A Modular Sewing Kit for Entanglement Wedges

A modular sewing kit for entanglement wedges

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Citation Czech, Bartlomiej et al. "A modular sewing kit for entanglement wedges." Journal of High Energy Physics 2019, 11 (November 2019): 94

As Published http://dx.doi.org/10.1007/jhep11(2019)094

Publisher Springer Science and Business Media LLC

Version Final published version

Citable link https://hdl.handle.net/1721.1/128445

Detailed Terms https://creativecommons.org/licenses/by/4.0/ JHEP11(2019)094 Springer July 19, 2019 : d October 29, 2019 : November 15, 2019 : Received Accepted Published and for sufficiently smooth , /N and Lampros Lamprou Published for SISSA by c https://doi.org/10.1007/JHEP11(2019)094 [email protected] , Dongsheng Ge b ´ Ecole Normale ENS, Supérieure, UniversitéPSL, [email protected] , . 3 Jan de Boer, a 1903.04493 The Authors. c AdS-CFT Correspondence, Gauge-gravity correspondence

We relate the Riemann curvature of a holographic spacetime to an entangle- , [email protected] E-mail: [email protected] University of Amsterdam, PO BoxLaboratoire de 94485, Physique 1090GL, de Amsterdam, l’ TheCNRS, Netherlands Sorbonne Université,UniversitéParis-Diderot, Sorbonne Paris Cité,24 rue Lhomond,Center 75005 for Paris, Theoretical France Physics, MassachusettsCambridge, Institute MA of 02139-4307, Technology, U.S.A. Institute for Advanced Study, TsinghuaBeijing University, 100084, China Institute for Theoretical Physics and Delta Institute for Theoretical Physics, b c d a Open Access Article funded by SCOAP ArXiv ePrint: on entanglement is aand gravitation. promising new tool for connecting theKeywords: dynamics of entanglement The modular Berry connection encodesits the bulk relative dual, bases of restrictedresponding nearby to CFT entanglement subregions the wedges. while codeHRRT surfaces, subspace, At the relates leading modular the orderordinate Berry edge-mode in systems connection frames simply covering 1 of sews neighborhoods together the of the cor- HRRT orthonormal surfaces. co- This geometric perspective Abstract: ment property of the dual CFT state: the Berry curvature of its modular Hamiltonians. Bartlomiej Czech, A modular sewing kit for entanglement wedges JHEP11(2019)094 18 (1.1) ]. The cornerstone of 4 the area operator of the – 1 A and 10 ρ 15 9 log ] bulk mod 3 − 13 H = + 23 N mod A G 9 H 4 – 1 – ) to make precise how boundary entanglement 4 11 11 = 1.1 20 1 CFT mod ) of its dual CFT subregion [ 2 ] by sewing together entanglement wedges to produce its H 25 O ) ( 11 9 , A 3.8 3 10 ] within the code subspace [ 2 23 7 vacuum – 2 5 ] bounding the entanglement wedge. 9 , 8 2.4.1 CFT 2.4.2 Null deformations and modular inclusions and operator algebra ρ In this paper, we utilize relation ( 3.4 Example: pure AdS 3.1 Modular3.2 zero modes in Relative the edge-mode3.3 bulk frame as a Bulk connection modular curvature and parallel transport 2.2 Gauging the2.3 modular zero modes Comment on2.4 two-sided modular Hamiltonians Modular Berry holonomy examples 2.1 A toy example ‘builds’ the bulk spacetime [ global geometry. In ordinary differential geometry,gluing spacetime is small constructed by patches consistently of Minkowski space, the local tangent spaces of a base manifold. sides of the duality [ with the modular HamiltoniansHRRT defined surface as [ 1 Entanglement as aSubregion connection duality has taughtgeometry, quantum us state that and the dynamicsstate of physics local in quantum a fields, canthis bulk be important entanglement recovered insight wedge, from was the i.e. the equivalence its between the modular Hamiltonians on the two B Modular connection for CFT vacuum C Solution to equation ( 4 The proposal and implications A Berry connection 3 Entanglement wedge connection Contents 1 Entanglement as a connection 2 Modular Berry connection JHEP11(2019)094 : A i ). The , these relative Q encode A 2 3.14 ( − that they are ) 4 . Our proposal of the local coor- 3.12 2.1 with modular operator A gauge symmetry in the space ] which we define for arbitrary 13 . 3 relative embedding = 0 (1.2) . that relates the Lorentz frames of nearby ,A 3.3 ] of the extremal surface and, as we show in and the CFT connection of section mod 3 – 2 – 16 and ) we, therefore, propose in section , ,H A i 15 1.1 , bulk modular zero modes consist of large diffeo- 3.2 ]. In close analogy to the ordinary geometric connec- Q N 12 G spacetime connection . The relative modular frame is then encoded in the connection . of different modular Hamiltonians physical, establishing a map between 2.2 ]. All correlation functions within a CFT subregion 12 ). At leading order in , they consist of internal diffeomorphisms and local boost transformations on the ] or asymptotic symmetries [ is indexing the zero-mode subalgebra. For a physicist with access only to 14 1.1 3.1 i are invariant under the unitary evolution generated by modular zero-modes The gravitational connection of section The bulk meaning of this zero-mode ambiguity of CFT subregions follows from re- Our central idea is to treat entanglement as a quantum notion of connection between A mod The central idea underlyingfor this subsystems work of is a that quantum entanglement state plays [ the role of a connection this work. 2 Modular Berry connection 2.1 A toy example the relations of modularby Hamiltonians identical on sets of the rules. tworelated sides By by virtue of duality. of AdS/CFT ( This and providesthe are a Berry direct constructed curvature holographic for link the betweendiscussion the modular of bulk Hamiltonians a curvature of number and the of CFT conceptual state. and technical We conclude applications with of a the tools developed in spacetime, the bulk modular connectiondinate systems becomes about the the surfaces, whichcurvature is of a this central result connection inas our includes we paper the ( demonstrate bulk in Riemann detail tensor in section as one of its components, morphisms that do notmodes [ vanish at thesection HRRT surface.normal These 2-D plane. are While the thethe gravitational edge-mode bulk frame edge spacetime for every allows givencurvature us wedge of can to the be compare chosen HRRT at frames surface will, of is different small wedges. compared to When the the Riemann extrinsic curvature of the bulk of modular Hamiltonians on the relevant bundle —thestates modular in Berry section connection [ lation ( symmetries of her localframe. state On translate thezero-mode to frame other a hand, freedom ofthe entanglement algebras in choice localized of the in her globalis different to overall CFT subregions, zero-mode think state as of we renders this explain the zero in mode section ambiguity of subregions as a subsystems [ H where tangent spaces and endows spacetimehow holographic with spacetimes its are curvature. assembled bygeometric Adopting the connection, this set which of spirit, we entanglement propose we wedges is bystructure explain determined means of microscopically of by a the the dual entanglement the CFT bulk state. curvature in The a curvature way we of make this precise entanglement in connection section reflects Central to this task is the JHEP11(2019)094 . 1 in a (2.3) (2.4) (2.5) (2.1) (2.2) ]. B 11 are invari- and B as discussed ρ A A σ . This reflects the and B A U ρ : and can be thought of as the A B i B A . i i ij 1 U or j | ∀| − jl W ∗ ) , a global pure state can likewise ij A ∗ , . B W ), however, expectation values of . W 1 B . ( i AB . j 2 i j i X and i | B, lj can be interpreted as an open Wilson ψ B A | A ∗ A,kl U A = i U † A U ij i σ | kl σ i B i A | ∗ W ij ki σ i σ AB W h i † † A B ) are the Pauli operators that generate the W W A ψ U U | h h i † – 3 – A, ik = = ij h unitary frame of the subsystems. More precisely, X U = = A B x, y, z i i i = i ˜ B,ij | = i → A i B ˜ σ = B σ σ B AB ij h is simply a matter of convention. h ˜ σ i i i ( relative i ˜ ), the matrix W ) that the map between the two Hilbert spaces transforms ψ ⇒ B | → between the two Hilbert spaces 2.5 → | 2.3 A or ij i A,B i i A σ | W A i i A | σ ) and ( 2.4 a cyclic and separating vector for the algebra of operators acting on are a simple example of the mirror operators of B ), ( SU(2) and AB σ i 2.3 ψ ∈ | . B A ,U A defines a linear map are not invariant under independent unitary rotations ], with U Our main interest is in applying these observations to holographic duality; see figure It follows from definition ( Due to the entanglement of the two qubits in ( The simplest illustration of this idea involves a system of two qubits j B 17 AB σ i i A ψ line between the Hilbertof spaces entanglement. of From thegravitational a Wilson two line qubits, heuristic threading with ER=EPR the a viewpoint, quantum wormhole form connecting dictated theIf by qubits we divide [ the a holographic pattern CFT into two subregions under the action of a local SU(2) symmetry on each qubit as: By virtue of ( The operators ˜ in [ subsystem that can be usedspaces: to define an anti-linear map between the operators on the two Hilbert the local unitary frame for σ fact that the global state fixes| the Here algebra of observables for thea corresponding ‘local’ qubit. SU(2) Each symmetry; qubit in is, the therefore, absence endowed of with any external system of reference, the choice of The reduced density matrix ofant each under qubit is unitary maximally transformations mixed.symmetry on of Both the expectation respective values for Hilbert operators space, localized which in translates to a structure of entanglement defines thesubsystems. relation between the Hilbert space bases of different maximally entangled state: tion of General Relativity which relates the Lorentz frames of nearby tangent spaces, the JHEP11(2019)094 . A ρ . log 2.3 − = of a bipartite ij mod W . The focus of this H W via . ), i.e. as a Wilson line. A 2 2.5 that is localized in it, and A A ). We can think of this matrix as ): the global state 2.1 is well-defined in the continuum and 2.5 ¯ A )–( A 2.1 – 4 – that is analogous to ( , the matrix also transforms as in ( which encodes the reduced state in and its commutant ij s and comment on its two-sided version in subsection B A W mod mod A H H and in A of a CFT selects an algebra of operators A . A holographic representation of eqs. ( We emphasize that our discussion is inherently two-sided. In order to postpone a few The remainder of this section is devoted to formulating the quantum notion of connec- modular Hamiltonians of any rigorous construction should directly refer to these two-sidedminor operators. subtleties for conceptual clarity, however, wein choose terms to phrase