Entanglement Negativity in Flat Holography
Debarshi Basu, Ashish Chandra, Himanshu Parihar, Gautam Sengupta
aa r X i v : . [ h e p - t h ] F e b Prepared for submission to JHEP
Entanglement Negativity in Flat Holography
Debarshi Basu, a Ashish Chandra, a Himanshu Parihar a and Gautam Sengupta a a Department of Physics,Indian Institute of Technology,Kanpur 208 016, India
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We advance holographic constructions for the entanglement negativity ofbipartite states in a class of (1 + 1) − dimensional Galilean conformal field theories dualto asymptotically flat three dimensional bulk geometries described by Einstein Gravityand Topologically Massive Gravity. The construction involves specific algebraic sums ofthe lengths of bulk extremal curves homologous to certain combinations of the intervalsappropriate to such bipartite states. Our analysis exactly reproduces the correspondingreplica technique results in the large central charge limit. We substantiate our constructionthrough a semi classical analysis involving the geometric monodromy technique for the caseof two disjoint intervals in such Galilean conformal field theories dual to bulk EinsteinGravity. ontents
43 Entanglement measures in GCFT
64 Entanglement in flat holography 7 c M M L c M limit 266.2 Holographic entanglement negativity for two disjoint intervals in proximity 276.2.1 Two disjoint intervals in vacuum 296.2.2 Two disjoint intervals at a finite temperature 296.2.3 Two disjoint intervals in a finite-sized system 30 – 1 – Introduction
In recent years quantum entanglement has emerged as a fundamental issue connectingdiverse areas of physics from many-body condensed matter systems to black holes andquantum gravity. It is well known in quantum information theory that bipartite pure stateentanglement is characterized by the entanglement entropy which is the von Neumannentropy of the corresponding reduced density matrix. However the entanglement entropyis not a valid measure for mixed state entanglement due to contributions from irrelevantcorrelations. To address this significant issue several entanglement and correlation measureswere introduced in quantum information theory. However most of these were not easilycomputable as they involved extremization over LOCC protocols. Vidal and Werner [1]in a classic work introduced a computable measure for mixed state entanglement termed entanglement negativity (logarithmic negativity) which was defined as the trace norm of thepartial transpose of the density matrix with respect to one of the subsystems and providedan upper bound to the distillable entanglement. Despite its non-convexity [2], entanglementnegativity was proved to be an entanglement monotone and is widely used to characterizemixed state entanglement.For extended quantum many-body systems with infinite degrees of freedom such en-tanglement measures are usually computationally intractable although a formal definitionmay be attempted. Significantly, it was shown in [3, 4] that the entanglement entropy ofbipartite states in (1 + 1) -dimensional relativistic conformal field theories (CFT ) maybe explicitly computed through a replica technique. Remarkably the replica technique de-scribed above could also be modified to compute the entanglement negativity of bipartitestates in such relativistic CFT described in [5–7].Over the last decade there has been intense focus on the holographic characterizationof entanglement in conformal field theories dual to bulk AdS geometries in the frameworkof the AdS/CFT correspondence [8]. This was pioneered by the classic work of Ryu andTakayanagi (RT) in [9, 10] where it was conjectured that the universal part of the entan-glement entropy of a subsystem in a relativistic CFT d was proportional to the area of abulk static codimension two minimal surface homologous to the subsystem. A covariantgeneralization of the above holographic conjecture was proposed by Hubeny, Rangamaniand Takayangi (HRT) for relativistic CFT d dual to bulk non-static AdS geometries in [11].The above conjectures were subsequently proved in a series of significant works in [12–18].In the above context, it was natural to seek a corresponding holographic characteri-zation for the entanglement negativity of such bipartite states in dual CFT d s. This wasinitially attempted for the pure vacuum state of dual CFT d s in [19]. Subsequently a com-prehensive holographic construction for the entanglement negativity of both pure and mixedstates in dual CFT s was advanced in the context of the AdS /CFT [20–23] scenario.These proposals were substantiated by a large central charge analysis of the entanglementnegativity for CFT s utilizing the monodromy technique in [13, 