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Gravity Dual of Connes Cocycle Flow PHYSICAL REVIEW D 102, 066008 (2020) Editors' Suggestion Gravity dual of Connes cocycle flow † ‡ Raphael Bousso,* Venkatesa Chandrasekaran, Pratik Rath , and Arvin Shahbazi-Moghaddam § Center for Theoretical Physics and Department of Physics, University of California, Berkeley, California 94720, U.S.A. and Lawrence Berkeley National Laboratory, Berkeley, California 94720, U.S.A. (Received 2 July 2020; accepted 14 August 2020; published 24 September 2020) We define the “kink transform” as a one-sided boost of bulk initial data about the Ryu-Takayanagi surface of a boundary cut. For a flat cut, we conjecture that the resulting Wheeler-DeWitt patch is the bulk dual to the boundary state obtained by the Connes cocycle (CC) flow across the cut. The bulk patch is glued to a precursor slice related to the original boundary slice by a one-sided boost. This evades ultraviolet divergences and distinguishes our construction from a one-sided modular flow. We verify that the kink transform is consistent with known properties of operator expectation values and subregion entropies under CC flow. CC flow generates a stress tensor shock at the cut, controlled by a shape derivative of the entropy; the kink transform reproduces this shock holographically by creating a bulk Weyl tensor shock. We also go beyond known properties of CC flow by deriving novel shock components from the kink transform. DOI: 10.1103/PhysRevD.102.066008 I. INTRODUCTION (ANEC) and quantum null energy condition (QNEC) [14,15]. The Tomita-Takesaki theory, the study of modular The AdS=CFT duality [1–3] has led to tremendous flow in algebraic QFT, puts constraints on correlation progress in the study of quantum gravity. However, our functions that are otherwise hard to prove directly [16]. understanding of the holographic dictionary remains lim- Recently, an alternate proof of the QNEC was found ited. In recent years, quantum error corrections were found using techniques from the Tomita-Takesaki theory [17]. to play an important role in the emergence of a gravitating The key ingredient was the Connes cocyle (CC) flow. (“bulk”) spacetime from the boundary theory [4–6]. The Given a subregion A and global pure state ψ, the Connes study of modular operators led to the result that the boundary relative entropy in a region A equals the bulk cocycle flow acts with a certain combination of modular operators to generate a sequence of well-defined global relative entropy in its entanglement wedge EWðAÞ [7]. The ψ → ∞ ’ combination of these insights was used to derive subregion states s. In the limit s , these states saturate Wall s “ant conjecture” [18] on the minimum amount of energy in duality: bulk operators in EWðAÞ can, in principle, be 0 reconstructed from operators in the subregion A [8]. the complementary region A . This proves the ant con- The relation between bulk modular flow in EWðAÞ and jecture, which, in turn, implies the QNEC. boundary modular flow in A has been used to explicitly CC flow also arises from a fascinating interplay between reconstruct bulk operators, both directly [9,10] and indi- quantum gravity, quantum information, and QFT. Recently, rectly via the Petz recovery map and its variants [11–13]. the classical black hole coarse-graining construction of Thus, modular flow has shed light on the emergence of the Engelhardt and Wall [19] was conjecturally extended to the bulk spacetime from entanglement properties of the boun- semiclassical level [20]. In the nongravitational limit, this dary theory. conjecture requires the existence of flat space QFT states → ∞ Modular flow has also played an important role in with properties identical to the s limit of CC flowed proving various properties of quantum field theory states. This is somewhat reminiscent of how the QNEC was (QFT), such as the averaged null energy condition first discovered as the nongravitational limit of the quantum focusing conjecture [21]. Clearly, CC flow plays an important role in the connection between QFT and gravity. *[email protected] † Our goal in this paper is to investigate this connection at a [email protected] ‡ [email protected] deeper level within the setting of AdS=CFT. §[email protected] In Sec. II, we define the CC flow and discuss some of its properties. If ∂A lies on a null plane in Minkowski space, Published by the American Physical Society under the terms of operator expectation values and subregion entropies within the Creative Commons Attribution 4.0 International license. 