of our single-sided initial presentation 2.2 Gauging the modularEvery subregion zero modes a modular Hamiltonian Strictly speaking, density matricesand become are meaningful not only well-defined in objects the in presence quantum of field a UV theory cutoff. In contrast, the sum of the tion when the subsystemsall of geometric interest problems are that involve subregionsis a of that connection, a the of correct fiber CFT, mathematical bundles.well in formalism as A arbitrary here a reminder states. description of of the As relevant the in concepts fiber from bundle differential at geometry, hand, as is given in figure bulk spacetime do theyOne probe? highlight We is will that answerof the these the holonomies dual questions of spacetime. in the the Weof entanglement interpret body connection subsystems this probe and of as the the the anbe pattern curvature indication paper. considered of that the on physical pattern equal Wilson of footing. line entanglement dressing in gauge theories ought to be represented as a matrix being prepared by achanges tensor of network bases that in fillsWhat a are spatial slice the of corresponding the Wilson bulk loops? spacetime. Are Under they non-trivial and what feature of the Figure 1 holographic CFT is prepared(orange). by The a division tensorpanels of network show the that two CFT general fills is examplespaper a illustrated of will spatial with ‘gauge be transformations’ a slice on of red of those ‘Wilson line gauge the line’ transformations, that bulk which cuts spacetime localize through on the HRRT bulk. surfaces. The JHEP11(2019)094 (2.7) (2.8) obeying and, as a result, A i Q selects the basis of mod H U . It is a gauge freedom, defines an automorphism i A Q , A ∈ A O ∀ U, = 0 (2.6) Hermitian operators ∆ ) † i preserve the form of ,A s i U ( Q Q mod = – 5 – : the transformation ,H OU A mod A i ) i H Q s ( † Q U → . The unitary flow generated by O of subregion , maps the algebra into itself while preserving the expectation values i s i ) is only determined up to a gauge transformation consisting of right Q i 2.8 P symmetry i − in ( e = modular zero modes U . The fiber bundle studied in this paper. The base comprises different modular Hamilto- Q that is a U A A useful way of describing the zero-mode ambiguity is by switching to a ‘Schrödinger A it can be decomposed as where a diagonal matrixeigenvectors. Transformations ∆ generated by encodesthe the basis spectrum and a unitary carry no information about itsirrelevant overall for zero all mode frame. measurementswhich This or spans local computations the ambiguity vertical restricted is, (fiber) of to directions course, of our bundle. picture.’ The modular Hamiltonian is a Hermitian operator on the CFT Hilbert space, so where of all of its elements in the given state. As a result, physical data localized in a subregion are called of Figure 2 nians and the fibers are modular zero mode frames. Modular zero modes as local symmetries. JHEP11(2019)094 ]. has ) + 25 λ (2.9) ( (2.10) mod H mod H . In particular, Consider now a λ , in which Q and their modular λ UU ∼ U ] and holographically [ 18 U, ˙ ∆ † ) is organized as U λ for a detailed comment. . ( Q ] + 2.3 mod UU H mod , which are well-defined operators on the full = λ -dependence of all operators for clarity. This 0 λ – 6 – U U, H † ˙ → U U .) The relations between these modular Hamiltonians = [ 3 -derivative of mod λ ˙ H may cause some discomfort to the careful reader, since the density ), the 2.8 ). (See figure 1 mod : λ ]. The trick is to think of the shape deformation as sourcing a stress- Q ( H the complement of region U 18 c mod λ H and we have suppressed the λ ∂ . A closed trajectory in the space of CFT regions. To avoid clutter and to clarify the ), with ≡ c λ ( In other words, there is an equivalence class of CFT bases defined by 1 mod the discussion can be entirelyH expressed in terms ofCFT the Hilbert full space, modular and operators refer them to subsection identical matrix elements. transformation of the region’s boundary, however,as this computation was is shown in in fact under [ tensor control, insertion at the subregion’sof boundary. the The calculation cutoff requires but aIn it delicate case treatment yields this sensible results comment both does in not the alleviate CFT the [ reader’s distress, we emphasize again that where ˙ shape-derivative of matrices of different subregions formally live in different Hilbert spaces. For an infinitesimal continuous family ofHamiltonians connected CFT subregionscan parametrized be by conveniently expressedchange in of terms spectrum of andusing two another decomposition families the ( precession of of operators: the one basis describing as the we vary multiplication by Modular Berry connection as the relative zero-mode frame. Figure 3 holographic application, here we display the family of corresponding RT surfaces in the bulk of AdS. JHEP11(2019)094 ) D onto (2.13) (2.14) (2.15) (2.11) (2.12) λ 0 P spacetime d relative zero ) the relative are the SU( i T 2.14 s ) λ , ( | . ) and ( , where 0 a i } i . mod Q ] 2.10 ,T )] and define the projector: that encodes the E, q iH I V λ i − { ih ( ij K 0 a δ Q , parametrized by a set of coordi- V e . mod s K Tr[ ) )] E, q ) = | λ ) and a commuting set of zero modes j λ 1 ( ) encodes the change in the spectrum D ) then equates the spectrum changing ( λ V Q | ( U, H | † a i i mod ˙ 2.10 mod X U 2.10 Q iH ˙ mod : H stands for the shape and location of the subregion’s E, q = [ H λ i i 0 ih ] = [ λ ds e mod a is a 1-form in P Q – 7 – ˙ ) Λ H Λ = − mod . One can, therefore, find an orthonormal basis of V | E, q ˙ Z i ij | H U [ δ ) only up to addition of zero modes. This ambiguous 0 a Q ] = 0 and ( ˙ λ 1 0 ∆ λ ( ,q 2Λ † ( a form a vector space and P i ] = ] = 0, it belongs to the local algebra of modular zero X U mod ), in turn, encodes the change of basis accompanying an U ). The latter can formally be constructed in terms of j X which necessarily have imaginary eigenvalues, leading to exponential − ) E,q respectively. For systems with finite-dimensional Hilbert D λ T →∞ λ ( lim i ,H mod a ( Λ ≡ i T κV † q 2.10 ] = ). ] mod ) = ˙ Q ≡ U mod ˙ V V H [ ] on both sides of eq. ( Tr [ H ] = [ U, H ,[ H 0 2.11 λ 1 } 0 ˙ V and D λ 0 P ∆ i [ ) in ( ,V P † λ P λ 0 Q E ( U P { mod ) fixes ) = U j H ) T λ | ( 2.15 i with dim( modular Berry connection † Consider the space of CFT subregions T ˙ U H here can be discrete or continuous. For subregions of quantum field theories in are simultaneous eigenstates of 2 i i a The and, since [ 3 . i E, q | λ mod An application of The second term on the right hand side of ( This formula should be taken with a grain of salt sinceThe there index can be Hermitian eigen-operators of the H , with eigenvalues 2 3 a modular Hamiltonian [ contributions to the integral ( dimensions, which is ourboundary. main focus in this paper, As is apparent, ( zero mode component is preciselymodular the Berry information connection: we seek and it leadsDefinition us 1. to introduce the nates infinitesimal shape variation ofbasis the operator region. is defined Combining as eq. the solution ( to equation: operator to the zero mode component of The operator modular zero modes where Q spaces, the zero-modeHilbert projector space takes anothergenerators, useful form form. ainner complete product Hermitian basis ( operators which on is a orthonormal with respect to the Frobenius or in a Hilbert space representation simply as: of modes. We can isolatethe this zero-mode spectrum sector changingmodular piece of flow by introducing a projector JHEP11(2019)094 ) ) i ). λ δλ ( δλ (2.19) (2.20) (2.16) (2.17) (2.18) 2.12 + . mod i λ U H ( ) or ( mod H . 2.11 i δλ . ) reduces to the ] δλ (1). The operator and U ) transforms as: † ) Q 2.16 ˙ i U U mod [ λ i 0 ( λ , we solve this transport 2.16 H λ 0 . ∂ given by ( P C † Q mod ) 2.4 ) ). This covariant derivative 0 U i H ] λ λ ( ( − (0) = 2.16 ). For an infinitesimal step , U mod Q i 0 mod U mod λ ,H δλ ( ) + H Γ ) H ] ). Therefore, we can compute the U λ † , Q U Granted a connection on a bundle, we ( δλ ) ) for the Berry connection unfamiliar † r U 2.9 ) to the modular Hamiltonians ( δλ + . U λ i V ) = = 0 ( λ 0 ), the connection ( 2.16 0 λ ∂ ˜ + Γ λ U ( [ Γ ( λ λ 2.9 0 ] = [ U ) from the left, we observe that the operator ˜ U ∂ )] = 0 0 P → An important comment is in order. In a typical – 8 – λ λ ( = ( mod Γ ) = † ˙ ) δλ 0 ) = H r ˜ i [ U ( λ V λ [ λ ⇒ ( 0 D λ 0 P U ) , δλ P i λ λ ] when applied to a family of pure states. − ( λ Q ) with 20 Γ( , U ) assigns a basis δλ D mod ) 0 we include a short illustration of how ( ˙ 19 λ λ H 2.19 . Any charged object, parallel transported along a closed loop ( ( ) + A U U 0 1], which forms a closed loop λ the modular Berry transport. In section , ( ) = [0 U λ ( ∈ ), which encodes the local choice of basis in every subregion, is charged ≈ 0 is equal to define ) U λ 2.8 ) ˜ U 1], with the initial conditon δλ that generates the parallel transport of the basis obeys the conditions: ), → , , λ 0 of infinitesimally separated modular Hamiltonians + ) ˜ ( [0 2.20 U λ is an appropriate representation of connection ( λ 0 -dependent gauge transformation ( parallel transport is the projector onto the zero-mode sector of ( δ itself or the zero-modes of any globally conserved charges — or they are phase ) λ δλ ∈ λ r ( mod λ † U ( 0 ˜ λ ˜ U P H U mod Parallel transport of Consider now a continuous 1-parameter family of modular Hamiltonians of a QFT For the readers who may find expression ( ) from eq. ( = H λ ( , returns to its starting point transformed by the holonomy of δλ problem in two tractable examples and compute theWhat modular are curvature. the modularCFT zero state, modes? the only symmetriesby of the modular Hamiltonianrotations of of a individual subregion modular are generated eigenstates. However, in anticipation of a connection to Equations ( away from Multiplying both sides ofV ( U under the zeromodular modes Berry with holonomy of transformation our rule closed loop ( by solvingfor the all transport problem for where Γ generates C state, Modular parallel transport andcan holonomies. define a covariant derivative or confusing, in appendix standard Berry connection [ and is given by: where Under a mode frame JHEP11(2019)094 are ) + i λ ( ), and (2.23) (2.21) (2.22) ψ | λ ( “sponta- ) are ob- mod i full H 2.6 has a much ψ | H ≡ full ) H λ ( full H . 4 on a circle. This was computed bulk . 2 H ]. In connecting our CFT discussion + . The fiber bundle associated to the 3 ] = 0 . 