Changing states in holography: From modular Berry curvature to the bulk symplectic form

Bartłomiej Czech; Jan De Boer; Ricardo Espíndola; Bahman Najian; Jeremy Van Der Heijden; Claire Zukowski

doi:10.1103/PhysRevD.108.066003

Changing states in holography: From modular Berry curvature to the bulk symplectic form

Bartłomiej Czech, Jan De Boer, Ricardo Espíndola, Bahman Najian, Jeremy Van Der Heijden, Claire Zukowski

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We present a new perspective on bulk reconstruction using Berry phases in the boundary conformal field theory (CFT). Our parallel transport of modular Hamiltonians is associated to a trajectory in the space of states, which we obtain from the insertion of a source in the Euclidean path integral. Using a modular version of the extrapolate dictionary and the equivalence between modular flow in the boundary and the bulk, we show that the expectation value of the modular Berry curvature on the boundary agrees with an appropriately defined bulk symplectic form associated to the entanglement wedge. In addition, we derive a quantum information metric on the space of density matrices from the Berry curvature, which is related to the canonical energy in the bulk. We also explore the case where a state change reduces to a shape change, uncovering the coadjoint orbit structure of kinematic space in higher dimensions.

Original language	English (US)
Article number	066003
Journal	Physical Review D
Volume	108
Issue number	6
DOIs	https://doi.org/10.1103/PhysRevD.108.066003
State	Published - Sep 15 2023
Externally published	Yes

Bibliographical note

Funding Information:
It is a pleasure to thank Tom Faulkner, Alex May and Erik Verlinde for discussions. B. C. and R. E. are supported by the Dushi Zhuanxiang Fellowship. R. E. also acknowledges a Shuimu Scholarship as part of the “Shuimu Tsinghua Scholar” Program. R. E. was supported by the ERC Consolidator Grant QUANTIVIOL. B. N. is supported by the Spinoza Grant of the Dutch Science Organisation (NWO). J. dB. and J.vdH. are supported by the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013), ERC Grant agreement ADG 834878. C. Z. is supported by a UMD Higholt Professorship, and acknowledges a Heising-Simons Fellowship as part of the “Observational Signatures of Quantum Gravity” collaboration grant No. 2021-2818. This work is supported by the Delta ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW).

Publisher Copyright:
© 2023 authors. Published by the American Physical Society.

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abstract = "We present a new perspective on bulk reconstruction using Berry phases in the boundary conformal field theory (CFT). Our parallel transport of modular Hamiltonians is associated to a trajectory in the space of states, which we obtain from the insertion of a source in the Euclidean path integral. Using a modular version of the extrapolate dictionary and the equivalence between modular flow in the boundary and the bulk, we show that the expectation value of the modular Berry curvature on the boundary agrees with an appropriately defined bulk symplectic form associated to the entanglement wedge. In addition, we derive a quantum information metric on the space of density matrices from the Berry curvature, which is related to the canonical energy in the bulk. We also explore the case where a state change reduces to a shape change, uncovering the coadjoint orbit structure of kinematic space in higher dimensions.",

author = "Bart{\l}omiej Czech and {De Boer}, Jan and Ricardo Esp{\'i}ndola and Bahman Najian and {Van Der Heijden}, Jeremy and Claire Zukowski",

note = "Funding Information: It is a pleasure to thank Tom Faulkner, Alex May and Erik Verlinde for discussions. B. C. and R. E. are supported by the Dushi Zhuanxiang Fellowship. R. E. also acknowledges a Shuimu Scholarship as part of the “Shuimu Tsinghua Scholar” Program. R. E. was supported by the ERC Consolidator Grant QUANTIVIOL. B. N. is supported by the Spinoza Grant of the Dutch Science Organisation (NWO). J. dB. and J.vdH. are supported by the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013), ERC Grant agreement ADG 834878. C. Z. is supported by a UMD Higholt Professorship, and acknowledges a Heising-Simons Fellowship as part of the “Observational Signatures of Quantum Gravity” collaboration grant No. 2021-2818. This work is supported by the Delta ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). Publisher Copyright: {\textcopyright} 2023 authors. Published by the American Physical Society.",

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Changing states in holography: From modular Berry curvature to the bulk symplectic form

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