PHYSICAL REVIEW D 102, 066008 (2020)

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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]. The result of Jafferis et al. (JLMS) [7] has j si¼ð ΩÞ ð Ωjψ Þ j i: ð2:4Þ conventionally been understood as a relation that holds for a small code subspace of bulk states on a fixed background Thus far, the definitions have been purely algebraic. In order spacetime. However, results from a quantum error correc- to elucidate the intuition behind CC flow, let us write out the tion suggest that this code subspace could be made much modular operators in terms of the left and right density ψ ψ 1 – operators, ρ ¼ Tr 0 jψihψj and ρ 0 ¼ Tr jψihψj, larger to include different background geometries [5,27 A0 A0 A0 A0 29]. Our proposal then follows from an extended version of the JLMS result, which includes nonperturbatively different 1We follow the conventions in [30] where complement background geometries. Equipped with this understanding, operators are written to the right of the tensor product.

066008-2 GRAVITY DUAL OF CONNES COCYCLE FLOW PHYS. REV. D 102, 066008 (2020) Z Z ψ Δ ρΩ ⊗ ρ −1 V0 ψ Ω ¼ ð 0 Þ : ð2:5Þ 0 j A0 A0 Δ −2π − hKV0 i¼ dy dv½v V0ðyÞhTvviψ ; ð2:14Þ −∞ One finds that the CC operator acts only in A0 , 0 0 and similarly for KV0. Thus, KV0 is simply the boost ψ Δ0 is Δ0 −is ρΩ is ρ −is ∈ A0 generator about the cut V0ðyÞ in the left Rindler wedge. ð ΩÞ ð Ω ψ Þ ¼ð A0 Þ ð A0 Þ 0: ð2:6Þ j 0 0 That is, it generates a y-dependent dilation,

It follows that the reduced state on the right algebra satisfies 2πs v → V0ðyÞþ½v − V0ðyÞe : ð2:15Þ ψ ψ ρ s ¼ ρ : ð2:7Þ A0 A0 This allows us to evaluate Eq. (2.11) explicitly for local 0 O ∈ A operators at u ¼ . For example, the CC flow of the stress Therefore,expectationvalues ofobservables 0 remain tensor is invariantunderCC flow. Theseheuristicargumentswouldbe valid only for finite-dimensional Hilbert spaces [30],but hψ sjTvvjψ sij Eq. (2.6) can be derived rigorously [17]. v

We can write the full modular Hamiltonian as Moreover, the “left” von Neumann entropy is defined as

ˆ 0 0 0 ψ ψ ⊗ 1 − 1 ⊗ ψ − 0 ρ ρ KV0 ¼ KV0 KV0 : ð2:13Þ S ð ;VÞ¼ trA 0 log 0 : ð2:20Þ V AV AV

Let Δ denote vacuum subtraction, ΔhOi¼hOiψ − hOiΩ. With these definitions in hand, one can decompose the Then, for arbitrary cuts of the Rindler horizon, we have [31] relative entropy as

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0 ψ Ω; Δ 0 − Δ 0 δ Srelð j VÞ¼ hKVi S ðVÞ: ð2:21Þ Here, oð Þ designates the finite (nondistributional) terms. These are determined by Eq. (2.16) and by its trivial At this point, we drop the explicit vacuum subtractions, as counterpart in the v>V0 region. we will only be interested in shape derivatives of the This s-dependent shock is a detailed characteristic of the vacuum subtracted quantities, which automatically annihi- CC flowed state. As such, reproducing it through the late the vacuum expectation values. In particular, one can holographic dictionary will be the key test of our proposal 0 directly compute shape derivatives of KV, of the bulk dual of CC flow (see Sec. IV). Z δ 0 V0 hKViψ D. Flat cuts and the precursor slice ¼ 2π dvhTvviψ : ð2:22Þ δV −∞ V0 For the remainder of the paper, we further specialize to flat cuts of the Rindler horizon, so that ∂A0 corresponds to Hence, the transformations of both K0 and its derivative V u ¼ v ¼ 0. We therefore set V0 ¼ 0 in what follows. We simply follow from Eq. (2.16). take C to be the Cauchy surface t ¼ 0, so that A0 (t ¼ 0, Combining Eq. (2.18) and Eq. (2.14), as well as x>0) and A0 (t ¼ 0, x<0) are partial Cauchy surfaces for 0 ψ 0 Eq. (2.19) and Eq. (2.22), we see that S ð ;VÞ and its the right and left Rindler wedges. derivative transform as is In this case, ΔΩ is a global boost by rapidity s about ∂A0 [37]. Thus, it has a simple geometric action not only on the 0 ψ 0 ψ −2πs − S ð s;VÞ¼S ð ;V0 þ e ðV V0ÞÞ; ð2:23Þ null plane u ¼ 0, but everywhere. CC flow transforms 0 is observables in A by Δ and leaves invariant those in A0. δ 0 δ 0 0 Ω S −2π S ¼ e s : ð2:24Þ For a flat cut, this action can be represented as a geometric δV δV −2πs ψs;V ψ;V0þe ðV−V0Þ boost in the entire left Rindler wedge. This allows us to characterize the CC flowed state jψðsÞi on C very simply in The respective properties of the complement entropy follow terms of a different state on a different Cauchy surface from purity. which we call the “precursor slice”. This description will motivate the formulation of our bulk construction in C. Stress tensor shock at the cut Sec. III A. By Eq. (2.11), the CC flowed state on the slice C, CC flow generates a stress tensor shock at the cut V0, proportional to the jump in the variation of the one-sided − jψ ðCÞi¼ðΔ0 ÞisðΔ0 Þ isjψðCÞi; ð2:28Þ von Neumann entropy under deformations, at the cut [20]. s Ω Ωjψ To see this, let us start with the sum rule derived in [17] for satisfies null variations of relative entropy,2 ψ C O ψ C ψ C O ψ C δS0 ψ Ω; V h sð Þj Aj sð Þi¼h ð Þj Aj ð Þi; ð2:29Þ 2π − −2πs −2πs − 1 relð j Þ ðPs e P0Þ¼ðe Þ δ ; ð2:25Þ V V0 −is is hψ sðCÞjΔΩ OA0 ΔΩjψ sðCÞi ¼ hψðCÞjOA0 jψðCÞi; ð2:30Þ where where OA and OA0 denote an arbitrary collection of local Z 0 ∞ operators that act on spacelike half slices A and A of C, 3 P ≡ dvTvv ð2:26Þ respectively. In the second equality above, we used the fact −∞ is that ΔΩ acts as a global boost to move it to the other side of is the averaged null energy operator at u ¼ 0, and the equality, compared to Eq. (2.11). We work in the Schrödinger picture, where the argument P ≡ ψ P ψ P0 ≡ ψ P ψ s h sj j si, so in particular h j j i. (There is C one such operator for every generator, i.e., for every y.) should be interpreted as the time variable. The fact that Δis ∂ Inserting Eq. (2.21) and Eq. (2.22) into Eq. (2.25), and Ω acts as a boost around A0 motivates us to consider the making use of Eq. (2.16), we see that there must exist a time slice, shock at v ¼ V0ðyÞ, C 0 ∪ ∂ ∪ s ¼ As A0 A0; ð2:31Þ 1 δS0 ψ ψ 1 − −2πs δ − δ h sjTvvj si¼ð e Þ 2π δ ðv V0Þþoð Þ: where V V0 0 ð2:27Þ As ¼ft ¼ðtanh 2πsÞx; x < 0g: ð2:32Þ

2For type I algebras, one can derive the analogous sum rule 3More precisely, one would have to smear the operator in a from simpler arguments [36]. codimension 0 neighborhood of points on the slices.

