The first law of differential entropy and holographic complexity

Published for SISSA by 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: , 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 Open Access, c The Authors. 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 conformal group C.2 From the boundary limit of the boost Killing vector 57 57 58 D Variation of coupling constants in the first law of causal diamonds 63 –i– JHEP11(2020)004 2 A first law in CFT2 2.1 Review of kinematic space 2.1.1 Differential entropy 2.1.2 Volume formula 2.1.3 Boundary dual of finite bulk volume 2.2 First law of differential entropy and holographic complexity 2.2.1 Varying differential entropy 2.2.2 Varying holographic complexity 2.2.3 Combining the variations 2.3 Limiting cases: small and large boundary intervals 1 Introduction –1– JHEP11(2020)004 Deriving gravitational thermodynamics of black holes [1–3] from a microscopic perspective remains one of the guiding principles in the quest for quantum gravity. The microscopic state counting of black hole entropy [4] is considered to be one of the major successes of string theory. Later, this microscopic derivation of black hole entropy was reinterpreted [5] in terms of the Anti-de Sitter (AdS)/ Conformal Field Theory (CFT) correspondence [6], where the entropy of three-dimensional AdS black holes [7, 8] matches with the thermodynamic entropy in two-dimensional CFTs [9]. In higher dimensions, it has also been argued that the mass, entropy and temperature of AdS black holes can be identified with the energy, entropy and temperature of a thermal state in the dual CFT at high temperature [10]. Furthermore, the correspondence between gravitational entropy and CFT entropy can be extended to the entanglement entropy of subregions on the conformal boundary of AdS. The Ryu-Takayagani (RT) formula [11, 12] states that the entanglement entropy of a subregion R in the CFT is, to leading order in Newton’s constant, dual to the Bekenstein-Hawking entropy A/(4G) of the minimal bulk surface which intersects the conformal boundary at ∂R. The entanglement entropy satisfies a first law-like relation, which is the quantum generalization of the first law of thermodynamics [13, 14]. An important result in AdS/CFT shows that the linearized gravitational dynamics in the bulk emerge from the RT formula and the first law of entanglement on the boundary [15]. More recently, the area of non-extremal codimension-two surfaces in three-dimensional AdS spacetime, which are not necessarily homologous to the boundary, was related to the notion of differential entropy in 2d CFTs, via equation (2.3) [16, 17]. The authors discovered that closed curves in a spatial slice of AdS3 can be reconstructed by adding and subtracting boundary-anchored geodesics tangent to the curve. Since RT surfaces in AdS3 are boundary-anchored geodesics, they were able to express the length (‘area’) of the closed curve in terms of an integral over entanglement entropies, associated to the boundary intervals subtended by the geodesics, which they dubbed ‘differential entropy’. This new field theoretic quantity can be qualitatively interpreted as the uncertainty about the global state for local observers who make measurements for a finite time in the CFT, because the exterior of a bulk closed curve is naturally associated to a time strip in the dual CFT. The formalism of differential entropy was extended to higher dimensions [18–20], covariant setups [21, 22], bulk curves near horizons or singularities [23], bulk points and distances [24], the Poincaré and Rindler wedges of AdS [25, 26], and it was reinterpreted in terms of kinematic space in [27], reviewed in section 2.1. In the present work, in similarity to the first law of entanglement, we derive a first law of differential entropy for a holographic CFT 2 . To construct the first law of differential entropy we find inspiration from the bulk side, where gravitational thermodynamics has been extended to spherical causal diamonds in maximally symmetric spacetimes (hence including in AdS) [28, 29]. Spherically symmetric causal diamonds are defined as the future and past domain of dependence of spherical, codimensi (...truncated)