24–26]. Subsequently,the covariant extension of the holographic entanglement negativity constructions describedabove were advanced for bipartite states in CFT s dual to non-static AdS backgroundsfollowing the HRT construction [11] in [23, 27–29]. Higher dimensional generalizations– 2 –f the above holographic constructions for bipartite states described by configurations ofsubsystems with long rectangular strip geometries in CFT d s dual to bulk static AdS d +1 ge-ometries were proposed in [30–32]. We should mention here that an alternate holographicconstruction based on the entanglement wedge cross-section [33, 34], for the entanglementnegativity of bipartite states in the AdS d +1 /CFT d scenario was developed in [35, 36]. It hasbeen shown in [37] that this proposal is completely equivalent to the earlier construction forthe holographic entanglement negativity upto certain overall multiplicative factors arisingfrom the backreaction of cosmic branes associated with bulk conical defects. In a separate context, a class of (1 + 1) dimensional field theories with Galilean con-formal symmetries obtained through a parametric İnönü-Wigner contraction of the usualrelativistic conformal algebra were investigated in [40–54]. The authors of [43, 44] devel-oped a replica technique for computing the entanglement entropy of such Galilean conformalfield theories (GCFT ). Following this a replica technique to compute the entanglementnegativity of bipartite states in a class of such GCFT was established in [55].The above class of GCFT s was proposed as possible holographic duals to bulkthree-dimensional gravity in asymptotically flat space-times [56] in the framework of flatspace holography [57, 58]. The asymptotic symmetry algebra of the bulk geometry was de-scribed by the infinite dimensional Bondi-Metzner-Sachs (BMS ) algebra isomorphic to theGalilean conformal algebra in dimensions (GCA ). The authors of [59] computed theholographic entanglement entropy of a single interval in the corresponding dual BMS fieldtheory located at the null infinity of the bulk asymptotically flat geometry. Interestinglyin [60], the authors established a holographic construction for the entanglement entropyin the dual BMS field theories described above, through a generalization of the covariantHRT construction [11]. From a different perspective, the authors of [43] obtained the aboveflat space holography results utilizing the Chern-Simons formulation of three-dimensionalgravity [61] and the Wilson line prescription [62].The above developments bring the critical issue of a holographic description of mixedstate entanglement for these dual GCFT into sharp focus. In this article we addressthis issue through the BMS /GCA correspondence [58–60]. In this context we establishholographic constructions to compute the entanglement negativity of bipartite states inGCFT s dual to bulk asymptotically flat (2 + 1) dimensional Einstein Gravity and Topo-logically Massive Gravity (TMG) [59, 60, 63–66], following the corresponding constructionsfor relativistic CFT s described in [20, 21, 55]. Interestingly our results match exactlywith the universal parts of the corresponding replica technique results obtained in [55]. Forthe mixed state of disjoint intervals in proximity we substantiate our results through a rig-orous geometric monodromy analysis [67] to obtain the corresponding large central chargelimit.This article is organized as follows. In section 2 we briefly recollect the salient featuresof GCFT s and the BMS /GCA correspondence. The replica techniques developed in[43, 44, 55] for computing the entanglement entropy and negativity respectively in suchGCFT s are reviewed in section 3. In section 4 we describe the covariant construction For more recent developments see [38, 39]. – 3 –or computing the entanglement entropy in [59, 60]. In particular, we apply this covariantprescription to obtain the entanglement entropy for a single interval in a GCFT describ-ing a finite-sized system and find perfect agreement with [43, 44]. In section 5, we establishour flat-holographic constructions for computing the entanglement negativity for a singleand two adjacent intervals in GCFT s dual to Einstein gravity in the bulk asymptoticallyflat spacetimes. The holographic construction for computing the entanglement negativityfor the case of two disjoint intervals along with the large central charge analysis is describedin section 6. In section 7 we generalize the above constructions to the case of GCFT sdual to bulk geometries described by TMG. The special case of the entanglement negativityin flat chiral gravity is discussed in appendix A. We conclude in section 8 with a summaryof our results and discuss future open issues.