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By Eqs. (2.29) and (2.30), each side of the CC-flowed state ds2 ¼½Θðt˜ þ x˜Þþe2πsð1 − Θðt˜ þ x˜ÞÞ jψ ðC Þi is simply related to the left and right restrictions of s s × e−2πsΘ t˜ − x˜ 1 − Θ t˜ − x˜ −dt˜2 dx˜2 the original state on the original slice, ½ ð Þþð ð ÞÞð þ Þ þ dd−2y; ð2:41Þ hψ sðCsÞjOAjψ sðCsÞi¼hψðCÞjOAjψðCÞi; ð2:33Þ and the precursor slice corresponds to t˜ ¼ 0. ψ C O 0 ψ C ψ C O 0 ψ C “ ” h sð sÞj As j sð sÞi¼h ð Þj A j ð Þi: ð2:34Þ In these tilde coordinates, the stress tensor shock of Eq. (2.27) takes the form,4 0 O 0 In the second equation, As denotes local operators on As 0 1 ∂ 2 δ which are analogous to O 0 on A . More precisely, because v −2πs S A ψ T ˜ ˜ ψ 1 − e δ v o δ : 0 0 h sj v vj si¼2π ∂ ˜ ð Þ δ ð Þþ ð Þ the intrinsic metric of A and As are the same, there exists a v V V¼0 natural map between local operators on A0 and A0 . s ð2:42Þ In words, Eqs. (2.33) and (2.34) say that correlation C ψ C functions in each half of in the state j ð Þi are equal to Recall that the entropy variation is evaluated in the state C the analogous correlation functions on each half of s in the ψ . By Eq. (2.24), ψ C C j i state j sð sÞi. This justifies calling s the precursor slice C since the CC flowed state on arises from it by time δS δS evolution. ¼ ˜ ; ð2:43Þ δV ψ δV ψ We find it instructive to repeat this point in the less s rigorous language of density operators. In the density where V˜ ðyÞ is a cut of the Rindler horizon in the v˜ operator form of CC flow, coordinates. Thus, we may instead evaluate the entropy ψ ψ C ρΩ is ρψ −is ψ C variation in the state j si on the precursor slice. This will be j sð Þi¼ð 0 Þ ð 0 Þ j ð Þi; ð2:35Þ A0 A0 convenient when matching the bulk and boundary. The Jacobian in Eq. (2.42) has a step function in it, as Ω is it is evident that the action of ðρ 0 Þ can be absorbed into a A0 will the Jacobian coming from δðvÞ. A step function C → C change of time slice s, multiplying a delta function is well-defined if one averages the left and right derivatives, ψ C ρψ −is ψ C j sð sÞi¼ð 0 Þ j ð Þi: ð2:36Þ A0 2 ∂v 1 ∂v ∂v δðvÞ¼ þ δðv˜Þ: ð2:44Þ Tracing out each side of ∂A0 implies ∂v˜ 2 ∂v˜ 0− ∂v˜ 0þ ψ ψ ρ s ρ A0 ¼ A0 ; ð2:37Þ Thus Eq. (2.42) becomes ψ ψ ρ s ρ 1 δS A0 ¼ A0 : ð2:38Þ s 0 hψ sjTv˜ v˜ ðv˜Þjψ si¼ sinhð2πsÞ δðv˜ÞþoðδÞ: 2π δ ˜ ˜ V ψ s;V¼0 The first equality is trivial and was already discussed in ψ is ð2:45Þ Eq. (2.7). The second equality follows because ðρ 0 Þ A0 ψ ↔ commutes with ðρ 0 Þ. This is the density operator version Since we are dealing with a flat cut, the symmetry s A0 − ↔ of Eqs. (2.33) and (2.34). s; v u implies that CC flow also generates a Tuu shock ψ 0 Equation (2.36) should be contrasted with the one-sided in the state j si at u ¼ v ¼ , ϕ C ρψ −is ψ C modular-flowed state j ð Þi¼ð 0 Þ j ð Þi. The latter A0 1 δS state would live on the original slice C, but it is not well- ψ ψ 2π δ ˜ δ h sjTu˜ u˜ j si¼ sinhð sÞ ˜ ðuÞþoð Þ: 2π δU ψ ˜ 0 defined since it would have infinite energy at the entangling s;V¼ surface. ð2:46Þ It will be useful to define new coordinates adapted to the C precursor slice s. Let (Note that δ=δV goes to −δ=δU.) The linear combination, ˜ Θ −2πs 1 − Θ v ¼ v ðvÞþe vð ðvÞÞ; ð2:39Þ 1 δS ψ T˜ ˜ t;˜ x˜ ψ sinh 2πs δ x˜ h sj t xð Þj si¼2π ð Þ δ ˜ ð Þ X ψ ;X˜ 0 u˜ ¼ e2πsuΘðuÞþuð1 − ΘðuÞÞ; ð2:40Þ s ¼ ˜ þhψjTtxðt ¼ t; x ¼ x˜Þjψi; ð2:47Þ ˜ 1 where Θð:Þ is the Heaviside step function. Let t ¼ 2 ðv˜ þ u˜Þ 1 ˜ ˜ − ˜ 4 and x ¼ 2 ðv uÞ. In these coordinates, the Minkowski We remind the reader that oðδÞ refers to any finite (non- metric takes the form, distributional) terms.

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μ μ will be useful in Sec. IV. The last term was obtained from Here, Pa is the projector from M onto Σ, and t is the unit Eqs. (2.37) and (2.38); it makes the finite piece explicit. norm timelike vector field orthogonal to Σ. Indices a; b; … Note that these equations are valid in the entire left and are reserved for directions tangent to Σ. For matter fields, right wedges, not just on Cs. initial data consist of the fields and normal derivatives, for a μ a example, ϕðw Þ and ½t ∇μϕðw Þ, where ϕ is a scalar field a Σ III. KINK TRANSFORM and w are coordinates on . By the Einstein equations, the initial data on Σ must In this section, we introduce a novel geometric trans- satisfy the following constraints: formation called the kink transform. The construction is 2 − ab 16π μ ν motivated by thinking about what the bulk dual of the rΣ þ KΣ ðKΣÞabðKΣÞ ¼ GTμνt t ; ð3:2Þ boundary CC flow would be in the context of AdS=CFT. a − 8π ν As we discussed in Sec. II, CC flow boosts observables in D ðKΣÞab DbKΣ ¼ GTbνt ; ð3:3Þ DðA0Þ and leaves observables in DðAÞ unchanged. μ Subregion duality in AdS=CFT then implies that the bulk where Da ¼ Pa∇μ is the covariant derivative that Σ inherits ψ dual of the state j si has to have the property that the from ðM;gμνÞ, rΣ is the Ricci scalar intrinsic to Σ, and KΣ entanglement wedges of D A , and D A0 will be diffeo- ab ð Þ ð Þ is the trace of the extrinsic curvature: KΣ ¼ðhΣÞ ðKΣÞ . ψ ab morphic to those of the state j i but are glued together with Let Σ be a Cauchy slice of M containing R and smooth “ ” a one-sided boost at the HRT surface. In a general in a neighborhood of R. The kink transform is then a geometry, a boost Killing symmetry need not exist. The map of the initial data on Σ to a new initial data set, kink transform appropriately generalizes the notion of a parametrized by a real number s analogous to boost one-sided boost to any extremal surface. rapidity. The transform acts as the identity on all data In Sec. III A, we formulate the kink transform. In Sec. III except for the extrinsic curvature, which is modified only at B, we describe a different but equivalent formulation of the the location of the extremal surface R, as follows: kink transform and show that the kink transform results in the same new spacetime, regardless of which Cauchy → − 2π δ R ðKΣÞab ðKΣ Þab ¼ðKΣÞab sinh ð sÞxaxb ð Þ: surface containing the extremal surface is used for the s construction. In Sec. IV, we will describe the duality ð3:4Þ between the bulk kink transform and the boundary CC Here, xa is a unit norm vector field orthogonal to R and flow in AdS=CFT and provide evidence for it. tangent to Σ, and we define

A. Formulation δðRÞ ≡ δðxÞ; ð3:5Þ Consider a d þ 1—dimensional spacetime M with where x is the Gaussian normal coordinate to R in Σ metric gμν satisfying the Einstein field equations. (We will (∂ ¼ xa). Thus, the only change in the initial data is in the discuss higher curvature gravity in Sec. VI E.) Let Σ be a x component of the extrinsic curvature normal to R.An Cauchy surface of M that contains an extremal surface R equivalent transformation exists for initial choices of Σ that of codimension 1 in Σ. (That is, the expansion of both sets are not smooth around R though the transformation rule of null geodesics orthogonal to R vanishes.) will be more complicated than Eq. (3.4). We will discuss Initial data on Σ consist of [38] the intrinsic metric ðhΣÞ ab this later in the section. and the extrinsic curvature, Let Σs be a time slice with this new initial data, as in M Σ μ ν Fig. 1, and let s be the Cauchy development of s. That ðKΣÞ ¼ PaP ∇ μtν : ð3:1Þ ab b ð Þ is, Ms is the new spacetime resulting from the evolution of

FIG. 1. Kink transform. Left: a Cauchy surface Σ of the original bulk M. An extremal surface R is shown in red. The orthonormal a a a vector fields t and x span the normal bundle to R; x is tangent to Σ. Right: the kink transformed Cauchy surface Σs. As an initial data set, Σs differs from Σ only in the extrinsic curvature at R through Eq. (3.4). Equivalently, the kink transform is a relative boost in the normal bundle to R, Eq. (3.21).

