The holographic dual of the entanglement wedge symplectic form
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Springer
Received: October 7, 2019 Accepted: December 28, 2019 Published: January 14, 2020
Josh Kirklin Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Cambridge, U.K.
E-mail: [email protected] Abstract: In this paper, we find the boundary dual of the symplectic form for the bulk fields in any entanglement wedge. The key ingredient is Uhlmann holonomy, which is a notion of parallel transport of purifications of density matrices based on a maximisation of transition probabilities. Using a replica trick, we compute this holonomy for curves of reduced states in boundary subregions of holographic QFTs at large N , subject to changes of operator insertions on the boundary. It is shown that the Berry phase along Uhlmann parallel paths may be written as the integral of an abelian connection whose curvature is the symplectic form of the entanglement wedge. This generalises previous work on holographic Berry curvature. Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence ArXiv ePrint: 1910.00457
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)071
JHEP01(2020)071
The holographic dual of the entanglement wedge symplectic form
Contents 1 Introduction
1
2 Uhlmann holonomy and fidelity
5 8 10 15 19
4 Symplectic form of the entanglement wedge 4.1 Subregion deformations and edge modes 4.2 Resolution of the boundary ambiguity
25 26 29
5 Conclusion
30
1
Introduction
Most research stemming from the discovery of the AdS/CFT correspondence [1, 2] can be loosely sorted into two categories. The first involves using the duality to translate a hard question about quantum field theory into an easier one about gravity, or vice-versa. This translation makes use of the so-called holographic dictionary, i.e. the collection of 1-to-1 maps between concepts in the bulk gravity theory and the boundary field theory. But many pages of the dictionary remain empty, and the second category of research endeavours to fill these pages with new entries, in order to both deepen our understanding of holography, and widen the scope for its potential applications. In recent years a coherent picture of a particular section of the dictionary, under the heading ‘subregion duality’, has emerged [3– 14]. The entries in this section make precise the relationship between boundary locality and bulk locality by identifying properties of a given subregion of the boundary with those of an associated subregion of the bulk. The current consensus is that the bulk dual of a boundary subregion with Cauchy surface A is its ‘entanglement wedge’, which is the domain of dependence of a codimension 1 surface in the bulk joining A with its HubenyRangamani-Ryu-Takayanagi (HRT) surface (i.e. the codimension 2 surface homologous to A in the bulk with extremal area). The standard depiction of the entanglement wedge is given in figure 1. This paper makes an argument for a new entry in this section of the dictionary. To explain our new entry, consider
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