1 c N i − λ A G ρ ψ code 4 | ) to the subgroup that preserves the state: P ⊗ λ = . This is important because it is = ( λ λ i mod ρ – 9 – ψ full | H code i H ˜ Q P U code ] holds within a code subspace mod 7 ,P i H Q [ ) to work and explicitly compute the modular curvature in ) which is shared between the two-sided and single-sided code but preserve implies that there is a globally consistent gauge in which the P c 2.16 . These generate unitary transformations on the entire Hilbert λ i 2.23 full ρ ˜ Q H , λ ρ the complement of region s. c λ vacuum mod ) proposed by JLMS [ 2 H ), that generate a well-defined unitary flow in continuum quantum field theories. 1.1 λ ( ), with c λ mod ( mod Nevertheless, the modular Berry holonomies associated to a given global state First note that the zero-modes of the single-sided modular Hamiltonians ( H H mod 2.4 Modular Berry2.4.1 holonomy examples CFT We now put our definitiona ( tractable, illustrative example: the vacuum of a CFT subgroup of zero-modesmodular ( operators.zero-mode The directions vanishing of of therelevant projection Berry of curvature the connection componentsthe vanishes one along everywhere presented and the in the the extra computation previous reduces section. to identical for the twoneously problems. breaks” the The symmetry reason group is of that the Hilbert space vector As a result, parallel transport will only generate holonomies valued in the much smaller plementary subregions. Intuitively,the they are density matrices transformations thatfull are modular Hamiltonians allowed has, tosingle-sided therefore, change a much larger gauge group than the one for the not viously a subset oflarger zero-modes set of of zero-modes thespace full that modular do operator. not necessarily However, factorize to products of unitary operators on the two com- 2.3 Comment onHaving two-sided modular concluded Hamiltonians the presentationmodular Hamiltonians, of we our wish to CFTin illustrate formalism terms that in the of construction theH the can language be full phrased of modular directly subregion Hamiltonians of CFT bipartitions, ones, constructed by the requirement thatof they commute with the code subspace projection We discuss the importance of this point in more detail in section operators ( as articulated in theto error the correction bulk framework we of are, [ therefore, not interested in exact zero-modes but only in approximate holography, it is important to recall that the equivalence of the bulk and boundary modular JHEP11(2019)094 = ]. µ R that 23 [ x (2.24) (2.26) . + and has δa B ) and the V R ) and x B.4 − , a and + decomposes to a a 0 L 2 x L = ( . More details of the . Their coefficients are µ L + 1 ] = 2 x , 1 0 , K . − 1 + 2 2 − a / / ,L ∂ ¯ L ) ) 1 ) + − ] (2.25) L [ a a πi + 2 and + half-sided modular inclusions − − − / , 1 K ,K K , + − − 0 + b b 1 , ( ( 1 K − K 2 2 − L ), we need to find an operator + + L a + sin sin ∂ + [ , b ] = K 1 πi πi + 1 1 πi 1 − 2 2 – 10 – = 2 da − − ,L . = 0 2) symmetry algebra of a CFT + , mod L B + [ ] = ] = + H a + − K ( + + a , δb , δb ∂ 1 K − + L δa δa − [ [ ]. R R 1) subalgebras, which act on left-moving and right-moving null ) and , 13 ] = + 1 , respectively. The commutation relations are ). Using the explicit form of the modular Hamiltonian ( − , b ,L x are linear combinations of + . 0 i a L − ( ¯ 2.20 [ L ] by exploiting the geometry of the space of CFT intervals, or kinematic + K and 13 K + ]. This subsection establishes the consistency of the general rules proposed x and vacuum. The origin of the simplification in this case is not conformal symmetry 22 , d + 21 ) is the generator of the boost transformation that preserves K − The modular Berry curvature can now be computed straightforwardly: In order to compute modular Berry holonomies, we need to solve the parallel transport The modular Hamiltonian of the interval with endpoints at The (two-sided) vacuum modular Hamiltonian of an interval is an element of the , b + b 2.4.2 Null deformationsThe and modular solution inclusions toexample: the families modular of modular Berryin Hamiltonians transport a for CFT subregions becomes with tractable nullbut, separated more in boundaries, interestingly, another an algebraic interesting QFT theorem for This exercise can alsonians be of applied ball-shaped to regions the in computation the of vacuum holonomies of for higher modular dimensional Hamilto- CFTs. calculation, as well asCFT intervals, parallel are transport given along in appendix more general trajectories in the space of conformal algebra, we find that so in this case parallel transport is generated by (1 functions of the endpoint coordinates of the interval; weproblem derive for them the in basis of appendix Hamiltonians the modular Hamiltonian. For example,solves given equations two ( nearby modular ( the form where coordinates and similarly for space [ here with the results of [ conformal algebra. Thepair global of commuting SO(2 SO(2 previously in [ JHEP11(2019)094 – if 5 mod (3.2) (3.1) H (2.27) (2.28) (2.29) (2.30) ]. The i dy 24 , ⊗ 4 α ]. 3 dx ) directly implies: ) 2 x modular inclusion ( 2.28 O is, therefore, an eigen- + ) modular times: ). The parallel transport x β ( mod . x δu 0 δH ) 2.20 y ( . β . the modular Hamiltonian of the s > | positive ) ) ∀ iα 1 x mod ( σ H 2 bulk 2 mod bulk mod ) δu δH − H j x H are said to form a mod ( 2 + ⊂ A dy πi 1 + H β δu ) δH ( s ⊗ N that satisfies ( ( dx i = 2 πi 1 A πi 1 G , ⊂ A 2 4 ⊗ πi dy – 11 – 2 α = mod ) = A ,H ] = 2 into itself for all 2 U 1 dx ) is that, for all holographic states of interest, ) x 2 δu 2 ( x V ) mod A ( mod A ,H and 3.1 2 O ) 2 H s x δu 1 ( δH ( + H [ 1 , A O α maps x + † ) ) their algebras are indeed included and ( mod 1 , y γ U ( e.s reduces to its vacuum expression in the same neighborhood, im- x α x | ) mod ( y ij µ ( H k bulk mod u γ | H + αβ ) + vacuum, when two subregions are related by an infinitesimal null defor- y w a geometric boost generator at the edge of the entanglement wedge. µ e.s. ( d with non-zero eigenvalue 2 x ij + γ mod → H αβ η H + is the HRRT surface area operator and µ e.s. x = A that asymptote to a homogenous boost near the RT surface. Moreover, for finite g To make our discussion concrete, we partially fix the gauge to be orthonormal to the An important consequence of ( In the CFT Two operator subalgebras M ]. The modular Hamiltonian of a boundary subregion is holographically mapped to: mod HRRT surface area operator in Einstein gravityζ is identified with the Noetherenergy charge bulk for states, diffeomorphisms plementing the above boost transformation onnon-local, the matter fields. This renders the, generally bulk QFT state inwithin the the associated subspace entanglement ofexcitations wedge. the about a CFT This given Hilbert operator spacetime space equivalence background, called holds that the corresponds codeadmits to subspace a effective [ geometric field description in theory a small neighborhood of the HRRT surface [ The link between our CFT7 discussion and the bulk gravity theory is the JLMS relation [ where 3 Entanglement wedge connection 3.1 Modular zero modes in the bulk The null functional derivativeoperator of of the modular Hamiltonian operator for null deformations is then: mation modular evolution by The half-sided modular inclusionincluded theorem algebras then satisfy the states commutator: that the modular Hamiltonians of JHEP11(2019)094 = 0. (3.4) (3.5) (3.6) (3.3) normal bulk mod x ,H i a class of zero Q mod 1 where H ) are non-trivial. In a . |− 3.4 . ij ) β ) is satisfied. The modular K = 0 |− ζ − ij α 3.5 M ∇ K ζ , x − + N αβ | ] ∂ ij β , x γ 30 x + K | N mod √ + ij ζ αβ x 6= 0 and ( K . RT − π + Z 2 0 x ( N 0 0 M mod ∼ ∼ ζ O 1 – 12 – α α πG Noether ζ N x x + −−−→ −−−→ 4 ∂ Q N − M ζ ∂ α i mod mod ) = ζ ζ i = y ( As is the case for the modular Hamiltonian itself, the zero M ] M ζ Noether ζ mod = = 0. The boost generated by the CFT modular Hamiltonian in Q α Q ζ, ζ [ x ]. Beyond this regime, modular flow gets modified by generically is some choice of coordinates along the minimal surface directions 4 . ), which translates to the condition 3.3 ). We will discuss how the corrections restrict the regime of validity of our 3.3 1 parametrize distances along two orthogonal transverse directions, with , . . . d , ) reads: 3.3 = 2 3.2 = 0 , i i , α y α x The symmetry group selected by the above requirements consists of diffeomorphisms An entanglement wedge has two boundaries: the standard asymptotic boundary used and constitute bulk zero-modesboost when is, of course,surface one in of Planck them, units. with a Noether charge equalalong to the minimal the surface area directions of and location-dependent the boosts extremal in its normal plane. In and they are nottrivially relevant on for the us HRRT here. surface have On Noether the charge other [ hand, diffeomorphisms that act non- spacetime with no boundary,a all result spacetime of transformations the have constraintphisms equations vanishing that of generators, act gravity. as When non-trivially boundaries on exist, them however, are diffeomor- endowed withto non-vanishing define Noether CFT charges. correlatorsmorphisms and that the do boundary not selected by vanish the asymptotically HRRT give surface. rise to Large diffeo- the boundary conformal group modular boost ( Moreover, we demand that the diffeomorphisms generated by ( Near the extremal surface,modes however, will due reduce to to the generators of geometric spacetime action transformations: of These need to preserve the location and area of the HRRT surface and commute with the proximation ( results in section Bulk modular zero modes. modes will generally be non-local operators in the bulk wedge, defined by This approximation for the modular flownormal is extrinsic valid within curvature a of neighborhoodlightlike the with size coordinates HRRT set surface [ by the non-local contributions. Our entire discussion in this section assumes the validity of ap- and the extremal surface at the gauge ( where JHEP11(2019)094 ) λ ; x (3.7) ( M mod ζ is the bulk 0 ). As in CFT, ] and the ana- ) is simply an P δλ 14 3.7 )] (3.8) + λ )). The vector field ; λ x ; ( x 3.14 ( M mod ζ M mod λ ζ δ ). It follows that the class of [ ) generate symmetries of the λ 0 3.5 P 3.7 ) and + x λ β · ; , whose HRRT surfaces are infinites- M x x ( )] δλ λ αβ ; + ) + 0 M mod ) x It is instructive to proceed in parallel with λ y ζ ( y ( ( i 0 -variation of the modular Hamiltonian in the ω ζ mod λ and – 13 – , ζ . 0 0 λ ) ] where our RT surface replaces their black hole ) in our CFT discussion. ∼ ∼ α α x x 16 3.2 −−−→ −−−→ . The λ, δλ 2.15 i = 0, as demanded by ( α ( ; below zooms in on a small fragment of four extremal ζ ζ x ( 2.2 4 U =0 ξ † α ˙ x U