In this section we review the basics of (1 + 1) dimensional Galilean conformal field the-ories (GCFT ) [40–54]. Interestingly the Galilean conformal algebra (GCA ) may beobtained via an İnönü-Wigner contraction of the usual relativistic conformal algebra in twodimensions: t → t , x → ǫx , (2.1)with ǫ → . This is equivalent to the non-relativistic small velocity limit v ∼ ǫ. The Galileanconformal transformations acts on the coordinates as t → f ( t ) , x → f ′ ( t ) x + g ( t ) , (2.2)which can be thought of as diffeomorphisms and t -dependent shifts, respectively. These aregenerated by the Nöether charges which, in the plane representation, are given by L n = t n +1 ∂ t + ( n + 1) t n x∂ x , M n = t n +1 ∂ x , (2.3)which obey the Lie algebra with different central extensions in each sector : [ L n , L m ] = ( m − n ) L n + m + c L
12 ( n − n ) δ n + m, , [ L n , M m ] = ( m − n ) M n + m + c M
12 ( n − n ) δ n + m, , [ M n , M m ] = 0 , (2.4)where c L and c M are the central charges for the GCA. The cylinder and plane representa-tions are related via the transformation [45, 67] t = e iφ , x = iu e iφ . (2.5)The maximally commuting subalgebra is that of the generators { L , M } and the repre-sentations are labelled by their eigenvalues (the conformal weights) h L and h M in order toconstruct the highest weight representation. L | h L , h M i = h L | h L , h M i , M | h L , h M i = h M | h L , h M i . (2.6) Note that we are working in the plane representation which differs from the familiar cylinder represen-tation used in [41, 42] by a negative sign in the GCA. – 4 –he two point correlator of primary fields may be written down utilizing the Galileanconformal symmetry as [42, 55] (cid:10) V ( x , t ) V ( x , t ) (cid:11) = C (2) δ h L h L δ h M h M t − h L exp (cid:18) − h M x t (cid:19) , (2.7)where ( h L , h M ) and ( h L , h M ) are the weights of the primary fields V and V respectively, C (2) is a normalization constant and x = x − x , t = t − t . In a similar manner it iseasy to determine the three point function of primary fields in a GCFT to be [42, 55] h V ( x , t ) V ( x , t ) V ( x , t ) i = C (3) t − ( h L + h L − h L )12 t − ( h L + h L − h L )23 t − ( h L + h L − h L )13 × exp h − ( h M + h M − h M ) x t − ( h M + h M − h M ) x t − ( h M + h M − h M ) x t i , (2.8)where the V i ’s are primary fields with weights { ( h iL , h iM ) } and x ij = x i − x j , t ij = t i − t j with ( i = 1 , , respectively and C (3) is a constant. Similarly, the four-point function ofprimary fields in the GCFT may be expressed as [55] * Y i =1 V i ( x i , t i ) + = t h L + h L t h L + h L t h L + h L t h L + h L t h L + h L t h L + h L exp h x t ( h M + h M ) + x t ( h M + h M ) − x t ( h M + h M ) − x t ( h M + h M ) − x t ( h M + h M ) − x t ( h M + h M ) i G ( t, xt ) , (2.9)where { ( h iL , h iM ) } are the weights of the primary fields V i ( x i , t i ) with ( i = 1 , , , and t = t t t t , xt = x t + x t − x t − x t , (2.10)are the non-relativistic counterparts of the cross ratio x in the relativistic CFT s. Ineq. (2.9), G ( t, xt ) is a non-universal function of the cross ratios that depends on the fulloperator content of the specific field theory.Interestingly the GCFT s are equivalent to the BMS field theories at the levelof the algebra [58]. This leads to a conjectured GCA /BMS correspondence between theasymptotic symmetry algebra of three dimensional Minkowski spacetime at null infinity andthe above class of GCFT [44, 45, 58]. Note that the central charges of these contractedalgebras are related with the parent Virasoro central charges as [58] c L = c + ¯ c , c M = ǫ ( c − ¯ c ) , (2.11)for GCA , and as c L = ǫ ( c − ¯ c ) , c M = c + ¯ c , (2.12)for BMS . Also, the kinematics in the two sectors are related by the replacement x ←→ t [44]. We will be using the BMS /GCA correspondence for the computations in the contextof flat holographic entanglement in sections 5 to 7.– 5 – Entanglement measures in GCFT
