The first law of differential entropy and holographic complexity
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Springer
Received: September 4, Revised: September 30, Accepted: October 1, Published: November 4,
2020 2020 2020 2020
Debajyoti Sarkara and Manus Visserb a
Indian Institute of Technology Indore, Khandwa Road, Simrol 453552 Indore, India b University of Geneva, Department of Theoretical Physics, 24 quai Ernest-Ansermet, 1211 Geneve 4, Switzerland
E-mail: [email protected], [email protected] Abstract: We construct the CFT dual of the first law of spherical causal diamonds in three-dimensional AdS spacetime. A spherically symmetric causal diamond in AdS 3 is the domain of dependence of a spatial circular disk with vanishing extrinsic curvature. The bulk first law relates the variations of the area of the boundary of the disk, the spatial volume of the disk, the cosmological constant and the matter Hamiltonian. In this paper we specialize to first-order metric variations from pure AdS to the conical defect spacetime, and the bulk first law is derived following a coordinate based approach. The AdS/CFT dictionary connects the area of the boundary of the disk to the differential entropy in CFT 2 , and assuming the ‘complexity=volume’ conjecture, the volume of the disk is considered to be dual to the complexity of a cutoff CFT. On the CFT side we explicitly compute the differential entropy and holographic complexity for the vacuum state and the excited state dual to conical AdS using the kinematic space formalism. As a result, the boundary dual of the bulk first law relates the first-order variations of differential entropy and complexity to the variation of the scaling dimension of the excited state, which corresponds to the matter Hamiltonian variation in the bulk. We also include the variation of the central charge with associated chemical potential in the boundary first law. Finally, we comment on the boundary dual of the first law for the Wheeler-deWitt patch of AdS, and we propose an extension of our CFT first law to higher dimensions. Keywords: AdS-CFT Correspondence, Conformal Field Theory ArXiv ePrint: 2008.12673
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)004
JHEP11(2020)004
The first law of differential entropy and holographic complexity
Contents 1 Introduction
1 4 5 5 7 12 14 15 17 19 23
3 A first law in AdS3 3.1 AdS3 with a conical defect 3.2 First law of causal diamonds in AdS3 3.2.1 Area variation 3.2.2 Volume variation 3.2.3 Bulk matter Hamiltonian variation 3.2.4 Combining the variations
25 27 30 30 32 34 36
4 Matching the boundary and bulk first laws 4.1 First law in holographic CFT2 from first law in AdS3 4.2 Extension of the boundary first law to higher dimensions 4.2.1 Comparison with extended first law of entanglement
39 39 41 44
5 Conclusion and outlook
46
A Embedding formalism and coordinate systems for AdS3 geometries A.1 Pure AdS A.2 Conical AdS
49 50 51
B Geodesics in AdS3 geometries B.1 Geodesic equation for conical AdS B.2 Chord length
52 53 54
C Conformal isometry of causal diamonds on the cylinder C.1 From the conform
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