i ζ ) = [ α λ ) are the gravitational edge modes discussed in [ ∂ ; x ( 3.7 M mod ), which map one extremal surface to the other, allowing us to relate the ζ λ x = 0 in a given wedge and will be treated as local gauge transformations map the HRRT surface to itself while preserving its normal frame, a fact ( δ i is the difference between the two modular generators and α ∂ M x i ξ ζ + ) the zero-modes read: M mod th component, ) is a diffeomorphism rotating the basis of the modular Hamiltonian, which in ζ M λ − 3.2 δ x i λ, δλ The key idea now is that the geometry of the global spacetime enables us to compare Transformations ( → ; x M ( where projector onto zero modes discussedξ in more detail below (see eq. ( Mapping the modular boostour generators. CFT construction ofbulk section becomes the difference ofthis the can vector generally fields be organized into two contributions as follows the two zero modex frames. What makes thiscoordinate possible systems in is their the neighborhoods.of existence the Bulk of relative diffeomorphisms, diffeomorphisms basis therefore, operator play the role imally separated from eachgenerating other. the corresponding Each modular wedgecomes flow is near with equipped its its with HRRT own a surface.internal arbitrary vector coordinate Moreover, field each choice system wedge of onits the zero-mode transverse extremal frame, 2D surface which plane. andsurfaces given a and ( Figure hyperbolic displays their angle zero-mode coordinate frames. on discussion on them. 3.2 Relative edge-mode frameConsider now as two a entanglement connection wedges, horizon. As inphysics near our CFT discussion,on the the space vector ofzero-modes fields entanglement as wedges. ( well, We e.g. shouldof edge-modes note the of modular that, bulk Berry in gaugeare connection. general, fields, universally there that The present exist generate gravitational in extra other edge-modes holographic components discussed theories here, and for however, this reason we choose to focus our of the zero modes that will play a crucial role in section logue of the horizon symmetries of [ where in the second line we chose to explicitly emphasize the vanishing transverse derivative gauge ( JHEP11(2019)094 ) ) is 3.8 (3.10) (3.11) 1 with , ) of the λ, δλ ; x ) fixes the = 0 ( ) in normal ξ λ, δλ a 3.9 ; i 3.3 y . The undeter- , which ensures ( = 0 (3.9) ij α M ; ) is precisely our α a β y ) is the direct bulk δx ( ). K ]. Eq. ( a δn δx ω in an arbitrary gauge ), where 3.8 = ij x under a local Lorentz 2 ij 25 ξ γ α 3.10 β b δn M ( = K , δx a ) δx ij O α ; y ( δn α M ab K among the family of Lorentz i 0 ) + ζ K y . M ( + Γ M − i α δ α M β β M n a ) then leads to the following solution up to additive contributions by zero- + a . ) δx b ). A detailed derivation of y j ξ δn ) ( x δn µ ∂B is determined simply by the deformation b ω + 3.2 αβ λ, δλ generates boosts on the normal 2-D plane a α ( δx ; i + − M a x δx R δ ( a α M a . The arbitrariness in ξ mod ij δ β ξ ζ ) δx γ – 14 – y ) = δn ( + M a δ ω δλ β of the two entanglement wedges. In order to define → − + + δx α j λ ) therefore needs to also have the form ( β ; . Crucially, ∇ ) = The ambiguous edge-mode part in the solution of ( y i δλ λ δn ). ( ∇ + M ) are arbitrary functions of the minimal surface coordinates ij a λ, δλ y λ 2.15 γ ( ( ; n ). A formal but explicit solution to the general problem can be i 0 . The expression for the diffeomorphism . To express this requirement, we introduce a pair of normal αβ M η , ζ is also extremal, as discussed in detail in [ x mod C δλ 3.7 ) ( ζ − y α relative to is the normal extrinsic curvature and ( + M ), in turn, expresses our right to pick the coordinate system on the = ξ ω δx λ y ij δλ can, therefore, be absorbed into the definition of α ( g ) determines the diffeomorphism i 0 α + K ζ ω L at will. 3.8 λ Imposing both conditions on δx δ 4 = relative zero-mode frame in terms of its boundary condition . δλ are the Christoffel symbols in gauge ( ab α ij ): ; + η α δx M NK λ 3.8 = K of the boundary subregion the surface is anchored at. This follows from equation b The quantities Second, we recall that the vector field Equation ( n There is of course no unique choice. There is a continuous family of normal vectors related by local · 4 µ ∂B a Bulk modular connection. encodes the the bulk modular connection we, therefore, needLorentz transformations, to which perform will be a important later zero-mode on. projection of The zero-mode mined function surface representing the edge-modefreedom ambiguity in in selectingequivalent a pairs, pair as of canboost orthonormal on be the vectors seen surface’s transverse by plane: the transformation of where Γ given given in appendix can, of course, be obtained simply by a change of coordinates in ( vectors on the new surface n of eq. ( the new surface at form of of the HRRT surface. coordinates about HRRT surface δx where mode projection describes theanalogue change of in CFT the equation spectrum. ( Condition ( mode transformations ( obtained as follows. First, we introduce the transverse location the geometric regime is simply the local coordinate system. On the other hand, the zero- JHEP11(2019)094 . − . δλ α δλ 4 ∂ with β + (3.13) (3.14) (3.15) (3.16) (3.12) x λ ], as any β λ modular α ] mod ξ and ξ, ζ Ω[ 2-dimensional − λ − defines ] = d ξ [ K )] P β 2 ) of the modular flow that n ξ δλ , γ 3.3 i Γ( Z δx , L ) 1 M γK ) Γ Ω i 1 δλ ) is, by definition, the spin con- L Z . δλ L αM RT i ( ) ]