In this section we briefly review the replica techniques employed to compute the entan-glement entropy and entanglement negativity, in the special class of GCFT describedabove. As in the case of relativistic CFT s [3, 4], the entanglement entropy for a bi-partite state in these GCFT s may be computed using a replica technique developed in[43, 44]. To this end, one considers n -copies of the GCFT plane sewed together alongcuts describing the intervals (subsystems) under consideration. The partition function onthis replica manifold then computes the Renyi entropy S ( n ) A for the boosted interval A , interms of the two-point function of twist fields Φ ± n inserted at endpoints ∂ i A of the interval A as (1 − n ) S ( n ) A = T rρ nA = h Φ n ( ∂ A )Φ − n ( ∂ A ) i , (3.1)where the twist fields are primary fields of the GCFT with scaling dimensions ∆ n = c L (cid:18) n − n (cid:19) , χ n = c M (cid:18) n − n (cid:19) , (3.2)and ρ nA is the reduced density matrix corresponding to the subsystem A . The entanglemententropy for the bipartite state corresponding to the interval A in the GCFT may nowbe obtained by taking the replica limit n → as S A = lim n → S ( n ) A = lim n → ∂ n h Φ n ( ∂ A )Φ − n ( ∂ A ) i . (3.3)Interestingly it was possible to compute the entanglement negativity for mixed states inrelativistic CFT s through a related replica technique [5–7]. To define the entanglementnegativity in quantum information theory a tripartite system in a pure state consistingof subsystems A , A and B is considered. Subsequently the degrees of freedom of thesubsystem B are traced over to obtain the reduced density matrix of the mixed stateconfiguration described by A = A ∪ A , as ρ A = Tr B ρ , where ρ describes the tripartitestate A ∪ B . The entanglement negativity of the bipartite mixed state described by thereduced density matrix ρ A is then defined as the trace norm of the partially transposeddensity matrix ρ T A [1, 5–7] E = ln Tr || ρ T A || , (3.4)where the trace norm is defined as the sum of absolute eigenvalues of ρ T A . The operationof partial transpose is described as D e (1) i e (2) j | ρ T A | e (1) k e (2) l E = D e (1) i e (2) l | ρ A | e (1) k e (2) j E , (3.5)where | e (1) i i and | e (2) j i are the basis elements for the Hilbert spaces H and H correspondingto A and A , respectively. Note that in the case of GCFT s one cannot consider subsystems at a fixed time slice due to the lack ofLorentz invariance. Therefore one must consider Galilean boosted intervals of the form A = [( x , t ) , ( x , t )] [43, 55]. – 6 –ext we briefly discuss the replica construction for computing the entanglement neg-ativity of bipartite states in a GCFT developed in [55] which closely follows [5, 6] forrelativistic CFT .As for the relativistic CFT , in this case one considers a replicated manifold describedby n e -copies (with n e even) of the GCFT plane glued together in an appropriate fashion[55]. The entanglement negativity for the bipartite mixed state configuration A ≡ A ∪ A may then be obtained through a replica technique as E = lim n e → log Tr( ρ T A ) n e . (3.6)In eq. (3.6), we have used the replica limit n e → and the quantity Tr ( ρ T A ) n e can beexpressed in terms of a four-point correlator of twist fields Φ ± n e inserted at the endpointsof the intervals asTr ( ρ T A ) n e = h Φ n e ( x , t )Φ − n e ( x , t )Φ − n e ( x , t )Φ n e ( x , t )) i . (3.7)The authors of [55] computed the entanglement negativity for various bipartite pure andmixed state configurations involving a single interval and two adjacent intervals in a GCFT .In the subsequent sections, we will