066008-6 GRAVITY DUAL OF CONNES COCYCLE FLOW PHYS. REV. D 102, 066008 (2020) the kink-transformed initial data. Since the intrinsic metric 2 − ab 16π μ ν rΣs þðKΣs Þ ðKΣs ÞabðKΣs Þ ¼ GTμνt t : ð3:13Þ of Σs and Σ are the same, they can be identified as d manifolds with a metric; the subscript s merely reminds us To check the vector constraint Eq. (3.3), we separately of the different extrinsic data they carry. In particular the consider the two cases of b ¼ x and b ¼ i where i, j surface R can be so identified; thus, Rs has the same represent directions tangent to R, intrinsic metric as R. It also trivially has identical extrinsic Σ a − a − data with respect to s. In fact, we will find below that like DaðKΣ Þx DxKΣ ¼ DaðKΣÞx DxKΣ R M R M s s in , s is an extremal surface in s. However, the ðxÞ þ BR sinh ð2πsÞδðxÞ trace-free part of the extrinsic curvature of Rs in Ms may a − 8π ν have discontinuities. ¼ DaðKΣÞx DxKΣ ¼ GTxνt ; We will now show that the constraint equations hold on ð3:14Þ Σs; that is, the kink transform generates valid initial data. R This need only be verified at since the transform acts as a a ν D ðKΣ Þ − D KΣ ¼ D ðKΣÞ − D KΣ ¼ 8πGT νt ; the identity elsewhere. Here, we will make essential use of a s i i s a i i i the extremality of R in M, which we express as follows. ð3:15Þ The extrinsic curvature of R with respect to M has two independent components. Often these are chosen to be the where the second line of the first equation follows from the two orthogonal null directions, but we find it useful to extremality of R. consider We conclude that the kink transform is a valid modifi- cation to the initial data. For both constraints to be satisfied ðtÞ μ ν∇ after the kink, it was essential that R is an extremal surface. ðBR Þij ¼ Pi Pj ðμtνÞ; ð3:6Þ Thus, the kink transform is only well-defined across an μ R ⊂ Σ ðxÞ ν∇ extremal surface. Note also that s s is an extremal ðBR Þij ¼ Pi Pj ðμxνÞ: ð3:7Þ surface in Ms. By Eq. (3.10), μ Here, i, j represent directions tangent to R, and P is the i ðtÞ a b ðtÞ ðtÞ M R R M ðBR Þ ¼ P P ðKΣ Þ j ¼ðBR Þ ⇒ BR ¼ 0: ð3:16Þ projector from to . Extremality of in is the s ij i j s ab Rs ij s statement that the trace of each extrinsic curvature compo- nent vanishes In the second equality, we used Eq. (3.4) as well as the fact that all relevant quantities are intrinsic to Σs,soRs can be ðtÞ γ ij ðtÞ 0 identified with R. Moreover, BR ¼ð RÞ ðBR Þij ¼ ; ð3:8Þ ðxÞ ðxÞ ðxÞ ðxÞ ðxÞ ðBR Þ ¼ðBR Þ ⇒ BR ¼ 0; ð3:17Þ γ ij 0 s ij ij s BR ¼ð RÞ ðBR Þij ¼ ; ð3:9Þ since this quantity depends only on the intrinsic metrics of γ a b R where ð RÞij ¼ Pi Pj ðhΣÞab is the intrinsic metric on . Σ and Σ , which are identical. μ μ μ a μ s Orthogonality of t and x implies that Pi ¼ Pi Pa, and hence, B. Properties

ðtÞ a b We will now establish important properties and an ðBR Þ ¼ P P ðKΣÞ : ð3:10Þ ij i j ab equivalent formulation of the kink transform. Σ Since xa is the unit norm orthogonal vector field at R, the Let us write as the disjoint union, trace of ðKΣÞ at R can be written as ab Σ ¼ a0 ∪ R ∪ a: ð3:18Þ a b ij ðtÞ a b KΣ x x KΣ γR B x x KΣ : jR ¼ ð Þab þð Þ ð R Þij ¼ ð Þab The spacetime M contains DðaÞ and Dða0Þ, where Dð:Þ ð3:11Þ denotes the domain of dependence. The kink transformed 0 slice Σs contains regions a and a with identical initial data, 0 A little algebra then implies so Ms also contains DðaÞ and Dða Þ. Because Σs has different extrinsic curvature at R, the two domains of 2 ab 2 ab ðKΣ Þ − ðKΣ Þ ðKΣ Þ ¼ðKΣÞ − ðKΣÞ ðKΣÞ : M s s ab s ab dependence will be glued to each other differently in s, M ð3:12Þ so the full spacetime will differ from in the future and past of R. This is depicted in Fig. 2. We will now derive an alternative formulation of the kink Moreover, we have rΣ ¼ rΣs since the two initial data slices transform as a one-sided local Lorentz boost at R. The unit have the same intrinsic metric. Thus, Eq. (3.2) implies that μ vector field t normal to Σ is discontinuous at R due to the kink-transformed slice satisfies the scalar constraint Σs s equation, the kink. Let

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Cauchy slices that are not smooth around R, but it reduces to Eq. (3.4) in the smooth case. This observation applies equally to any other vector field ξμ in the normal bundle to R,ifξμ has a smooth extension into Dða0Þ and DðaÞ in M. The norm of ξμ and its inner μ μ products with ðt Þ and ðt Þ are unchanged by the kink Σs L Σs R μ transform. Hence, in Ms, the left and right limits of ξ to R will satisfy

ξμ Λ μξν R ¼ð 2πsÞν L: ð3:22Þ

Now let Ξ ⊃ R be another Cauchy slice of DðΣÞ. Since FIG. 2. The kink-transformed spacetime Ms is generated by Ξ contains R, its timelike normal vector field ξμ (at R) lies the Cauchy evolution of the kinked slice Σs. This reproduces the left and right entanglement wedges DðaÞ and Dða0Þ of the in the normal bundle to R. We have shown that Eq. (3.21) is original spacetime M. The future and past of the extremal equivalent to the kink transform of Σ, that Eq. (3.22) surface R are in general not related to the original spacetime. is equivalent to the kink transform of Ξ, and that Eq. (3.21) is equivalent to Eq. (3.22). Hence, the kink transform μ μ of Σ is equivalent to the kink transform of Ξ. In other ðtΣ ÞR ¼ lim tΣ ; ð3:19Þ s x→0þ s words, the spacetime resulting from a kink transform about R does not depend on which Cauchy surface containing R μ μ we apply the kink transform to. ðtΣ ÞL ¼ lim−tΣ ð3:20Þ s x→0 s The kink transform (with s ≠ 0) always generates physically inequivalent initial data. However Ms need M Σ be the left and right limits to R. The metric of M is not differ from . They will be the same if and only if s is s M continuous since it arises from valid initial data on Σs. an initial data set in . There is an interesting special case Therefore, the normal bundle of 1 þ 1-dimensional normal where this holds for all values of s. Namely, suppose M spacetimes to points in R is well-defined. The above vector has a Killing vector field that reduces to a boost in the μ μ R Σ ⊂ M fields ðt Þ and ðt Þ belong to this normal bundle. normal bundle to . Then s (as a full initial data Σs R Σs L Therefore, at each point on R, the two vectors can only set), for all s. For example, the kink transform maps straight differ by a Lorentz boost acting in 1 þ 1-dimensional to kinked slices in the Rindler or maximally extended Minkowski space. The kink transform, Eq. (3.4), implies Schwarzschild spacetimes (see Fig. 3). We can also consider the kink transform of matter fields on a fixed background spacetime with the above symmetry. μ Λ μ ν ðtΣ ÞR ¼ð 2πsÞνðtΣ ÞL; ð3:21Þ s s Geometrically, M ¼ Ms for all s, but the matter fields will differ in M by a one-sided action of the Killing vector μ s where ðΛ2πsÞν is a Lorentz boost of rapidity 2πs. In this field. For example, let M be Minkowski space, with two sense, the kink transform resembles a local boost around R. balls at rest at x ¼1, y ¼ z ¼ 0 (see Fig. 4), and let R Alternatively, we can view Eq. (3.21) as the definition of given by x ¼ t ¼ 0. In the spacetime Ms obtained by a the kink transform. This definition can be applied to kink transform, the two balls will approach with velocity

FIG. 3. Straight slices Σ (red) in a maximally extended Schwarzschild (left) and Rindler (right) spacetime get mapped to kinked slices Σs (blue) under the kink transform about R.

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We neglect end-of-the-world branes in this discussion [42,43]. The generalized entropy is given by

AreaðRÞ S ¼ þ SðaÞþ…; ð4:2Þ gen 4Gℏ where SðaÞ is the von Neumann entropy of the region a and the dots indicate subleading geometric terms. The entan- glement wedge is also referred to as the Wheeler-DeWitt patch of A. There is significant evidence [6,8] that EWðρAÞ repre- sents the entire bulk dual to the boundary region A. That is, all bulk operators in EWðρAÞ have a representation in the algebra of operators A associated with A; and all simple correlation functions in A can be computed from the bulk. FIG. 4. On a fixed background with boost symmetry, the kink In other words, the entanglement wedge appears to be the transform changes the initial data of the matter fields. In this answer [41] to the question [44–47] of “subregion duality.” example, M is Minkowski space with two balls relatively at rest A bulk surface R is called quantum extremal (with respect (red).The kink transform is still Minkowski space, but the balls to A in the state ρ) if it satisfies the first two criteria, and R collide in the future of (blue). quantum RT if it satisfies all three. When the von Neumann entropy term in Eq. (4.2) is neglected, R is called an tanh 2πs and so will collide. The right and left Rindler extremal or RT surface, respectively. This will be the case wedge, DðaÞ and Dða0Þ, are separately preserved; the everywhere in this paper except in Sec. VI B. collision happens in the past or future of R. We now specialize to the setting in which CC flow was considered in Sec. II. Recall that the pure boundary state jψðCÞi is given on a boundary slice C corresponding to IV. BULK KINK TRANSFORM = BOUNDARY t ¼ 0 in standard Minkowski coordinates, and that we CC FLOW 0 regard C as the disjoint union of the left region A0 (x<0), In this section, we will argue that the kink transform is ρψ ∂ 0 with reduced state A0 , the cut A0 (x ¼ ), and the right the bulk dual of boundary CC flow. We will show that the ψ 0 region A0 (x>0), with reduced state ρ 0 . Let a0 and a0 be kink transform satisfies two nontrivial necessary condi- A0 arbitrary Cauchy surfaces of the associated entanglement tions. First, in Sec. IVA, we show that the left and right ψ ψ wedges EWðρ 0 Þ and EWðρ Þ. bulk region are the subregion duals to the left and right A0 A0 boundary region, respectively. In Sec. IV B, we show that The entanglement wedges of nonoverlapping regions are the bulk kink transform leads to precisely the stress tensor always disjoint, so shock at the boundary generated by boundary CC flow, ψ ψ EWðρ 0 Þ ∩ EWðρ Þ¼∅: ð4:3Þ Eq. (2.47). (In Sec. V, we will show that the kink trans- A0 A0 form predicts additional shocks in the CC flowed state, which have not been derived previously purely from QFT For the bipartition of a pure boundary state ψ, entanglement methods.) wedge complementarity holds