i n 1 Z ξ [ λ, δλ Z i L + 2 ] δλ N ) + [Γ( Z ξ 1 ξ [ δλ M i Ω( preserves the normal frame, as explained + 2 δλ ∆ β βN Z i Ω n n δλ Z Γ( − + the approximation ( L 2 L δλ ] ∆ ) ξ 2 δλ M ∆ ∂ . − δ ∂λ ∂ – 15 – δλ ) ( 2) are the corresponding tangents. We also in- 2 − M RT i δλ αM when

). We illustrate the modular curvature in figure ) n α Z δλ − 2 ) = Ω[ N = 1 n of the nearby minimal surfaces before comparing the ξ δλ Ω( δλ 3.15 δλ i N αβ αβ 1 t ∆ Γ( λ, δλ ∆ ), the bulk modular curvature follows from the standard δλ 1 2 1 2 1 − , . . . , d ∆ Γ( δλ + = = = 5 δ 3.12 ) ) = = = 1 ), and ). 2 i 2) are two unit normal vectors on the extremal surface , mapping between the two coordinate systems at ( δλ , , is itself a zero-mode and it annihilates the vector Lie bracket [ 3.7 1 λ, δλ λ, δλ M 3.15 P ( ( ) for the boost component of Γ( δλ ξ iN ξ ξ t = R = 1 i Ω P 3.13 Z ), and, therefore, provides a canonical map between normal vectors at α ◦ are the Lie derivatives generating the corresponding asymptotic symmetries for the ‘internal’ covariant derivative associated to the and i P 3.7 )( Z δλ λ is given by expression ( αβ ( L η , M δλ Ω = α L satisfies n β align the internal coordinates i n Expression ( ∂ ] It is straightforward to confirm that the definition of the zero-mode projector · ξ 5 [ i α where ∆ Z consistent projector should. definition. It reads: computes the bulk Riemannis curvature justified. We explain this proviso in more3.3 detail in the Bulk next modular subsection. Equipped curvature and with parallel connection transport ( covariant derivative ( nection for the normal frame ofis the to HRRT surface. Thenormal role frames of at the the covariant derivative ‘same ∆ location’. Thus, the curvature of our modular Berry connection diffeomorphism subgroup It is very important herearound that the eq. zero-mode ( different locations on the same HRRT surface, allowing the construction of the internal Here n troduced ∆ of the HRRT surface ( In covariant form, theconnection zero-mode in the component bulk of reads: the vector field where the diffeomorphism JHEP11(2019)094 ) α | α ij : δx K Ω α (3.18) | L ij K ( O ) of the mod- ) where |− 3.3 ij K − , x + | ij ) is required by covariance ) (3.17) . K 1 ) 1 + 3.18 δλ x ( ( δλ i ). Since the modular curvature ) (which transform under orthogonal O Z in ( Ω( 2 2 ) for the modular Berry connection is δλ δλ λ, y , ∂K δλ ( 2 ∆ ∆ M α K 3.13 n ( − − ) O ) 2 2 ) while the mismatch between their directions is ) directly probes the bulk Riemann curvature. δλ δλ ( i – 16 – 3.17 Ω( Z 3.18 1 1 ). δλ δλ Expression ( 3.18 = ∆ = ∆ 2 2 δλ δλ 1 ) 1 Z (Ω) δλ ( δλ R R ) can be decomposed into two contributions: the curvature of the 3.16 component of the curvature ( Ω . Modular Berry curvature in the bulk. The modular zero mode frames are marked L ), which affect its curvature at orders At this point it is important, however, to recall that in approximation ( The curvature ( 3.13 ular flow as a boost generatoris we neglected the terms normal of order extrinsic curvaturebutions of to the the HRRT surface. bulkto These, modular ( generally Hamiltonian non-local, can contri- generate corrections of order The appearance of the internalbecause covariant derivative the ∆ orthogonal boosts are non-trivially fiberedRelation over to the the surface bulk diffeomorphisms. curvature. the spin connection for the normalthat frame the of the extremal surfaces. This immediately implies and the curvature of the abelian subgroup of local transverse boosts generated by the boost component of the curvature ( non-abelian group of surface diffeomorphisms boosts); the distances between neighboringWe pairs parallel reflect the transport extremal a surfacedifferent diffeomorphism zero paths frame. (red mode and frameture. blue); from the The the mismatch mismatch bottomdiffeomorphism between between surface component the the of to resulting locations the the frames curvature of top is ( the the surface red modular along and curva- two blue arrows on the top is the surface Figure 4 with pairs of arrows that stand for the normal vectors JHEP11(2019)094 . 0 M λ x ) we (3.22) (3.20) (3.19) λ = ( 1] that , γ 0 ) for the M λ [0 x ∈ 3.14 . λ =0 ). The modular ), ( α ˜ x

3.3 , parallel transport i ) from the extremal 3.13 ξ M λ ) with x M i 1) measures distances λ /K δ , ( (1 γ − M λ O = 0 ˜ x λβ ˜ x α ( i λ . αβ ˜ y ∂ α λ ) ∂ x ) M α δ ) (3.21) ) is an approximation to the modular expansion. λ, dλ ). =0 ; ), given an initial condition ˜ ) at every step is the bulk Riemann cur- α λ, δλ x 3.14 ˜ ( ), we can illustrate the modular parallel x λ λ, δλ i (

( 3.3 3.19 γ Z M ), ( ), where , ∂K . δ ξ 3.19 i λ 2 + ξ + Γ( γ + , y K α λ 3.13 ∂ /R, ∂K/R δ α λ – 17 – ˜ x ∂ δ δλ 2 M λ x ∂ γ x K = λβ R 1 2 = = ( λ ˜ x ∇ δλ αβ Assuming ( M λ + x ) is under control only when there is a hierarchy between ) M λ ) + is of course subject to the zero mode ambiguity, which is ), we can define a covariant derivative x to every surface (1). In a neighborhood of every minimal surface ξ 3.10 γ M λ M λ λ, δλ , the parallel transported frame becomes: λ, δλ 3.12 ˜ x x ; λ δλ Ω( x (˜ ∇ ) and in the third step we used the formulas ( M (0) = + δλ ξ 3.7 γ ) can be intuitively understood as follows: the geometric approximation + + δλ + ): M λ M λ M λ 3.19 x x x 3.10 = ˜ = = ˜ δλ ) along two orthogonal directions. These are simply local choices for the edge- + λ M λ ( ˜ x γ Given the connection ( Condition ( In the secondorthonormal step gauge we ( used the explicit form of the zero-mode generators in the local assigns a canonical frameFor ˜ an infinitesimal step which generates parallel transport. Applied to the coordinate frames morphisms ( The ‘gluing’ diffeomorphism the focus of this paper. transport geometrically. Considerform a family a of closed minimalcan loop surfaces define a coordinatefrom system mode frames of thesection, corresponding these entanglement different wedges. localized As coordinate we patches are explained related in to the each previous other by the diffeo- surface is comparable to the correctionsBerry neglected curvature, in therefore, the reliably approximation measures ( RT the surface bulk when curvature in the the surfaces neighborhood considered of obey an ( Modular parallel transport. Berry connection at leading order in a to the bulk modular flowsurface. confines If us this within distance aeffectively is distance flat, also of small and order compared the to boost the of Riemann the curvature, normal spacetime frame looks resulting from parallel transporting the vature, our computation of ( the bulk Riemann curvature and the normal extrinsic curvature of the HRRT surface The curvature of the geometric connection ( computed employing by approximation na¨ıvely ( JHEP11(2019)094 . 3 (3.26) (3.24) . ) reveals 3 3.22 , which shows ) is a solution 4 M ) to ( . In doing so, we B.12 i ) δλ λ 3.14 ; + x ( λ mod , ζ ). ) 1). , ) (3.23) 2.20 λ, δλ to interval ) hold for the corresponding Killing ; SO(2 λ, δλ x λ ( ; . ˜ ξ × x we identify the operator that generates B.13 h )] = 0 (3.27) (˜ λ B 1) M ; = 0= 0 (3.25) , in the boundary discussion — it too lives ˜ ξ x that are orthogonal to the 0-eigenspace. In )] = ( + RT RT δλ λ