develop holographic constructions to compute the entan-glement negativity for such configurations in a GCFT . Furthermore, in section 6 we willdescribe a geometric monodromy technique to obtain the universal part of the four-pointtwist correlator in (3.7) from which it is possible to establish a holographic constructionfor the entanglement negativity of the mixed state configuration of two disjoint intervals inproximity. In this section we review the salient features of the covariant construction in [59, 60] forcomputing entanglement entropy in flat holography in the spirit of the HRT prescription [11]in the usual AdS/CFT scenario. The entanglement entropy of a bipartite state describedby a single interval in the BMS /GCA field theory located at the null infinity of the dualasymptotically flat bulk geometry will be given by the length of a bulk extremal geodesichomologous to the interval. We first consider the case of the BMS /GCA field theorydual to bulk asymptotically flat (2 + 1) -dimensional Einstein Gravity for which the Brown-Henneaux symmetry analysis at null infinity leads to the infinite dimensional BMS /GCA algebra. For the appropriate boundary conditions, the general solution to Einstein equationsin the Bondi gauge is [59] ds = Θ( φ ) du − dudr + 2 h Ξ( φ ) + u ∂ φ Θ( φ ) i dudφ + r dφ , (4.1)where u = t − r in the (retarded) Eddington-Finkelstein time, r is the holographic co-ordinate, and Θ( φ ) and Ξ( φ ) are arbitrary functions of the angular coordinate φ . It isinteresting to note that by construction the holographic direction is null.As stated earlier the flat space holographic principle requires a dual BMS /GCA field theory located at the null infinity of the bulk asymptotically flat spacetime. The– 7 –orresponding central charges for this dual field theory are obtained from the asymptoticsymmetry analysis as [59, 67–69] c L = 0 , c M = 3 G . (4.2)It is interesting to note that the global subalgebra of the BMS group is identical to thePoincare algebra. Therefore the corresponding conformal weights ∆ and χ which labelthe representations of the BMS /GCA must correspond to the quadratic Casimirs of thePoincare algebra. This indicates the presence of a massive particle with spin propagating inthe bulk geometry. For Einstein gravity in the bulk however the equations (3.2) and (4.2)indicate that ∆ = 0 , which corresponds to the propagation of a spinless massive particle inthe bulk spacetime [60]. We start with the holographic computation of the entanglement entropy for a single intervalin the vacuum state of a GCFT . To this end we consider the dual geometry of the bulkflat (2 + 1) dimensional Minkowski spacetime in Eddington-Finkelstein coordinates whichis given as ds = dr − du + r dφ , (4.3)where the coordinates are as described earlier. We consider an interval A = [( u ∂ , φ ∂ ) , ( u ∂ , φ ∂ )] on the dual GCFT plane located at the null infinity of the flat spacetime. It was shownin [60] the length of the bulk extremal curve joining the endpoints ∂ i A ( i = 1 , of theinterval, is given by L extrtot = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ∂ tan φ ∂ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4.4)Note that the bulk extremal curve consists of two null curves descending from the endpoints ∂ i A which do not intersect and a third extremal curve is required to connect them. Recallthat for Einstein gravity in the bulk we have c L = 0 from eq. (4.2). Therefore, as describedin [60], in the large c M limit, the twist fields inserted at the endpoints of the intervalcorrespond to a bulk propagating particle of mass m n = χ n . Consequently the two pointcorrelator (3.1) of these twist fields can be expressed as the exponential of the on-shellaction of such a particle propagating along an extremal trajectory