ψ C 0 ∪ R ∪ A. Matching left and right reduced states a½j ð Þi ¼ a0 a0; ð4:4Þ The entanglement wedge of a boundary region A in a where a ψ C is a Cauchy surface of EW ψ C .In ρ ½j ð Þi ðj ð ÞiÞ (pure or mixed) state A, particular, the left and right entanglement wedge share the same HRT surface R. ρ ρ EWð AÞ¼D½að AÞ ð4:1Þ Crucially, the classical initial data on a½jψðCÞi are 0 almost completely determined by the data on a0 and a0; is the domain of dependence of a bulk achronal region a 0 however, the data on R are not contained in a nor in a0.In satisfying the following properties [22,39–41]: 0 the semiclassical regime, the quantum state on a½jψðCÞi (1) The topological boundary of a [in the unphysical also includes global information (through its entanglement spacetime that includes the conformal boundary of structure) that neither subregion contains on its own. anti–de Sitter (AdS)] is given by ∂a ¼ A ∪ R. R Hence, in general, (2) SgenðaÞ is stationary under small deformations of . (3) Among all regions that satisfy the previous criteria, ψ ψ EWðjψðCÞiÞ ¼ D½EWðρ 0 Þ ∪ R ∪ EWðρ Þ ð4:5Þ ρ A0 A0 EWð AÞ is the one with the smallest SgenðaÞ.

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ψ ψ is a proper superset of EWðρ 0 Þ ∪ EWðρ Þ that also The above arguments establish that A0 A0 includes some of the past and future of R. ψ C D a ψ C ; : Now consider the CC-flowed state on the precursor slice EW½j sð sÞi ¼ ð ½j sð sÞiÞ ð4 10Þ jψ sðCsÞi. By Eqs. (2.37) and (2.38),wehave where a½jψ sðCsÞi is given by Eq. (4.8). In words, the bulk ψ ψ dual of the CC-flowed boundary state is the Cauchy s 0 EWðρ 0 Þ¼EWðρ 0 Þ¼Dða0Þ; ð4:6Þ As A0 development of the kink transform of a Cauchy slice containing the Hubeny-Rangamani-Takayanagi (HRT) sur- ψ ψ face R. Note that the classical initial data on this Cauchy EWðρ s Þ¼EWðρ Þ¼Dða0Þ: ð4:7Þ 0 A0 A0 surface are fully determined by the initial data on a0 and a0 inherited from the bulk dual of jψðCÞi, combined with the Since jψ si is again a pure state, EW½jψ sðCsÞi ¼ distributional geometric initial data consisting of the extrin- R Dða½jψ sðCsÞiÞ, where sic curvature shock at . The full spacetime geometry will differ from EW½jψðCÞi because of the different gluing at R. 0 a½jψ sðCsÞi ¼ a0 ∪ R ∪ a0: ð4:8Þ B. Matching bulk and boundary shocks We see that this initial data slice has the same intrinsic In Sec. III, we gave a prescription for generating bulk geometry as that of the original bulk dual. Indeed, by the geometries in AdS by inserting a kink on the Cauchy remarks following Eq. (4.4), the full classical initial data for surface, at the HRT surface. With the standard holographic 0 dictionary, the resulting geometry manifestly yields the the bulk dual to jψ si will be identical on a0 ∪ a0 and can only differ from the initial data for the original bulk at R. correct behavior of one-sided boundary observables under We pause here to note that a kink transform of a½jψðCÞi CC flow. This was shown in the previous subsection. ψ centered on R satisfies this necessary condition and hence, Another characteristic aspect of the CC flowed state j si is the presence of a stress tensor shock at the cut (Sec. II C), becomes a candidate for a½jψ sðCsÞi. However, this does not yet constrain the value of s. In order to go further, we proportional to shape derivatives of the von Neumann would now like to show that a kink transform of a½jψðCÞi entropy; see Eq. (2.47). We will now verify that this shock with a parameter s yields a bulk slice whose boundary is is reproduced by the kink transform in the bulk, upon applying the AdS=CFT dictionary. Notably, the shock is geometrically the precursor slice Cs. The bulk metric takes the asymptotic form [48],5 not localized to either wedge. Verifying kink/CC duality for this observable furnishes an independent, nontrivial check 1 of our proposal. 2 2 η A B d ds ¼ 2 ½dz þ ABdx dx þ Oðz Þ; ð4:9Þ We will now keep the first subleading term in the z Fefferman-Graham expansion of the asymptotic bulk met- ric [23,48], where ηAB is the metric of Minkowski space. Consider a R 1 stationary bulk surface anchored on the boundary cut 2 2 A B 0 R 0 ds ¼ 2 ðdz þ gABðx; zÞdx dx Þ; ð4:11Þ u ¼ v ¼ . At leading order, will reside at u ¼ v ¼ in z the asymptotic bulk, in the above metric [23]. (The first subleading term, which appears at order zd, will be crucial 16πG g ðx; zÞ¼η þ zd hT iþoðzdÞ; ð4:12Þ in our derivation of the boundary stress tensor shock in AB AB d AB Sec. IV B.) … Let Σ be a bulk surface that contains R and satisfies where indices A; B; correspond to directions along z ¼ 0 d const surfaces. t ¼ þ Oðz Þ in the metric of Eq. (4.9). Since the initial R data on each side of R are separately preserved (see Sec. III The location of the RT surface in the bulk can be described by a collection of (d − 1) embedding functions, B), Eq. (3.21) dictates that the kink transform Σs of Σ satisfies t ¼ 0 (x>0) and t ¼ x tanh 2πs (x<0), again up μ X ðy; zÞ¼ðz; XAðy; zÞÞ; ð4:13Þ to corrections of order zd. The corrections all vanish at 0 Σ C Σ C z ¼ , where is bounded by and s is bounded by s where ðy; zÞ are intrinsic coordinates on R. The expansion [see Eq. (2.32)]. Recall also that the kink transform is slice in z takes the simple form, independent. Thus, we have established that the kink ψ C transform of any Cauchy surface a½j ð Þi,bys along XAðy; zÞ¼zdXA þ oðzdÞ; ð4:14Þ R, yields a Cauchy surface bounded by the precursor ðdÞ C slice s. because the boundary anchor is the flat cut u ¼ v ¼ 0 of the Rindler horizon [23]. Stationarity of R can be shown to 5 l 1 We set AdS ¼ . imply [23]

066008-10 GRAVITY DUAL OF CONNES COCYCLE FLOW PHYS. REV. D 102, 066008 (2020) μ 4G δS The condition xμt 0 implies that A − ¼ XðdÞ ¼ A : ð4:15Þ d δX R 16πG zd−1 hT iþz−3xztz þ zd−1ðtx − xt Þ We consider a bulk Cauchy slice Σ ⊃ R for which ∂Σ d tx ðdÞ ðdÞ 0 −1 corresponds to the t ¼ slice on the boundary. Since the þ oðzd Þ¼0: ð4:25Þ subleading terms in Eqs. (4.12) and (4.14) start at zd, we are free to choose Σ so that it is given by Hence, we find

t ¼ zdςðxÞþoðzdÞ: ð4:16Þ ν − ν∂ d−18π d−1 ðKΣÞzνx x ðztνÞ ¼ z GhTtxiþoðz Þ: ð4:26Þ μ μ Recall that the vector fields t and x are defined to be We now apply the kink transform to Σ (viewed as an R orthogonal to , and, respectively, orthogonal and tangent initial data set). This yields a new initial data set on a slice to Σs. In FG coordinates, one finds Σs in a new spacetime Ms. We again expand in Fefferman- Graham coordinates, A A d A d t ¼ zðtð0Þ þ z tðdÞ þ oðz ÞÞ; ð4:17Þ 1 2 2 ˜ ˜ ˜A ˜B A A d A d ds ¼ 2 ðdz þ gABðx; zÞdx dx Þ; ð4:27Þ x ¼ zðxð0Þ þ z xðdÞ þ oðz ÞÞ; ð4:18Þ z