) and ( ; V ˜ ξ – 18 – M λ M mod x · mod ˜ ζ ( ξ x ζ i λ · t δ B.12 = ˜ [ β M mod λ 0 n ζ δλ . An application of the projector ( ), ( λ P i + ) generating parallel transport of the edge-mode frame δ [ ∂ M λ λ · ˜ 0 B.9 x ,Z 1) is also the algebra of the Killing vector fields of AdS P α , λ, δλ 3 n )–( ( − ) αβ M ˜ B.8 λ ξ SO(2 ; 2 1 x × ( 1) , M mod ζ λ ). We also know it has no zero mode component to be projected out δ 3.8 preserves the hyperbolic angles on the normal 2-D plane. maps between the two modular boost generatorsis (up always to orthogonal zero to modes), the extremal surface, But this SO(2 Following these rules we can transport the surface around a closed loop in the space We can covariantly express these conditions as follows: The modular parallel transport in the bulk can be summarized as a geometric flow, 3. 1. 2. of equation ( because — asin was eigenspaces the of case the for adjoint operator action of modular parallel transport fromonly boundary exploit interval the global conformal algebra SO(2 In particular, equations ( vector fields. As a consequence, the Killing vector field that represents ( But the picture is the same for larger3.4 loops, for example Example: the loop pureThis shown AdS in subsection figure mirrors thevacuum discussion of of a two-dimensional the CFT. boundary In modular appendix Berry connection in the of extremal surfaces, returningoriginal and to transported its coordinate originalboost frames location transformation in on in its its vicinity the normalThis will plane end. is and reveal the a a bulk diffeomorphism A modular location-dependent of comparison Berrythe the holonomy. computation internal of of We coordinates. saw the the an modular example curvature of — it that in is, figure the holonomy of an infinitesimal loop. They are direct bulk analogues of the CFT conditions ( which at every step: that the diffeomorphism indeed has vanishing zero mode components. components of the connection Ω JHEP11(2019)094 . t ∂ (3.30) (3.31) (3.32) (3.29) (3.28) ) to a ) . − b 2) θ 3.28 ∂ − , π/ , − isometry. The ) 2 ) a . ) on the bound- − 3 − b − π/ − . ∂ dλ K and intersect, bulk − + ) + − , 3 b 1 B.12 b 2 − ∂ dλ a + − − + − + , π/ dλ 2 + a 2 x − ( + b d K π/ generates the bulk modular sin is a global rotation about its − + . − a 3 3 ∂ δλ ) + dλ, π/ − + t V a + dλ + ( db ∂ − ) = ( d 1 2 2 + + R θ 1 ), the action of ( t K − π/ + ) + − − b − t − . : sin( B.3 R ∂ ∂ + 3 subspace of pure AdS θ ) − x ∂ , θ − 2 + . Mapping the special interval ( dλ, θ L a H t + + 1 sin δλ = 2 − − K 2 + − dλ sin( – 19 – + + L + − x δλ ) a b dλ V λ , θ ( db − ∂ d a 1 2 R sin − t to ( dλ, π/ − − + − dλ ]. to: λ ) + − πi 1 + R t dλ b λ da 13 2 2 + ∂ ( . 1 2 , θ + d 3 L π/ θ t + − − ) = + 1 + ) t + L sin( = ( θ x ) ( + B.12 + θ δλ a ≡ 1) algebra element that generates modular parallel transport, as an sin , ) + − + − dλ sin( . This rule for bulk modular parallel transport, dubbed ‘rotation without λ + -coordinates on the boundary, this becomes: ) dλ 2 , b b da t , we recognize the following rule of parallel transport: ( + SO(2 − 1 2 H λ d a equation ( × , a − − and + dλ 1) + θ If two geodesics live on a common , , b b 1 2 ( + d a Killing vector field. In the representation ( + − One option is to move from To understand bulk modular parallel transport geometrically, consider an initial HRRT 3 = ( 2 1 λ general initial Case 1. modular parallel transport is atheir rigid common rotation about theirslipping,’ intersection point was which first preserves explained in [ It is easy tocenter, see which that maps the the corresponding geodesic symmetry of AdS Let us survey what this solution means in the bulk of AdS In this case, parallel transport is carried out by this global conformal symmetry: Going to the SO(2 AdS ary is: The analysis for othertask initial is geodesics to interpret is identical up to an overall AdS parallel transport in pure AdS surface that is a diagonal of a static slice of AdS summary, the Killing vector field that corresponds to JHEP11(2019)094 , , ) ]. 29 and – /dλ 3.28 3 + (3.33) (3.34) 27 db , /dλ + gauge symmetry da . When we close . ) δλ ): in the language of dλ + . The characterization of λ − 2.26 δλ 2 subspace of pure AdS ]—is directly associated to + 2 12 λ and the longitudinal translation dλ, π/ − and − K ]. λ 2 to geodesic + 26 π/ λ + − . K t ∂ dλ, by a zero-mode transformation. The relative = 2 2 + – 20 – mod δλ . Once again, mapping the special interval ( V live on a common AdS H 3 to: dλ, π/ δλ isometry that maps the initial geodesic back to itself. λ + 3 2 + λ π/ − and λ = ( ] to the set of modular Hamiltonians. δλ 20 , we recognize the following rule of parallel transport: , . Altogether, these four basic cases span the four dimensions of kine- . This time translation also preserves that timelike geodesic in AdS, + ]. The most general case is of course a linear combination of the four. λ 2 19 λ /dλ 21 , − 13 db . , the orthogonal boost is generated by − Modular Berry holonomies are a promising quantity in this regard. One way in K If two geodesics 6 and − 2.4.1 isometry, which at each step maps geodesic + Along any trajectory in the space of geodesics, parallel transport is generated by an There are two other basic cases, which depend on the relative signs of Another case is to move from 3 /dλ Although there exist classification schemes that are customized to specific systems like qubits [ K 6 − multi-partite entanglement is ament, famously which unsolved is problem. entirely Unlikenot characterized two-partite known by entangle- what the quantities spectrum areglement. of sufficient the to modular classify Hamiltonians, different forms it of is multi-partite entan- zero-mode frame is thenThis promoted is to a a notion gauge ofthe connection curvature, with entanglement which a pattern — non-vanishing as of curvature. Wilczek first the and recognized state. Zee in [ [ It can beModular studied Berry by holonomies applying as the an ideas entanglement measure. of Berry, In this paper, webetween modular proposed Hamiltonians a of the linkary dual between is CFT that the state. the curvature Our setconsisting of key of of observation spacetime subregion on rotating modular the and the bound- Hamiltonians basis the is of relations endowed each with a geodesic. Of course, we havesection reached the same conclusion inby eqs. ( 4 The proposal and implications Its detailed geometric meaning will be discussed in [ AdS a loop, we generateSuch a isometries are finite spanned AdS by the orthogonal boost and rigid translation along the said their common AdS which connects the points of closest approach between da matic space [ This is a rigidto time a translation general in initial AdS Case 2. do not intersect, bulk modular parallel transport is a global time translation that preserves In this case, parallel transport is carried out by this global conformal symmetry: JHEP11(2019)094 local, . code is to introduce P exact code mod P H code P = ], while the algebra of modular ] clarified that the equivalence 3 mod ]. Our feeling is that there is an ) is actually the restriction of the 38 ], the construction of the physical , H 15 39 37 , 2.15 24 In the bulk, modular flow admits a simple ) holds within the code subspace, namely 1.1 ) and ( 2.6 – 21 – Our proposed holographic relation between mod- ) the modular Berry connection reduces to a geometric There is an aspect of our story that played a supporting ). The essential task of the projector 3.19 3.7 appearing in equations ( mod bulk requires a set of zero-modes that generate the asymptotic symmetry ). The latter is the only bridge between our CFT and bulk discussions. The H 1.1 itself and the conserved global charges of the CFT, if any, or they are simple phase The code subspace projection is more than just a technicality; it is directly responsible mod On the role ofrole soft in our modes. mainour presentation but knowledge, we use believe of deservestheir embedding more gravitational spacetime. attention. edge Edge This modes modestheir is of have relation the been to subregions subject new, soft to to to theorems a probe and lot the the of memory recent curvature effect studies of [ due to group of the HRRT surfacethe ( correct group of approximate zero-modes.independent way In the for absence identifying ofthe any the modular currently appropriate zero-mode known, algebra bulk- code and subspacesas the corresponding a in modular useful the Berry guiding holonomies boundary principle. can theory, serve for endowing the boundarytypical modular CFT Hamiltonian state, with the theH right symmetries zero-mode of algebra. therotations In modular of a Hamiltonian individual are modularsemiclassical either eigenstates. generated On by the other hand, the existence of a the subspace ofspecific the background. CFT Hilbert Itcontext space is the describing therefore bulk implicitexact low in CFT energy our modular excitations construction Hamiltonian to about that the in a code the subspace holographic Error correction and bulk locality. ular Berry curvature andrelation bulk ( spacetime curvatureerror hinges correction on framework for theof the validity bulk AdS/CFT of and dictionary boundary the [ modular JLMS Hamiltonians ( embedding of the internalof coordinates. the bulk Itsappeared Riemann curvature in curvature. the is, discussion therefore, of AHamiltonians holographic a was somewhat complexity [ used holographic different for probe CFToverarching bulk framework Berry reconstruction connecting in connection these [ results recently to the ideas we presented here. geometric description sufficiently closeus to to the translate corresponding the HRRTderive surface. CFT a rules This bulk of allowed avatar modular ofsurfaces the that parallel satisfy modular transport condition Berry to ( connection.connection entanglement wedges encoding Our and main the result spin is that connection for for HRRT the normal surface frame and the relative and study how theis resulting a bipartite description entanglement of variesholographic the as applications modular the of Berry-Wilson grouping the loop. evolves.uses modular for Because This Berry classifying the connection, entanglement focus to we the of leave future. this anModular paper exploration Berry is holonomies of in on its holography. which one might probe multi-system entanglement is to group the systems into two sets JHEP11(2019)094 ]. 