X µ ( s ) homologous tothe interval. With such an identification we write following [60]: h Φ n ( ∂ A )Φ − n ( ∂ A ) i = e − m n S on-shell , (4.5)where m n = χ n and S on-shell = q η µν ˙ X µ ˙ X ν = L extrtot . (4.6)Therefore the entanglement entropy for the single interval A in eq. (3.3) is given by theflat space analog of the HRT formula S A = 14 G L extrtot = 14 G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ∂ tan φ ∂ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.7)where we have used eq. (4.2). – 8 – .2 Holographic entanglement in global Minkowski orbifolds Next we focus on a GCFT compactified on a spatial circle of circumference L . The dualgeometry is the global Minkowski orbifold, which is described as the quotient of the usualMinkowski spacetime with the compact spatial circle [59]: ( u, φ ) ∼ ( u, φ + L φ ) . (4.8)The metric for global Minkowski orbifolds reads [59] ds = − (cid:18) πL φ (cid:19) du − du dr + r dφ . (4.9)The holographic entanglement entropy of the boosted interval A = [( u ∂ , φ ∂ ) , ( u ∂ , φ ∂ )] isobtained from the length of a bulk extremal curve homologous to the interval in the dualfield theory. Note that the bulk geodesics are not necessarily straight lines for this casewhich renders the analysis to be more involved than for the bulk flat Minkowski spacetime.To this end we compute the geodesic length in the Cartesian coordinates and map theendpoints to the global Minkowski orbifold through the the coordinate transformationswhich implements the quotienting [59, 60]. These coordinate transformations are given as r = 2 πL φ p x − t ,u = (cid:18) L φ π (cid:19) (cid:20) πiL φ y − r (cid:21) ,φ = L φ πi log (cid:20) πiL φ ( t − x ) r (cid:21) = L φ π sin − (cid:20) π ( t − x ) L φ r + L φ r π ( t − x ) (cid:21) . (4.10)Inverting these relations, we obtain x = L φ r π sin (cid:18) πφL φ (cid:19) , t = L φ r π cos (cid:18) πφL φ (cid:19) , y = L φ πi r − πiL φ u . (4.11)The length of the bulk geodesic from y to y obtained through this procedure is expressedas L ( y , y ) = L φ π " r r (cid:18) − cos 2 π ( φ φ ) L φ (cid:19) − π L φ ( r − r )( u − u ) − (cid:18) πL φ (cid:19) ( u − u ) / . (4.12)Similar to the previous case of the bulk pure Minkowski spacetime [60], we have null hyper-surfaces on which the null curves descending from the endpoints ( u ∂i , φ ∂i ) of the boundaryinterval lie: N i : 2 πL φ ( u ∂i − u i ) − r i sin (cid:18) π ( φ i − φ ∂i ) L φ (cid:19) = 0 . (4.13)The invariant length between y i ∈ N i and the boundary endpoint ∂ i A is given by L ( y i , ∂ i A ) = L φ π r i sin (cid:20) π ( φ i − φ ∂i ) L φ (cid:21) . (4.14)– 9 –he null lines now correspond to u i = u ∂i , φ i = φ ∂i which usually do not intersect andanother extremal curve connecting the null lines is required. The total length of the extremalcurve may then be expressed as follows L tot = L extr ( y , ∂ A ) + L extr ( y , y ) + L extr ( y , ∂ A ) = L extr ( y , y ) . (4.15)The extremization of the length in eq. (4.12) with respect to the position of the endpointsleads to ∂L tot ∂r i = 0 = ⇒ r = 4 π u ∂ /L φ − cos (cid:16) πφ ∂ L φ (cid:17) = − r . (4.16)Substituting this back into the expression (4.15) we obtain the length of the extremal curvehomologous to the interval as L extrtot = 2 πu ∂ L φ cot (cid:18) πφ ∂ L φ (cid:19) . (4.17)Consequently the holographic entanglement entropy for the interval A in the dual fieldtheory is given by S A = 14 G L extrtot = c M πu ∂ L φ cot (cid:18) πφ ∂ L φ (cid:19) , (4.18)where in the last expression we have used eq. (4.2). This matches with the c L = 0 part ofthe entanglement entropy of the single interval in the BMS /GCA field theory dual to theglobal Minkowski orbifold obtained in [43]. In this subsection we will consider a finite temperature GCFT with a compactifiedthermal