z d−1 z d−1 16π t z z t o z ; 4:19 d G ˜ d ¼ ð ðd−1Þ þ ð ÞÞ ð Þ g˜ ðx; zÞ¼η˜ þ z hT iþoðz Þ: ð4:28Þ AB AB d AB z d−1 z d−1 x ¼ zðz x −1 þ oðz ÞÞ: ð4:20Þ ðd Þ Here, η˜AB is still Minkowski space; any change in the bulk μ geometry will be encoded in the subleading term. z t The overall factor of is due to normalization. Note that ð0Þ The notation η˜AB indicates that we will be using the is a coordinate vector field but, in general, tμ is not. specific coordinates in which the metric of d-dimensional Individual coordinate components of vectors and tensors Minkowski space takes the nonstandard form given by μ μ Eq. (2.41). This has the advantage that the coordinate form are defined by contractions with tð0Þ and xð0Þ, respectively, t μ of all vectors, tensors, and embedding equations in Dða0Þ ∪ for example, t ≡ tμt 0 . ð Þ R ∪ DðaÞ will be unchanged by the kink transform, if we We now consider a contraction of the extrinsic curvature use standard Cartesian coordinates before the transform and tensor on Σ, the tilde coordinates afterwards. μ For example, the invariance of the left and right bulk b ν∇ ðKΣÞabx ¼ Pax ðμtνÞ: ð4:21Þ domains of dependence under the kink transform implies that Σ is given by We would like to further project the a index onto the z s direction. Deep in the bulk, the z direction does not lie ˜ d d μ t ¼ z ςðx˜Þþoðz Þ; ð4:29Þ entirely in Σ. However, note that gμzt → 0, in the limit z → 0, due to Eq. (4.19). Therefore, at leading order in z, μ μ with the same ς that appeared in Eq. (4.16). (In fact, this the z direction does lie entirely in Σ; moreover, P → δ as z z extends to at all orders in z.) As already shown in the z → 0. We will only be interested in evaluating Eq. (4.21) at ˜ 6 previous subsection, ∂Σs lies at t ¼ 0, z ¼ 0. leading order in z so we may freely set a ¼ z, which yields As another example, the coordinate components of the unit normal vector to Σ in M , t˜μ, will be the same as the ν − ν∂ ν Γγ s s ðKΣÞzνx x ðztνÞ ¼ x tγ νz ð4:22Þ components of the normal vector to Σ in M, tμ, and therefore, 2 t x z z d−1 ¼ z Γtxz þ zx Γttz þ zt Γxxz þ x t Γzzz þ oðz Þð4:23Þ ∂ ∂ ˜ ðztνÞjM ¼ ðztνÞjM : ð4:30Þ d − 2 16πG s ð Þ d−1 − −3 z z − d−1 x − t ¼ z hTtxi z t x z ðtðdÞ xðdÞÞ 2 d Below, we will use the convention that any quantity with a d−1 þ oðz Þ: ð4:24Þ tilde is evaluated in Ms, in the coordinates of Eq. (4.28). Any quantity without a tilde is evaluated in M, in the coordinates of Eq. (4.12). The only exception is the 6In d>2, the terms involving xztz will be higher order, by Eqs. (4.19) and (4.20) and need not be included. Since they extrinsic curvature tensor, where the corresponding dis- Σ Σ cancel out either way, we include them here to avoid an explicit tinction is indicated by the subscript s or , for consis- case distinction. tency with Sec. III.

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We now consider the extrinsic curvature of Σs.A form similar to Eq. (4.35). Thus, we expect that the calculation analogous to the derivation of Eq. (4.26) properties we find in holographic CC flow hold in non- implies holographic QFTs as well. To derive the hTxxi shock, we use the Gauss-Codazzi ν ν −1 −1 ˜ − ˜ ∂ ˜ d 8π ˜ ˜ d relation [49], ðKΣs Þzνx x ðztνÞ ¼ z GhTt x˜ iþoðz Þ: ð4:31Þ μ ν α β − From Eqs. (4.26) and (4.30), we find PaPbPcPdRμναβ ¼ KacKbd KbcKad þ rabcd; ð5:1Þ

1 where r is the intrinsic Riemann tensor of Σ.Itis d−1 ˜ d−1 − ν abcd z hTt˜ x˜ i¼z hTtxiþ ½ðKΣ Þzν ðKΣÞzνx Σ 8πG s important to note that this relation is purely intrinsic to . −1 Since Σ Σ as submanifolds, we can not only evaluate þ oðzd Þ; ð4:32Þ ¼ s Eq. (5.1) in both M and Ms but also meaningfully subtract the two. We emphasize that the following 2π d−1 sinh ð sÞ d−1 calculation is only nontrivial in d>2 (in d ¼ 2, the ¼ z hTtxi − δðRÞxz þ oðz Þ: ð4:33Þ 8πG Gauss-Codazzi relation is trivial). We comment on d ¼ 2

μ μ at the end. In the first equality, we used the fact that x and x˜ can be First, we evaluate Eq. (5.1) in M. We will only be identified as vector fields, and the extrinsic curvature interested in evaluating it to leading order in z in the Σ Σ tensors can be compared, in the submanifold ¼ s. Fefferman-Graham expansion. As argued in Sec. IV B, The second equality follows from the definition of the when working at leading order, we can freely set kink transform, Eq. (3.4). a c z −1 ¼ ¼ . We then compute the following at leading By Eq. (4.18), δðRÞ¼δðz x˜Þ¼zδðx˜Þ. The condition z μ order in : xμ∂zX ¼ 0 yields 2 Rzxzx ¼ KzzKxx − ðKxzÞ þ rzxzx: ð5:2Þ − d−2 ˜ d−2 xz ¼ dz XðdÞ þ oðz Þ; ð4:34Þ α We start by computing Kzz. First, we note that Γzztα ¼ 0 ˜ ˜ A → 0 identically. Therefore, where XðdÞ is the A ¼ x component of X . Taking z ðdÞ and using Eq. (4.15), we thus find δS ∂ 4 − 2 d−3 d−3 Kzz ¼ ztz ¼ Gðd Þz δ þ oðz Þ: ð5:3Þ T R ˜ 1 δS T˜ ˜ T sinh 2πs δ x˜ ; 4:35 h t xi¼h txiþ2π ð Þ δ ˜ ð Þ ð Þ X X˜ ¼0 We have made use of δ which agrees precisely with Eq. (2.47). d−2 S d−2 tz ¼ 4Gz þ oðz Þ; ð5:4Þ Note that this derivation applies to any boosted coor- δT R dinate system ðt;ˇ xˇÞ as well. Linear combinations of μ Eq. (4.35) with its boosted version reproduces both the which follows from tμ∂zX ¼ 0. Tu˜ u˜ shock of Eq. (2.46) and the Tv˜ v˜ shock of Eq. (2.45) Next, we compute holographically. x x x μ x μ Rzxzx ¼ ∂xΓzz − ∂zΓxz þ ΓxμΓzz − ΓzμΓxz: ð5:5Þ V. PREDICTIONS One finds Having found nontrivial evidence for kink transform/CC flow duality, we now change our viewpoint and assume the 1 16πG ∂ Γx ¼ ðd − 2Þðd − 1Þzd−2 hT iþoðzd−2Þ; duality. In this section, we will derive a novel property of z xz 2 d xx CC flow from the kink transform: a shock in the hTxxi component of the stress tensor in the CC flowed state. We ð5:6Þ do not yet know of a way to derive this directly in the 1 16π x z d−2 G d−2 quantum field theory, so this result demonstrates the utility ΓxzΓzz ¼ − ðd − 2Þz hTxxiþoðz Þ; ð5:7Þ of the kink transform in extracting nontrivial properties of 2 d CC flow. We further argue that hT i and hT i constitute xx tx with all other terms either subleading in z or identically all of the independent, nonzero stress tensor shocks in the vanishing, and hence, CC flowed state. Our holographic derivation only depends on near boun- −2 −2 R ¼ −8πGðd − 2Þzd hT iþoðzd Þ: ð5:8Þ dary behavior, and the value of the shock takes a universal zxzx xx

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Putting all this together, we have Evaluating the μ ¼ z components in the same way as in Sec. IV B, we find δS −8π − 2 d−2 4 d−3 − 2 Gðd Þz hTxxi¼ Gz ˜ Kxx ðKxzÞ ˜ − 0 δT R hTμ˜ y˜ i hTμyi¼ : ð5:16Þ r o zd−2 : 5:9 þ zxzx þ ð Þ ð Þ For s → ∞, the shocks derived in the previous two sections agree with those found to be required for the The analogous relation evaluated in M reads s existence of certain coarse grained bulk states in Ref. [20]. δ In that work, the cut was allowed to be a wiggly or flat cut d−2 ˜ d−3 S ˜ ˜ 2 −8πG d − 2 z T ˜ ˜ 4Gz K ˜ ˜ − K ˜ of a bifurcate horizon such as a Rindler horizon, and the ð Þ h x xi¼ δ ˜ x x ð xzÞ T R state could belong to any quantum field theory. d−2 þ r˜zxz˜ x˜ þ oðz Þ; ð5:10Þ Interpolation of these results suggests that the shocks we have derived here generalize to the case of CC flow for a ˜ where we have made use of Eq. (4.30) to set Kzz ¼ Kzz.We wiggly cut of the Rindler horizon, in general QFTs with a can now subtract these two relations. First, note that conformal fixed point. r˜abcd ¼ rabcd since it is purely intrinsic to Σ. Next, recall from the definition of the kink transform Eq. (3.4) that VI. DISCUSSION ˜ Kx˜ x˜ − Kxx ¼ −z sinhð2πsÞδðx˜Þ: ð5:11Þ A. Relation to JLMS and one sided modular flow The bulk dual of one-sided modular flow [25,26] d−2 Lastly, it is easy to check that Kxz ∼ oðz Þ; hence, its resembles the kink transform. CC flow yields a well- contribution to Eq. (5.9) is subleading and similarly for defined state, however, whereas a one sided modular Eq. (5.10). Thus, subtracting Eq. (5.10) from Eq. (5.9) flowed state is singular in QFT. Correspondingly, the kink yields transform defined here yields a smooth bulk solution, whereas the version implicitly defined in Ref. [26] results ˜ 1 δS in a singular spacetime (see also Ref. [17], footnote 4). We T ˜ ˜ − T sinh 2πs δ x˜ : 5:12 h x xi h xxi¼2π ð Þ δ ˜ ð Þ ð Þ T X˜ ¼0 will now explain this in detail. ρ Consider a boundary region A0 with reduced state A0 , 2 2 The above calculation only works in d> .Ind ¼ , dual to a semiclassical state ρa in the bulk entanglement since the boundary theory is a CFT, tracelessness of the wedge a associated to A0 as seen in Fig. 5. We denote by boundary stress tensor further implies that hTtxi is the only independent component of the stress tensor shock so there is no need for a calculation analogous to the one above. We expect that this argument is robust under relevant defor- mations of the CFT since the shock is highly localized and should universally depend only on the UV fixed point. Together with the hTtxi shock we reproduced in the previous section, and using Lorentz invariance of the boundary, this result determines the transformation of the stress tensor contracted with any pair of linear combi- nations of t and x, such as hTtti. This linear space contains all of the independent nonvanishing components of the shock. To see this, note that