40 , 16 ]. An excitation of the 43 ], the definition of entanglement 34 , 33 , ] can be thought of as the ‘experimental’ A particularly exciting question we leave 30 , 15 14 – 22 – with a bulk gauge field turned on. ], where we believe they may offer a useful framework An interesting playground for our ideas is the case of 3 14 7 ]. 42 , 41 ], and, more speculatively, to the black hole information problem [ ]. It will be illuminating to formulate our ideas more rigorously in the canon- ]. Intuitively, the shockwaves of [ 36 , 32 15 – 35 30 We also learned that we can ‘implant’ soft hair on the boundary of a subregion by One moral of our treatment is that soft modes are unphysical, gauge degrees of freedom We thank Beni Yoshida for this comment. 7 dynamical object. Whether theremains laws to governing this be evolution seen. takeit a treats It useful all is form, gauge worth however, fields,same noting including microscopic that gravity, origin on an in equal appealing the footing. feature CFT: All of the bulk our entanglement holonomies pattern approach have of the is the that state as encoded in Gravitation and gaugefor field future dynamics? study isconnections whether can our shed proposed light perspectiveCFT on on state the the gets emergence bulk imprinted of gravitational onaffects and their the the gauge dynamics modular modular [ Hamiltonians Berry in connection. its future As causal a cone result, and the thus latter is ultimately promoted to a bulk gauge field. Thereflected relevant in component the of the localis, modular field in Berry strength a curvature of sense, should the simplerfor then than gauge be the more field gravitational along computations. case an wecomputations discussed HRRT For we in surface. did example, this for work This it and pure setup would could AdS allow be an interesting exercise to repeat the the physics of chaos [ Bulk gauge field holonomies. holographic CFTs with globalof symmetries. modular zero-modes, The which conserved are charges holographically give mapped rise to to the a edge new modes set of the dual transport problem we formulatecould in apply this two shockwaves paper. along differentmode directions, To in holonomy construct two in different a this orderings. closed setupthe The loop measures edge- two the shockwaves. of soft It surfaces graviton wouldThe we component be appearance of interesting of the to shockwave commutators commutator understand also of this suggests heuristic an picture intriguing in possible detail. relation to transporting it around a closedoperational way loop. of exciting It soft modes is bysubregion interesting sending [ to shockwaves that compare cross the the boundary latterprotocol of with the for the shifting more the location of the horizon — an idealized version of which is the from the perspective of ainformation. given This subregion informs but the theirmodes recent holonomies [ discussion contain physical regarding geometric theical physical formalism significance along of the soft for lines describing of surface [ translations. Itare is not also asymptotically worthwhile to AdS. apply them in backgrounds that entropy [ In our work, thenew relative physical edge interpretation mode as frame athe of background gravitational infinitesimally spacetime. connection separated with regions curvature acquired that a depends on phase space of subsystems in gauge theories [ JHEP11(2019)094 ). . i ) A.1 (B.1) (B.2) (A.2) (A.3) δλ + ) onto the λ λ ( ( ψ V | and i ) λ . ( | respects equation ( is . Each state is invariant | ψ ψ | ) ] (A.1) λ V ih λ ( ψ | V, ρ ψ . . 1 ih i ) 1 | , which simply rotates the vector ) − ψ − λ | ψ | , with: ) = [ L ( ¯ | h ) L 1 ψ − λ 1 ψ λ ψ λ − | ∂ ( h − K ∂ s ( ), reads: t λ ψ i ∂ + + + ( ih ψ ) = ) 0 i + 0 2.12 i − h λ λ L ¯ ( ψ L ( K ψ | 0 ρ 0 λ | − | s ψ ]. t | = ∂ ) are therefore the modular zero modes in this + ψ ) | | + h λ + – 23 – λ 20 ψ ( ) 1 , ( ψ 1 h i h mod U ¯ L L iθ ) ψ 19 1 1 | i H s t λ ψ ∂ | = = )] = λ λ ∂ + − ( = ( K K V ) = exp [ = ( V λ λ 0 ρ ( P λ U ∂ ), the modular Berry connection is the projection of Γ = ). The operators λ ( 2.16 θ ) which, using the projector ( λ ( ρ by a phase i ) According to ( The variation of the state under an infinitesimal change of λ ( ψ B Modular connection forThe CFT two-sided modular vacuum Hamiltonian forterms an of interval the in conformal the generators CFT as vacuum can be written in zero modes of This is the familiar Berry connection [ where we defined the anti-Hermitian operator This is clearly notThis unique reflects since our any freedom addition to of independently zero-modes rotate to the phases of | simple example. Talents Program. LL is supported by the Pappalardo Fellowship. A Berry connection Consider a family ofunder normalized the pure transformation states at GGI Florence; BC, JdBfor and CFT LL thank and theBlack AdS” organizers Holes of held and workshop Holography” at “Modern heldFrom Techniques MITP at Simulations Mainz; TSIMF to Sanya BC Holography andsupported of II” thanks by conference held the the “Tensor at Networks: organizers Startup AEI of Fund Postdam from “Workshop and Tsinghua on University DESY and Zeuthen. by the BC Young is Thousand Faulkner, Matt Headrick, Hong Liu,Jieqiang Jakub Mielczarek, Wu, Tomasz Beni Trze´sniewski,Zizhi Wang, YoshidaBC and and LL Ellis thank YeStudy the Yuan University (Princeton), for of and interesting Pennsylvania, JdB BCWe discussions thanks all thanks and the thank MIT Institute feedback. the for for organizers Advanced hospitality of while the this workshop work “Entanglement in was Quantum completed. Systems” held would constitute a unified holographic description of gravitational andAcknowledgments gauge interactions. We thank Xiao-Liang Qi,and Lenny related Susskind work. and We Herman are Verlinde also for grateful fruitful to interactions Vijay Balasubramanian, Gavin Brennen, Tom the relative bases of modular Hamiltonians. A dynamical law of the sort we speculate here JHEP11(2019)094 (B.6) (B.7) (B.8) (B.9) (B.3) (B.4) (B.10) , 2) — a finite / + . ∂ + 2 2 mod / / ix 2 ) ) H − ie / − − ib − − ) a a ib − = − e e a − − 1 + + L − − − b b − − − live in orthogonal eigenspaces b ia ia e . − ), respectively. Working in the cot( cot( + e + − + π π K cot( 2 2 K and ] (B.5) , b K π . + + − + − − a b . a + is an element of the conformal algebra, and κ = = 2 = ,K ). K + + + 1 0 1 πi ∂ πi ∂ + + t t δa κE a 2 − K δa B.3 i∂ . V t ∂ − + V null coordinate, we find: a ) and ( = ] = πi ∂ 1 + preserve the left-moving and right-moving null − κ by demanding that exp( = [ 0 2 ] = 0 ] = ] = +2 x – 24 – ] = 0 , b L − + + + + ,E 2 + = + / K + K K K 2 2 a mod δa ) + ,K / / + + + V K + ) ) b H a a [ [ + + δa ) are: a 0 + + ∂ ib V + and , ∂ K ), i.e.: , ∂ a a P [ ib − − s on the and + + B.7 e e + i − − + ¯ K ). K L K [ 2.25 b [ + + + + b b + + B.7 + ia ia cot( ∂ e − cot( cot( π + e 2 π π ix 2 2 − are determined, up to an overall multiplicative constant, by the re- − i ie = = = , t 1 0 1 i = explicitly, it is convenient to decompose the conformal algebra into eigen- 1) transformation — map an interval to its complement. s s s − , 1 s + − ). The same fact guarantees that δa L V SO(2 2.20 × ) is automatically satisfied because 1) , B.6 To find The generator of modular parallel transport is defined by the conditions: Eq. ( of the eigenvalue equation ( This immediately implies eq. ( operators of the adjoint action of the modular Hamiltonian: The three solutions of equation ( be mapped toabsence any of other the spectrum usingequation changing ( conformal operator appearing transformations. onso the it This left is hand is a side linear the of combination the reason of general the for generators the ( In the vacuum of a two-dimensional CFT, any single-interval modular Hamiltonian can We found the overallSO(2 magnitude of with an identical action of the quirement that the generators coordinates of the intervalrepresentation endpoints ( The coefficients JHEP11(2019)094 , ) . κ x ( − b κE |− (C.1) (C.2) σ K is the ij (B.11) (B.12) (B.13) − ab a b σ K η σ ∂ ) ] = − ) κ x ( a , σ ,E σ /∂λ + | ) in the neigh- − i ij mod of the Christoffel − ∂b , y K H α + K M αβ x . The change in the σ − + b ), where δλ ∂ − i = ( . ) in the normal frame of − K + ) , y ) 3 ∂λ − M ∂b λ a x σ C.2 x ( ( ∂ ) is the generator of modular ) in the orthogonal directions. a + ) i O σ y − ( )) to + α K /∂λ B.12 − = ( β . − − a σ δσ , b M ∂ α − ∂a mod σ σ b − from the minimal surface and by ) , a H σ y ∂λ x λ b λ + + ∂a ( ∂ a + , b Consider now a nearby entanglement wedge M αβ − . π + K Γ + ] = a + 1 2 b . Therefore, ( K – 25 – = 2 = 0 ∂ − − + mod = ( ) ) ) b λ λ K ∂ a i ( ( λ M ,H ζ ζ + σ /∂λ will have the same form ( and a coordinate system δλ ) ∂λ + ) ∂b and V λ ) generating a geometric flow. About the surface, the [ δλ ) = ∂b σ i − + 3.8 ( + ) ) (here λ denotes distances along two directions orthogonal to the RT + K M , y ( λ M λ + ( ˜ ζ α ( α K ζ σ + x + ( a K mod ∂ M + -coordinates is that they set the components Γ x a H + σ ∂ ∂λ ) ) because it lives outside the zero-eigenspace of [ ∂a B.6 /∂λ πi 1 + is a choice of internal surface coordinates. 2 i ∂a y = ≡ δλ V -derivative of the modular boost. whose RT surface is separated from that of λ mod For as long as we focus on a small neighborhood of the RT surface ( Since we are ultimately interested in comparing the frames of two nearby extremal It is easy to consider a more general direction in kinematic space (space of CFT inter- δλ H λ + ∂ The λ Its modular boost generator 1) the action of thedescribed modular by Hamiltonian a is vectormodular expected field flow to generator be has local the and, form: therefore, it can be The advantage of the connection to zero, so they constitute the analog of the local inertial frame for a surface. surfaces, the formconvenient of to introduce normal the geodesic coordinates metricmeasures in the geodesic the distance vicinity ofunit a tangent of vector nearby to point the the same geodesic RTaround at the surface its starting surface, is point this on coordinate the important. surface. system In is: an It expansion is C Solution to equationConsider ( an entanglement wedge borhood of its RT surface. surface and It also satisfies ( the latter being generatedparallel by transport. solves vals.) Say we gomodular from Hamiltonian is: The operator JHEP11(2019)094 . λ ζ ) we (C.5) (C.7) (C.8) (C.9) (C.3) (C.4) 3.8 with ξ . . Then the map M a M δ a is not unique, since ] has a kernel. The b n a σ ) = s b ξ, ζ λ ) a ) into equation ( ( 2 -coordinates, we can use . a M ) M s i C.5 a , δn a y = 0 (C.6) 2 n (˜ ), we find: i 0 δn ζ . =0 . , δσ a + a C.5 ) 2 σ M i a c σ