cycle ( u, φ ) ∼ ( u + iβ u , φ + iβ φ ) . The corresponding holographic dual is anotherinteresting quotient of Minkowski spacetime called Flat Space Cosmology (FSC), with themetric [43–45] ds = M du − dudr + J dudφ + r dφ , (4.19)where the temperatures in the dual field theory at null infinity are related to the ADMmass and angular momentum of the spacetime as β u = πJ M − / and β φ = 2 πM − / . Forthis geometry a similar computation of the geodesic length as above yields the followingexpression for the geodesic length [60] L extrtot = √ M (cid:18) u ∂ + J M φ ∂ (cid:19) coth √ M φ ∂ ! − JM . (4.20)We are mainly interested in the non-rotating geometry, therefore putting J = 0 and writing β for β φ , we obtain L extrtot = 2 πu ∂ β coth (cid:18) πφ ∂ β (cid:19) , (4.21)and consequently the holographic entanglement entropy for the boundary interval A in thethermal GCFT is given by S A = c M πu ∂ β coth (cid:18) πφ ∂ β (cid:19) . (4.22)– 10 – Holographic entanglement negativity in flat Einstein gravity
In this section we detail the holographic constructions for computing the entanglementnegativity of bipartite states in the class of GCFT s dual to bulk asymptotically flatgeometries using results from the flat space holography described in the last section 4. Inparticular we will consider the asymptotically flat bulk spacetimes described by Einsteingravity for which the asymptotic symmetry analysis reveals that the dual GCFT s possessonly one non zero central charge c M (cf eq. (4.2)). We will first describe the holographicconstruction to compute the entanglement negativity of various bipartite states describedby a single interval in the dual GCFT . These include a single interval for a GCFT in its ground state, a GCFT describing a finite-sized system and a GCFT at a finitetemperature respectively. Next we turn our attention to the configuration of two adjacentintervals in the dual GCFT and establish holographic constructions to compute theentanglement negativity for the configurations described above using the results of flatspace holography. The case of the two disjoint intervals will require an analysis of thesemi-classical Galilean conformal blocks in the large central charge limit of the GCFT .We will postpone the discussion of such configurations till section 6. In this subsection we will consider various bipartite pure and mixed states consisting of asingle interval in a large system described by a GCFT . We start with the simplest con-figurations of bipartite pure states described by a single interval A ≡ [( x , t ) , ( x , t )] . Asdescribed in [55], the corresponding entanglement negativity involves a two-point correlatorof composite twist fields, given by E = lim n e → log (cid:10) Φ n e ( x , t )Φ − n e ( x , t ) (cid:11) . (5.1)We now apply the flat space holographic dictionary in eqs. (4.5) and (4.6) to obtain thefollowing form for the above twist correlator: (cid:10) Φ n e ( x , t )Φ − n e ( x , t ) (cid:11) = (cid:0)(cid:10) Φ n e / ( x , t )Φ − n e / ( x , t ) (cid:11)(cid:1) = e − χ ne/ L extr12 , (5.2)where χ n e / is the non-trivial scaling dimension of the twist fields Φ ± n e / and L extr12 is thelength of the bulk extremal curve homologous to the interval in question. In obtainingeq. (5.2), we have made use of the fact that for pure states the two point correlator ofcomposite twist operators factorizes into that of usual twist operators spanning half of thereplica geometry [55]. From eq. (3.2), in the replica limit n e → , we have χ n e / → − c M ,and therefore we obtain the following expression for the entanglement negativity of a purestate described by a single interval A in a holographic GCFT : E = 38 G L A , (5.3) Note that the negative scaling dimension of the twist fields Φ n e and Φ n e / in the replica