ν x ð∇νy˜μ − ∇νyμÞ¼0; ð5:13Þ

ν y ð∇νy˜μ − ∇νyνÞ¼0; ð5:14Þ FIG. 5. A boundary subregion A0 (pink) has a quantum R ν ˜ extremal surface denoted (brown) and an entanglement wedge y ð∇νtμ − ∇νtμÞ¼0; ð5:15Þ 0 denoted a. The complementary region A0 (light blue) has the entanglement wedge a0. CC flow generates valid states, but one- μ μ i i where y ¼ Pi y for any vector field y in the tangent sided modular flow is only defined with a UV cutoff. For bundle of R. Equations (5.13) and (5.14) follow trivially example, one can consider regulated subregions AðϵÞ (deep blue) from the fact that the prescription Eq. (3.22) only intro- and A0ðϵÞ (red). In the bulk, this amounts to excising an infrared duces a discontinuity in vector fields in the normal bundle region (gray) from the joint entanglement wedge resulting in a of R, while Eq. (5.15) simply follows from Eq. (3.4). regulated entanglement wedge DðΣϵÞ (yellow).

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K − log ρ and K − log ρ , the boundary and bulk ρ ˆ A ¼ A a ¼ a density matrix A0 , and hence, KA0 , do not exist. Physically, modular Hamiltonians, respectively. The JLMS result [7] ˆ the action of KA0 on a fixed boundary time slice would states that break the vacuum entanglement of arbitrarily short wave- ∂ ˆ length modes across A0; this would create infinite energy. ˆ A½R ˆ ˆ Therefore, any discussion of KA0 requires introducing a KA0 ¼ þ Ka; ð6:1Þ 4G ðϵÞ UV regulator. Consider the regulated subregions A0 and ˆ 0ðϵÞ where A½R is the area operator that formally evaluates the A0 shown in Fig. 5. The split property in algebraic QFT area of the quantum extremal surface R [41]. [30,53] guarantees the existence of a (nonunique) type I Suppose now that A0 has a nonempty boundary ∂A0. von Neumann algebra N nested between the algebras of ðϵÞ 0ðϵÞ Then there is an interesting asymmetry in Eq. (6.1). The subregion A0 and the complementary algebra of A0 , i.e., one-sided boundary modular operator appearing on the left- hand side is well-defined only with a UV cutoff. On the ϵ ϵ Að Þ ⊂ N ⊂ A0ð Þ 0: : other hand, at least the leading (area) term in the bulk 0 ð 0 Þ ð6 4Þ modular operator on the right-hand side has a well-defined action. Let us discuss each side in turn. With this prescription, one can define a regulated version ˆ ρ In Einstein gravity, the area operator A is the generator of of the reduced density matrix A by using the type-I factor one-sided boosts. To see this, let us restrict the gravitational N [54]. It has been suggested that there exists an N phase space to a truncated bulk region DðΣϵÞ. Σϵ is a partial consistent with the geometric cutoff shown in Fig. 5 bulk Cauchy slice that excludes asymptotic portions of the [53,55]: the quantum extremal surface R in the bulk is bulk near the entangling surface ∂A as seen in Fig. 5. There regulated by a cutoff brane B demarcating the entanglement a 0 ðϵÞ 0ðϵÞ exists a (nonunique) vector field ξ in Dða Þ ∪ R such that wedge of the subregion A0 ∪ A0 . The regulated area a ξ generates an infinitesimal one-sided boost at R [50,51]. operator Aˆ ½R=4G is well defined once boundary condi- This boost can be quantified by a parameter s in the normal tions on B are specified. bundle to R, as described in Sec. III B. The area functional Let us now specialize to the case for which we have A½R=4G is the Noether charge at R associated to ξa,given conjectured kink transform/CC-flow duality: the boundary 0 by the expectation value of the area operator in the slice C ¼ A0 ∪ A0 is a Cauchy surface of Minkowski space, semiclassical bulk state, and ∂A0 is the flat cut u ¼ v ¼ 0 of the Rindler horizon. We have just argued that the kink transformation is ˆ A½R¼hA½Ri: ð6:2Þ generated by the area operator through Eq. (6.3).By Eq. (6.1), the boundary dual of this action should be a Each point in the gravitational phase space can be one-sided modular flow, not a CC flow. By Eqs. (2.4) and specified by the metric in Dða0Þ, the metric in DðaÞ, and (2.6), these are manifestly different operations. Indeed, the boost angle s at R with which the two domains of unlike one-sided modular flow, Connes cocycle flow yields dependence are glued together [26,28,51,52]. The action of a well-defined boundary state for all s, without any UV divergence at the cut ∂A0: ψ C → ψ C . ˆ j ð Þi j sð Þi he2πisA½R=4Gið6:3Þ In fact, there is no contradiction. For both modular flow and CC flow on the boundary, a bulk-dual Cauchy surface on points in the gravitational phase space is to simply shift Σs is generated by the kink transform. The difference is in the conjugate variable, i.e., the relative boost angle between how Σs is glued back to the boundary. the left and right domains of dependence, by s. Note that the For modular flow, Σs is glued back to the original slice C. metrics in the left and right domains of dependence are Generically, this would violate the asymptotically AdS unchanged since the area functional acts purely on the boundary conditions, necessitating a regulator such as the phase space data at R. This is the classical analogue of excision of the grey asymptotic region in Fig. 5 and the statement that the area operator is in the center of the interpolation by a brane B. The boundary dual is an algebras of the domains of dependence [5]. Comparing appropriately regulated modular flowed state with energy with Sec. III B, we see that this action is equivalent to the concentrated near the cut ∂A0. This construction is possible kink transform of Σϵ about R by s. We stress that this action even if ∂A0 is not a flat plane, but the regulator is is well-defined even if R extends all the way out to the ambiguous and cannot be removed.7 conformal boundary, i.e., in the far ultraviolet from the For CC flow, Σs is glued to the precursor slice Cs as boundary perspective. discussed in Sec. III. This yields jψ sðCsÞi. Time evolution We turn to the left-hand side of Eq. (6.1), still assuming ∂ that A0 has a nonempty boundary A. Since the algebra of a 7There is evidence that a code subspace can be defined with an QFT subregion A0 is a type III1 von Neumann algebra, the appropriate regulator such that one-sided modular flow keeps the Hilbert space does not factorize across ∂A0 [30]. A reduced state within the code subspace [27–29].

066008-14 GRAVITY DUAL OF CONNES COCYCLE FLOW PHYS. REV. D 102, 066008 (2020) on the boundary yields jψ ðCÞi, the CC-flowed state on the a s ΔðD ðKΣÞ − D KΣÞ¼0; ð6:9Þ original slice C. a i i On the boundary, we can use the one-sided modular where Δ represents the difference in the constraint equa- C operator in two ways. As a map between states on [7,9] it tions between the original spacetime M and the kink C requires a UV regulator. As a map that takes a state on to a transformed spacetime M , and we have used Eqs. (6.5) C ψ C → ψ C s state on the precursor slice s, j ð Þi j sð sÞi,itis and (3.4). These are essentially the analogs of Eqs. (3.13) C equivalent to CC flow on by Eqs. (2.35) and (2.36). This and (3.14), and we have simplified the notation slightly. is a more natural choice due to its UV finiteness. But it is For the constraint equations to be solved, the kink ∂ available only if the vacuum modular operator for cut A is transform would have to generate the same change on geometric, so that the precursor slice is well defined. the right-hand side of the constraints. It would thus have to induce an additional stress tensor shock of the form, B. Quantum corrections 1 δS It is natural to include semiclassical bulk corrections to Δ 2π δ TTT ¼ 2π sinhð sÞ δ ðXÞ; ð6:10Þ all orders in G to our proposal. The natural guess would be T to perform the kink transform operation about the quantum 1 δS ΔT ¼ sinhð2πsÞ δðXÞ: ð6:11Þ extremal surface along with a CC flow for the bulk state. In TX 2π δX general, it is difficult to describe this procedure within EFT. In states far from the vacuum, the background spacetime Formally, these conditions agree precisely with the proper- changes under the kink transform, and it is unclear how to ties of CC flow discussed in Sec. II. Thus, we might expect map states from one spacetime to another. However, we a generalized bulk CC flow to result in shocks of precisely will find some evidence that suggests that the bulk this form. operation relating the two states is a generalized version In fact, the existence of semiclassical states satisfying the of CC flow in curved spacetime. above equations was conjectured in [20]; the fact that CC To see this, note that the quantum extremal surface R flow generates such states in the nongravitational limit was satisfies the equations, interpreted as nontrivial evidence in support of the con- jecture. Thus, we expect a kink transform at the quantum δ ðtÞ S extremal surface with a suitable modification of the state to B þ 4Gℏ ¼ 0; ð6:5Þ R δT provide the bulk dual of CC flow to all orders in G. At a more speculative level, we can also discuss the bulk δ ðxÞ S dual of CC flow in certain special states called fixed area BR þ 4Gℏ ¼ 0; ð6:6Þ δX states, which serve as a natural basis for modular flow [27– 29]. These are approximate eigenstates of the area operator ðtÞ ðxÞ where ðBR Þ and ðBR Þ denote the trace of the extrinsic and are therefore unlike smooth semiclassical states which curvature (expansion) in the two normal directions to R, are analogous to coherent states. The Lorentzian bulk dual μ μ δS δS i.e., t and x , respectively. Similarly, δT and δX are the of such states potentially involves superpositions over entropy variations in the tμ and xμ directions, respectively. geometries [56]. The classical kink transform involves an extrinsic cur- However, by construction, the reduced density matrix is vature shock at the classical RT surface. As shown in maximally mixed at leading order in G. Thus, the state jψi Sec. III, extremality of the surface ensures that the con- is unaffected by one sided modular flow, and the only effect straint equations continue to be satisfied after the kink of CC flow is that we describe the state on a kinematically transform in this case. However, the quantum extremal related slice Cs. Thus, the dual description must be invariant surface has nonvanishing expansion, the constraint equa- under CC flow up to a diffeomorphism. tions are not automatically satisfied when an extrinsic In such states, one could apply the semiclassical pre- curvature shock is added at the quantum RT surface. scription using Eq. (6.1). As discussed above, the action of More precisely, the left-hand side of the constraint the area operators results in a diffeomorphism of the equations on a slice Σ are modified by the kink transform geometric description, if it exists. From Eq. (6.1), the by remaining action of the boundary CC flow is to simply induce a bulk CC flow. δ ab 2 S ΔðrΣ − ðKΣÞ ðKΣÞ þðKΣÞ Þ¼8Gℏ sinhð2πsÞ δðXÞ; ab δT C. Beyond flat cuts ð6:7Þ Kink transform/CC flow duality can be generalized to other choices of boundary subsystems, so long as a δ a S precursor slice can be defined. The precursor slice is ΔðD ðKΣÞ − D KΣÞ¼4Gℏ sinh ð2πsÞ δðXÞ; ð6:8Þ a x x δX generated by acting on the original slice with the vacuum

