-Simons theory calculation which will appear in our next paper [1]. We find a realization of the Susskind-Uglum proposal identifying the entanglement entropy of closed strings with the thermal entropy of open strings ending on entanglement branes. We also comment on the BPS microstate counting of the entanglement entropy. Finally we relate the nonlocal aspects of our factorization map to analogous phenomenon recently found in JT gravity.

Edge modes as reference frames and boundary actions from post-selection

“…All we demand from g t is that g 0 = Id, and g T (x) = U −1 [A](x) for any x ∈ Γ. 39 Given those definitions, φ rad (where we again collectively denote pull-backs of fields on Γ by φ) can be recovered as the edge configuration induced by A T :…”

“…38 u is not unique, but at least locally in field-space, one can choose a smooth map U → u from group-to algebra-valued fields, defining local logarithmic coordinates. 39 Such a path necessarily exists since U , seen as a field on Γ, is connected to the identity. At is actually highly non-unique, but we will see that the resulting dressing of edge modes is independent from such ambiguity.…”

“…In a broader context, and already at the classical level, the emergence of edge modes was argued to be a necessary consequence of gauge symmetry in the presence of boundaries by Donnelly and Freidel [3]. This has led to a series of efforts to better understand the notion of edge modes in gauge theories [4][5][6][7][8][9][10][11][12][13][14][15], gravity [5, and string theory [38][39][40]. However, the genericity of this particular argument for edge modes, its interpretation and physical relevance, all remain debated in the literature.…”

Entanglement entropy in Jackiw-Teitelboim gravity with matter

“…For example, the modified trace of [41] may be explicable as a quantum trace. See [43], [44] for related recent results in the context of Gopakumar-Vafa duality.…”

Snowmass white paper: Quantum information in quantum field theory and quantum gravity

“…Attempts have been made to understand this question in the context of quasi-topological field theories, e.g. 2d large N Yang-Mills [163,164], and within the topological open/closed string duality [165][166][167]. An analysis of partition functions of weakly coupled CFT duals from the dual string perspective (in the AdS 3 /CFT 2 context), appears to suggest that worldsheet does not preferentially pick out a particular target space geometry in the semiclassical (large N ) limit [168].…”