0 Further distribution of this work must maintain attribution to the region A remain the same, whereas those in A trans- the author(s) and the published article’s title, journal citation, form analogously to a boost [17]. Further, CC flowed states 3 and DOI. Funded by SCOAP . ψ s exhibit a characteristic stress tensor shock at the cut ∂A, 2470-0010=2020=102(6)=066008(21) 066008-1 Published by the American Physical Society RAPHAEL BOUSSO et al. PHYS. REV. D 102, 066008 (2020) controlled by the derivative of the von Neumann entropy of we can distinguish our proposal from the closely related the region A in the state ψ under shape deformations of bulk duals of one-sided modular flowed states [25,26].We ∂A [20]. provide additional evidence for our proposal based on two As is familiar from other examples in holography, bulk sided correlation functions of heavy operators, and we duals of complicated boundary objects are often much discuss generalizations and applications of the proposed simpler [22,23]. Motivated by the known properties of CC kink transform/CC flow duality. flow, we define a bulk construction in Sec. III, which we In the Appendix, we derive the null limit of the kink call the “kink transform.” This is a one-parameter trans- transform and show that it generates a Weyl shock, which formation of the initial data of the bulk spacetime dual to provides intuition for how the kink transform modifies the original boundary state ψ. We consider a Cauchy gravitational observables. surface Σ that contains the Ryu-Takayanagi surface R of the subregion A. The kink transform acts as the identity II. CONNES COCYLE FLOW except at R, where a s-dependent shock is added to the extrinsic curvature of Σ. We show that this is equivalent to a In this section, we review Connes cocycle flow and its one-sided boost of Σ in the normal bundle to R. We prove salient properties; for more details, see [17,20]. We then that the new initial data satisfy the gravitational constraint reformulate the Connes cocycle flow as a simpler map to a equations, thus demonstrating that the kink transform state defined on a “precursor” slice. This will prove useful in later sections. defines a valid bulk spacetime Ms. We show that Ms is independent of the choice of Σ. We propose that Ms is the holographic dual to the CC- A. Definition and general properties ψ ∂ flowed state s, if the boundary cut A is (conformally) a Consider a quantum field theory on Minkowski space flat plane in Minkowski space. d−1;1 R in standard Cartesian coordinates ðt; x; y1; …;yd−2Þ. In Sec. IV, we provide evidence for this proposal. The Consider a Cauchy surface C that is the disjoint union of the 0 kink transform separately preserves the entanglement open regions A0;A , and their shared boundary ∂A0. Let 0 0 wedges of A and A , but it glues them together with a 0 A0; A denote the associated algebras of operators. Let jψi 2π 0 relative boost by rapidity s. This implies the one-sided be a cyclic and separating state on C and denote by jΩi the expectation values and subregion entropies of the CC ψ global vacuum (the assumption of cyclic and separating flowed state s are correctly reproduced when they are ψ M could be relaxed for j i, at the cost of complicating the computed holographically in the bulk spacetime s.We discussion below). The Tomita operator is defined by then perform a more nontrivial check of this proposal. By computing the boundary stress tensor holographically in α ψ α† Ω ∀ α ∈ A SψjΩ;A0 j i¼ j i; 0: ð2:1Þ Ms, we reproduce the stress tensor shock at ∂A in the CC- ψ flowed state s. The relative modular operator is defined as Having provided evidence for kink transform/CC flow duality, we use the duality to make a novel prediction for CC † Δψ Ω ≡ S Sψ Ω;A ; ð2:2Þ flow in Sec. V. The kink transform fully determines all j ψjΩ;A0 j 0 independent components of the shock at ∂A in terms of shape derivatives of the entanglement entropy. Strictly, our and the vacuum modular operator is results only apply only to the CC flow of a holographic CFT Δ ≡ Δ across a planar cut. However, their universal form suggests Ω ΩjΩ: ð2:3Þ that they will hold for general QFTs under CC flow. A Δ Moreover, the shocks we find agree with properties required Note that we do not include the subscript 0 on ; instead, to exist in quantum states under the coarse-graining proposal for modular operators, we indicate whether they were A A0 Δ Δ0 of Ref. [24]. Thus, our new results may also hold for CC constructed from 0 or 0 by writing or . ψ flow across general cuts of a null plane. The Connes cocycle (CC) flow of j i generates a one ψ ∈ R In Sec. VI, we discuss the relation of our construction to parameter family of states j si, s , defined by earlier work on the role of modular flow in AdS=CFT ψ Δ0 is Δ0 −is ψ [7,25,26].
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