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FIG. 6. An arbitrary spacetime M with two asymptotic boundaries is transformed to a physically different spacetime Ms by performing a kink transform on the Cauchy slice Σ. A piecewise geodesic (dashed gray line) in M connecting x and y with boost angle 2πs at R becomes a geodesic between xs and y in Ms. modular Hamiltonian; this is well-defined only if this in Sec. IV B. Notably, since ∂A ¼ ∅ in this case, there is no action is geometric. In Sec. II D, we ensured this by taking subtlety regarding boundary conditions for JLMS and thus, the boundary to be Minkowski space and choosing a planar one-sided modular flow makes sense without any regulator. cut. Precursor slices also exist in any conformally related Thus, our construction is simply kinematically related to choice, such as a spherical cut. the construction in [26]. But there are other settings where the vacuum modular An interesting situation arises for wiggly cuts of the Hamiltonian acts geometrically. This includes multiple Rindler horizon, i.e., u ¼ 0 and v ¼ VðyÞ. The modular asymptotically AdS boundaries where the boundary mani- Hamiltonian acts locally, but only when restricted to the fold has a time translation symmetry. For example, consider null plane [31]. Its action becomes nonlocal when extended a two-sided black hole geometry M with a compact RT to the rest of the domain of dependence. The properties of surface R as seen in Fig. 6. The boundary manifold is of the CC flow described in Secs. II A–II C all hold for this choice form C × R, where the first factor corresponds to the spatial of cut. This constrains one-sided operator expectation geometry and the second corresponds to the time direction. values on the null plane, subregion entropies for cuts The boundary Hilbert spaces factorizes; each boundary entirely to one side of VðyÞ, and even the Tvv shock at algebra is a type I factor. Thus, the version of CC flow the cut. Interestingly, all of them are matched by the kink defined in terms of density matrices in Eq. (2.6) becomes transform, by the arguments given in Sec. III. Even the rigorous in this situation. A natural choice of vacuum state expected stress tensor shock can still be derived, by taking a is the thermofield double [57,58]. The reduced state on null limit of our derivation as described in the Appendix. ρ ∼ −β each side is thermal, A0 expð HÞ. Thus, the modular One might then guess that the kink transform is also dual to Hamiltonian is proportional to the ordinary Hamiltonian on CC flow for arbitrary wiggly cuts. each boundary. This generates time translations and so is Even in the vacuum, however, the kink transform across geometric. a wiggly cut results in a boundary slice that cannot be Now, in any such geometry M one can pick a Cauchy embedded in Minkowski space, due to the absence of a slice Σ that ends on boundary time slices on both sides and boost symmetry that preserves the entangling surface. contains R. In obvious analogy with Sec. III, we conjecture Thus, the kink transform would have to be modified to that the domain of dependence of the kink transformed slice work for wiggly cuts. The transformation of boundary Σs in a modified geometry Ms is dual to the boundary observables off the null plane is quite complicated for state, wiggly cut CC flow. Thus, we also expect that regions of the entanglement wedge probed by such observables ψ C ρ−is ψ C j sð sÞiLR ¼ L j ð ÞiLR; ð6:12Þ should be drastically modified, unlike the case where the entangling surface is a flat Rindler cut. where we have used the notation of Eq. (2.36). However, the wiggly cut boundary transformation In such a situation, it is again manifest that the Wheeler- remains simple for observables restricted to the null plane. DeWitt patches dual to either side are preserved by argu- Thus, one could try to formulate a version of the kink ments similar to those made in Sec. IVA. However, since transform on Cauchy slices anchored to the null plane on there is no portion of R that reaches the asymptotic the boundary and the RT surface in the bulk. Perhaps a boundary, there is no analog of the shock matching done nontrivial transformation of the entanglement wedge arises

066008-16 GRAVITY DUAL OF CONNES COCYCLE FLOW PHYS. REV. D 102, 066008 (2020) from the need to ensure that the kink transformed initial shock exists everywhere on the RT surface. One could data be compatible with corner conditions at the junction solve for the position of the RT surface to further where the slice meets the asymptotic boundary [59].We subleading orders and relate the bulk shock to the boundary leave this question to future work. stress tensor. This would yield a sequence of relations that the stress tensor must satisfy in order to be dual to the kink D. Other probes of CC flow transform. In general, these relations may be highly theory dependent, but it would be interesting to see if some follow In Sec. IV, we provided evidence for kink transform/CC directly from CC flow or make interesting universal flow duality. The preservation of the left and right entan- predictions for CC flow in holographic theories. glement wedges under the kink transform ensures that all one-sided correlation functions transform as required. It E. Higher curvature corrections would be interesting to consider two sided correlation functions. However, these do not change universally and In Sec. IV, we argued that the bulk kink transform in a are difficult to compute in general. In the bulk, this is theory of Einstein gravity satisfies properties consistent manifested by the fact that the future and past wedges do with the boundary CC flow. However, this result is robust to not change simply and need to be solved for. the addition of higher curvature corrections in the bulk However, because of the shared role of the kink trans- theory. The preservation of the two entanglement wedges, 8 form, we can take advantage of the modular toolkit for one- i.e., Eq. (2.38), is a geometric fact that remains unchanged. ψ˜ ρ−is ψ Further, the matching of the stress tensor shock crucially sided modular flow [26]. Let j si¼ A j i be a family of states generated by one-sided modular flow as discussed in depended on two ingredients: firstly, Eq. (4.12), the holo- Sec. VI A. Then certain two sided correlation functions graphic dictionary between the boundary stress tensor and the bulk metric perturbation and secondly, Eq. (4.15), the hψ˜ sjOðxÞOðyÞjψ˜ si can be computed as follows. Suppose OðxÞ is an operator dual to a “heavy” bulk relation between the boundary entropy variation and the 1 l ≪ ≪ 1 l 1 l shape of the RT surface. Both of these relations are field with mass m such that = AdS m = P; = s. Correlation functions for such an operator can then be modified once higher curvature corrections are included computed using the geodesic approximation, [61,62]. However, it follows generally from dimension counting arguments that O x O y ≈ exp −mL ; 6:13 h ð Þ ð Þi ð Þ ð Þ 16π ðdÞ η G gij ¼ 1 hTiji; ð6:16Þ where L is the length of the bulk geodesic connecting d boundary points x and y. Now consider boundary points x 4 δ A G S and y such that there is a piecewise bulk geodesic of length X ¼ −η2 ; ð6:17Þ ðdÞ δ A Lðx; yÞ joining them in the spacetime dual to the state jψi. d X R This kinked geodesic is required to pass through the RT η η surface of subregion A with a specific boost angle 2πs as where 1 and 2 are constants that depend on the higher curvature couplings. Using the first law of entanglement, it seen in Fig. 6. (This is a fine-tuned condition on the set of η η points x, y.) Since single sided modular flow behaves can be shown that in fact 1 ¼ 2 [61,62]. Hence, the locally as a boost at the RT surface, it straightens out the boundary stress tensor shock obtained from the kink kinked geodesic such that it is now a true geodesic in the transform is robust to higher curvature corrections. spacetime dual to the state jψ˜ i. Thus, we have s F. Holographic proof of QNEC

hψ˜ sjOðxÞOðyÞjψ˜ si ≈ expð−mLðx; yÞÞ: ð6:14Þ A recent proof of the QNEC from the ANEC [17] considers CC flow for a subregion A on the null plane As discussed in Sec. II D, the CC flowed state can u ¼ 0 with entangling surfaces v ¼ V1ðyÞ and v ¼ V2ðyÞ equivalently be thought of as the single sided modular surrounding a given point p. From the transformation flowed state jψ˜ si on a kinematically transformed slice Cs. properties of the stress tensor under CC flow described Thus, the above rules can still be used to compute two in Sec. II B, Tvv → 0 as s → ∞. In addition, there are stress sided correlation functions in the CC flowed state jψ si¼ tensor shocks at ∂A, as described in Sec. II C, of weight u jψi as δS → s fðsÞ δVðyÞ jψ;∂A. In the limit, V1ðyÞ V2ðyÞ, computing the ANEC in the CC flowed state, one obtains contributions hψ sjOðxsÞOðyÞjψ si ≈ expð−mLðx; yÞÞ; ð6:15Þ

8 where x is the point related to x by the vacuum modular Here, we assume that the initial value formulation of Einstein s gravity can be perturbatively adjusted to include higher curvature flow transformation. corrections despite the fact that a nonperturbative classical We also note that the shock matching performed in analysis of higher curvature theories is often problematic due Sec. IV was a near boundary calculation. However, a bulk to the Ostrogradsky instability [60].