2 s δ b ( σ ) in the distances along b M ) b + O c , δn y a ζ a b ) in the local choice of normal vectors 2 (˜ λ s . C.2 + σ δn δ ω surface are δn b b b bc C.4 s , δσ a a λ + − η -coordinates and compute the diﬀerence of a 2 = 0 ) (and each other) by: . a b s a σ that can be straightforwardly solved to get 1) be two orthonormal vectors at every point ) ( M a b M a a , 2 ξ =0 δ δ δn δσ ∂ C.8 O a – 26 – ) δn σ i − ab +

M a + y , δn = 0 i The next step is to compute the zero mode com- b δ (˜ a 2 b → ζ 2 1 a s s By plugging the result ( ω − λ ) and we have used the fact that, in the orthonormal i a δσ − δ b λ b b , δσ ). )( + and denote by = has the form ( ( a 2 ) + s δn δn ) M y δλ M ( ) = ) = M (˜ δλ ξ M a a C.5 ξ δλ ζ ζ a + δ O + n λ λ + i + δ δ λ → ) to the right hand side of ( λ δσ y M − + ( ( − λ ; ˜ ζ i ) y Ω( M = = ˜ ) are related to ( = (˜ Z ξ i 3.14 a δλ M y σ M 3.8 a + ζ , at first non-trivial order in the separation of the two surfaces, is: n λ ) to map it back to the λ δ ) ( i ) is not unique, because the vector Lie bracket [ ˜ y M and subtract it to obtain an equation for the Lie bracket of C.3 , a a C.8 n ) contains no zero mode components, so no extra subtraction is necessary. s M ( ζ i ≡ λ C.5 δ , y ) M i a ˜ y , a δn s Since the vector field It is important to note that the choice of normal coordinates ( on the HRRT surface of a i σ ˜ As discussed in thefirst main term, text, corresponding thisdirections, to is can a simply be spatially an absorbed varying addition in boost the of along ambiguity modular ( the zero orthogonal modes. RT surface The obtain an equation for the diffeomorphism The solution ( family of solutions of ( Equation ( The bulk modular connection. Zero-mode component ofponent ( of Applying the projector ( transformation ( the two modular boost generators: will yield an equally acceptablein pair of the normal map directions. between Therewill is, the be therefore, normal important an frames ambiguity in what of follows. two nearby minimal surfaces. This ambiguity Here gauge we are using, the normal vectors on the any local Lorentz boost on the orthogonal plane y the new wedge. Let JHEP11(2019)094 ]. , M x 42 ] 61 (C.10) (2016) SPIRE IN 09 (2016) 068 ][ ]. 09 2-dimensional − (2017) 151 JHEP SPIRE d Commun. Math. , ]. Gen. Rel. Grav. , 07 IN Fortsch. Phys. JHEP , ][ , , ]. arXiv:1601.05416 arXiv:1807.04276 SPIRE [ , . JHEP IN , a ][ ]. x arXiv:1712.07123 ]. SPIRE [ ]. IN b ][ Relative entropy equals bulk relative δσ SPIRE SPIRE IN SPIRE M ab IN (2016) 021601 ][ IN ]. arXiv:1005.3035 ][ Modular Berry connection for entangled ][ A covariant holographic entanglement + Γ 117 ) back to the general normal gauge Quantum corrections to holographic M (2018) 091601 SPIRE a arXiv:1307.2892 IN C.8 [ δn ][ 120 – 27 – Entanglement holonomies ]. arXiv:0705.0016 − [ (2010) 2429] [ Bulk locality and quantum error correction in a Reconstruction of bulk operators within the entanglement hep-th/0603001 δσ ]. [ SPIRE arXiv:1411.7041 (2013) 074 Bulk locality from modular flow [ D 19 IN M a Cool horizons for entangled black holes Phys. Rev. Lett. arXiv:1512.06431 δ ]. ][ , ), which permits any use, distribution and reproduction in 11 [ ]. Local subsystems in gauge theory and gravity − (2007) 062 SPIRE Holographic derivation of entanglement entropy from AdS/CFT The gravity duals of modular Hamiltonians . The second term, in turn, allows the internal coordinate IN = Phys. Rev. Lett. 07 δλ ][ , SPIRE M arXiv:1607.03901 JHEP SPIRE to be related by an infinitesimal element of the ξ IN (2015) 163 + [ , Building up spacetime with quantum entanglement IN (2006) 181602 ][ λ (2016) 004 δλ ][ 04 CC-BY 4.0 JHEP 96 + , 06 This article is distributed under the terms of the Creative Commons λ Int. J. Mod. Phys. arXiv:1306.0533 [ The Ryu–Takayanagi formula from quantum error correction JHEP (2017) 865 [ , and ]. ]. JHEP λ , 354 ). The computation yields the following general solution (up to zero modes): arXiv:1601.04744 [ SPIRE SPIRE C.1 IN arXiv:1412.8465 arXiv:1704.05464 IN [ subregions in AdS/CFT 102 (2010) 2323 (2013) 781 entropy Phys. Rev. Lett. entropy proposal entanglement entropy [ AdS/CFT Phys. [ wedge in gauge-gravity duality [ It instructive to transform the result ( B. Czech, L. Lamprou, S. Mccandlish and J. Sully, W. Donnelly and L. Freidel, M. Van Raamsdonk, J. Maldacena and L. Susskind, B. Czech, L. Lamprou and L. Susskind, S. Ryu and T. Takayanagi, V.E. Hubeny, M. Rangamani and T. Takayanagi, T. Faulkner, A. Lewkowycz and J. Maldacena, D.L. Jafferis and S.J. Suh, D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, D. Harlow, T. Faulkner and A. Lewkowycz, X. Dong, D. Harlow and A.C. Wall, A. Almheiri, X. Dong and D. Harlow, [8] [9] [5] [6] [7] [3] [4] [1] [2] [13] [14] [10] [11] [12] References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. systems on diffeomorphim subgroup. using ( on the RT surface of JHEP11(2019)094 ] ]. , (2019) Phys. SPIRE 04 , A 61 IN ][ ]. , JHEP (2017) 204001 , (2017) 086016 SPIRE ]. arXiv:1604.03110 ]. 34 (2019) 182 IN [ Phys. Rev. , ][ Proc. Roy. Soc. London D 96 01 , SPIRE (2016) 038 (2019) 116 IN ]. ]. ][ 09 JHEP 03 arXiv:1703.10656 (2016) 129 ]. , [ Four qubits can be entangled in Phys. Rev. 07 SPIRE SPIRE , IN JHEP IN Entanglement, holography and causal JHEP SPIRE quant-ph/0109033 , ][ Modular Hamiltonians for Deformed , ][ IN arXiv:1310.6335 Class. Quant. Grav. JHEP ][ [ , , Superrotation charge and supertranslation (2017) 364001 ]. Distributed entanglement arXiv:1611.09175 ]. [ – 28 – A 50 SPIRE (2002) 052112 [ Modular Hamiltonians on the null plane and the Markov IN Flat entanglement spectra in fixed-area states of quantum (2014) 086010 ]. SPIRE ][ ]. A modular toolkit for bulk reconstruction Three qubits can be entangled in two inequivalent ways arXiv:1606.03307 IN arXiv:1802.10103 The holographic shape of entanglement and Einstein’s [ A 65 [ [ State-dependent bulk-boundary maps and black hole (2017) 161 arXiv:1811.05382 ]. ]. ]. ]. ]. J. Phys. D 89 [ SPIRE ]. , SPIRE Soft hair as a soft wig Observable supertranslations 05 IN IN [ ][ Appearance of gauge structure in simple dynamical systems SPIRE SPIRE SPIRE SPIRE SPIRE SPIRE IN IN IN IN IN (2016) 162 (2018) 147 IN Phys. Rev. JHEP [ ][ ][ ][ ][ ][ , , (2019) 240 08 Phys. Rev. (2000) 062314 05 . , A stereoscopic look into the bulk Lectures on the infrared structure of gravity and gauge theory quant-ph/9907047 10 [ (1984) 2111 Quantal phase factors accompanying adiabatic changes A 62 Unambiguous phase spaces for subregions JHEP Superselection sectors of gravitational subregions JHEP 52 , , ]. JHEP (1984) 45 , arXiv:1806.10560 [ SPIRE arXiv:1706.09280 arXiv:1901.09857 arXiv:1810.01802 arXiv:1706.00436 IN arXiv:1605.08072 [ gravity [ [ [ (2000) 052306 Phys. Rev. nine different ways equations diamonds property of the vacuum state 119 Rev. Lett. [ Half-Spaces and the Averaged Null[ Energy Condition A 392 arXiv:1703.05448 hair on black holes complementarity X. Dong, D. Harlow and D. Marolf, J. Kirklin, J. Camps, R. Bousso and M. Porrati, R. Bousso and M. Porrati, W. Dur, G. Vidal and J.I. Cirac, F. Verstraete, J. Dehaene, B. de Moor and H. Verschelde, A. Lewkowycz and O. Parrikar, B. Czech and Z. Wang, to appear. V. Coffman, J. Kundu and W.K. Wootters, H. Casini, E. Teste and G. Torroba, T. Faulkner, M. Li and H. Wang, F. Wilczek and A. Zee, B. Czech et al., J. de Boer, F.M. Haehl, M.P. Heller and R.C. Myers, T. Faulkner, R.G. Leigh, O. Parrikar and H. Wang, M.V. Berry, S.W. Hawking, M.J. Perry and A. Strominger, K. Papadodimas and S. Raju, A. Strominger, [33] [34] [30] [31] [32] [28] [29] [25] [26] [27] [23] [24] [20] [21] [22] [18] [19] [16] [17] [15] JHEP11(2019)094 31 , Phys. , 01 Phys. , JHEP , (2014) 067 (2016) 106 03 08 Class. Quant. Grav. , JHEP ]. , JHEP , SPIRE ]. IN ]. Black hole entropy and soft hair ][ ]. SPIRE ]. SPIRE IN IN ][ ][ SPIRE A bound on chaos SPIRE IN ]. IN ][ ][ ]. – 29 – The boundary dual of the bulk symplectic form Complexity and the bulk volume, a new York time SPIRE arXiv:1812.02915 [ IN SPIRE ][ IN Black holes and the butterfly effect ][ arXiv:1412.1895 ]. Entanglement entropy of electromagnetic edge modes [ ]. arXiv:1811.03097 Emergence of spacetime from the algebra of total modular [ (2019) 017 SPIRE arXiv:1806.10144 SPIRE arXiv:1810.01847 [ IN 05 [ IN ][ ][ (2019) 044 arXiv:1406.7304 (2015) 111603 JHEP [ Entanglement entropy and nonabelian gauge symmetry 03 , (2019) 71 arXiv:1510.07911 Wormholes, emergent gauge fields and the weak gravity conjecture [ 114 (2018) 098 12 JHEP B 789 , arXiv:1306.0622 arXiv:1503.01409 (2016) 122 JHEP [ [ story Hamiltonians (2014) 214003 Rev. Lett. Lett. D. Harlow, S.H. Shenker and D. Stanford, J. Maldacena, S.H. Shenker and D. Stanford, A. Belin, A. Lewkowycz and G. Sárosi, D. Kabat and G. Lifschytz, S. Haco, S.W. Hawking, M.J. Perry and A. Strominger, W. Donnelly and A.C. Wall, A. Belin, A. Lewkowycz and G. Sárosi, W. Donnelly, [43] [41] [42] [38] [39] [40] [36] [37] [35]