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ACKNOWLEDGMENTS We thank Chris Akers, Xi Dong, Thomas Faulkner, Daniel Jafferis, Ronak Soni, and Aron Wall for helpful discussions and comments. This work was supported in part by the Berkeley Center for Theoretical Physics; by the Department of Energy, Office of Science, Office of High Energy Physics under QuantISED Award No. DE- SC0019380 and under Contract No. DE-AC02- 05CH11231; and by the National Science Foundation under Grant No. PHY1820912.

APPENDIX: NULL LIMIT OF THE KINK TRANSFORM

FIG. 7. Holographic proofs. Left: Boundary causality is re- In this appendix, we apply the kink transform to a spected by the red curve that goes through the bulk in a spacetime Cauchy slice Σ that has null segments. In the null limit, we M; this is used in proving the ANEC. The RT surfaces R1 and express the kink transform in terms of the null initial value R2 must be spacelike separated; this is used in proving the problem. We then show that this leads to a shock in the M → ∞ QNEC. Right: In the kink transformed spacetime s as s , Weyl tensor for d>2. From this Weyl shock, we extract the QNEC follows from causality of the red curve, which only R R the boundary stress tensor shock. This serves a twofold gets contributions from the Weyl shocks (blue) at 1 and 2, and purpose. The first is that it provides direct intuition for how the metric perturbation in the region between them. the kink transform modifies the geometry. The second is that, as will be evident from the calculation below, the from the stress tensor TvvðpÞ in subregion A, and a derivation of the stress tensor shock from the Weyl shock δ2S contribution proportional to ψ from the shocks. δVðy1ÞδVðy2Þ j ;p works even for wiggly cuts of the Rindler horizon on the 9 Positivity of the averaged null energy in the CC flowed boundary. Σ R state then implies the QNEC in the original state. Let Nk be a null segment of in a neighborhood of , a Prior to the QFT proofs, both the ANEC and QNEC had and let k be the null generator of Nk. We now allow the been proved holographically [23,63]. The guiding principle boundary anchor of R to be an arbitrary cut V0ðyÞ of the behind both of these proofs was the fact that consistency of Rindler horizon, as considered in Sec. II B. Lastly, denote a i the holographic duality requires bulk causality to respect by Pμ and Pμ, the projectors onto Nk and cross sections of boundary causality as we demonstrate in Fig. 7. In the case of Nk (including the RT surface R), respectively. We can i the ANEC, one considers an infinitely long curve connecting compose these to obtain the projector Pa. points on past null infinity to future null infinity through the By Eq. (3.4), when Σ is spacelike in a neighborhood of bulk and demands that it respect boundary causality [63].In R, the kink transform can be contracted as follows: the proof of the QNEC, one requires that curves joining the a → a − 2π δ R RT surfaces of subregions vV2ðyÞ, x ðKΣÞab x ðKΣÞab sinh ð sÞxb ð Þ: ðA1Þ R R denoted 1 and 2, respect boundary causality [15,23]. In the null limit, both xa and tμ approach ka. Therefore, the There are two contributions to the light cone tilt of this bulk quantity in the lhs of Eq. (A1) has the following null limit: curve coming from the metric perturbation hvv in the near boundary geometry, and the shape of the Ryu-Takayanagi a null a x ðKΣÞ !k ∇ k : ðA2Þ (RT) surface Xμðy; zÞ. By the holographic dictionary, these ab a b contributions can be related to the boundary stress tensor Tvv The transformation of Eq. (A1) then becomes and the entropy variations δS as discussed in Sec. IV B. δV κ → κ − 2π δ λ Now performing the kink transform removes the con- sinh ð sÞ ð Þ; ðA3Þ tribution coming from the shape of the RT surface and puts where λ is a null parameter adapted to ka and κ is the it into a time advance/delay coming from shocks in the bulk inaffinity defined by Weyl tensor that we compute in the Appendix. Considering b a a the extended curve from past to future null infinity, we see k ∇bk ¼ κk : ðA4Þ that whether or not it respects boundary causality is determined entirely by the region between the entangling R R 9The results of this section do not apply when d ¼ 2, as the surfaces 1 and 2 since the bulk solution approaches the 2 → ∞ shear and the Weyl tensor vanish identically. However, in d ¼ , vacuum everywhere else in the limit s . Requiring there is no distinction between flat and wiggly cuts on the causality of the ANEC curve then results in the QNEC, boundary so we gain neither additional intuition nor generality making the connection to the boundary proof manifest. compared with the analysis in Sec. IV B.

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We refer to this transformation as the left stretch, as it arises use null coordinates ðu; vÞ and ðu;˜ v˜Þ on the boundary as ¯ a from a one-sided dilatation along Nk. This transformation defined in Sec. II D. To start with, we note that ka∂zX ¼ 0 ¯ a was originally described in [20] in the context of black hole since ∂zX is tangent to the RT surface. Evaluating this at coarse graining. leading order yields the relation, We now show that the left stretch generates a Weyl tensor − d−3U O d−4 shock at the RT surface. The shear of a null congruence is kz ¼ dz ðdÞ þ ðz Þ: ðA10Þ defined by We recall that σ a b∇ ij ¼ Pi Pj ðakbÞ: ðA5Þ 4 δ U − G S ðdÞ ¼ δ : ðA11Þ It satisfies the evolution equation [49], d V V0

L σ κσ σ kσ − μ μ a b Moreover, the projector is given by k ij ¼ ij þ i kj Pi Pj k k Caμbν: ðA6Þ μ ∂ ¯ μ a Pi ¼ iX : ðA12Þ Now let λ be a parametrization of Nk adapted to k , with λ 0 R λ ¼ corresponding to . In terms of , the evolution From this definition, one can check that equation can be written as z δz O d−1 k Pi ¼ i þ ðz Þ; ðA13Þ ∂λσij ¼ κσij þ σi σkj − Cλiλj: ðA7Þ A O d−1 Pi ¼ ðz Þ: ðA14Þ Consider now the new spacetime Ms generated by the left stretch. As in Sec. IV B, we denote quantities in Ms with tildes. We can then write the evolution equation Furthermore, in Ms, −1 ∇zkA; ∇Akz ∼ Oðz Þ; k ˜ ∇ ∼ O 1 ∂λ˜ σ˜ ij ¼ κ˜σ˜ ij þ σ˜ i σ˜ kj − Cλiλj: ðA8Þ AkB ð Þ; ∇ − − 2 U d−4 O d−5 a zkz ¼ dðd Þ ðdÞz þ ðz Þ; ðA15Þ Since k is tangent to Nk, and ðNkÞs ¼ Nk as submanifolds, a ˜a we can identify k with k . Thus, we can use the same where we have used that kA ∼ Oð1Þ. Hence, to leading λ σ parameter in both spacetimes. Since ij is intrinsic to Nk, order, we simply have we can identify σij and σ˜ ij for the same reason. Comparing σ − − 2 U d−4δzδz O d−5 Eqs. (A7) and (A8), and inserting Eq. (A3), we find that ij ¼ dðd Þ ðdÞz i j þ ðz Þ: ðA16Þ there is a Weyl shock, Finally, a straightforward but tedious calculation of the ˜ Cλiλj ¼ Cλiλj − sinhð2πsÞσijδðλÞ: ðA9Þ Weyl tensor yields ˜ − 8π − 2 ˜ We now show that the Weyl shock Eq. (A9) reproduces Cvi˜ vj˜ ¼ Cvivj Gðd ÞðhTv˜ v˜ i the near boundary shock Eq. (2.44) but not for wiggly cuts − d−4δzδzδ ˜ − O d−5 hTvviÞz i j ðv V0Þþ ðz Þ; ðA17Þ of the Rindler horizon. To do this, we evaluate both σij and Cλiλj in Fefferman-Graham coordinates to leading non- where we have used that λ → v; v˜ as z → 0 in M; Ms, trivial order. The Fefferman-Graham coordinates for M respectively. Putting this together yields the desired shock and Ms are defined exactly as in Sec. IV B, except we now for wiggly cuts of the Rindler horizon.

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