limit n e → has to be understood only in the sense of an analytic continuation . – 11 –here we have made use of eq. (4.2). In the following, we will employ our holographicproposal in eq. (5.3) to compute the holographic entanglement negativity in some purequantum states in a holographic GCFT . Particularly we will investigate the case of asingle interval in the ground state of the GCFT , which is dual to the asymptotically flatpure Minkowski spacetime. Then we will turn our attention to the pure state described bythe single interval in a finite-sized system described by a GCFT compactified on a spatialcylinder, which is dual to the boost orbifold of Minkowski spacetime. We will find that theresults obtained using our holographic formula will reproduce the universal behaviour ofthe entanglement negativity for both of these configurations [55]. Later, in subsection 5.1.3we will consider the mixed state configuration of a single interval at a finite temperaturewhich involves a particular four-point twist correlator in the large central charge limit. To obtain the entanglement negativity in the bipartite pure state configuration describedby a single boosted interval in a GCFT (cf. footnote 3) at zero temperature we usethe results from the flat space holography reviewed in section 4 . At this point, we recallthat the computation of the length of the extremal geodesic in the dual gravity theory incylindrical coordinates ( u, φ ) results in eq. (4.4) [60]. In the planar coordinates in eq. (2.5)[45, 60] this translates to L extr = 2 x t . (5.4)Therefore, using the above expression for L extr , we obtain the entanglement negativity fora single interval in a GCFT at zero temperature from eq. (5.3) to be E = 38 G L A = c M x t . (5.5)This is precisely the result obtained in [55] using field theory methods, for c L = 0 . It isinteresting to note that we may recast the above expression for entanglement negativity inthe form E = 32 S A , (5.6)using the flat space analogue of the HRT formula in eq. (4.7), where S A is the entanglemententropy for the single interval A in the GCFT vacuum. This indicates that for pure statesthe holographic entanglement negativity is given by the Rënyi entropy of order half as inthe case of quantum information theory [6]. Next we turn our attention to the computation of holographic entanglement negativity forthe pure state configuration of a single boosted interval in a finite-sized system admittingperiodic boundary conditions described by a GCFT defined on an infinite cylinder withcircumference L φ . The bulk gravity dual is the global Minkowski orbifold described by themetric in eq. (4.9). The extremal geodesic length was computed in section 4 and is givenby L extr ij = 2 πu ij L φ cot (cid:18) πφ ij L φ (cid:19) , (5.7)– 12 –here u ij = u i − u j and φ ij = φ i − φ j are the differences in the coordinates of the endpointsof the boundary interval.We may now employ our holographic proposal in eq. (5.3) to compute the holographicentanglement negativity for the single boosted interval in a finite-sized system. Utilizingeq. (5.7) we obtain E = c M π u L φ cot (cid:18) πφ L φ (cid:19) , (5.8)which matches exactly with the universal part of the dual field theory result for c L = 0 [55].Again using the flat holographic HRT formula in (4.7) we may express the above result inthe form (5.6). The mixed state configuration described by a single interval in a finite temperature GCFT requires a more careful analysis. To start with we recall that a GCFT at a finite tem-perature is defined on an infinite cylinder of circumference equal to the inverse temperature β . The corresponding entanglement negativity involves a four-point twist correlator on theinfinite cylinder arising from the configuration of a single interval sandwiched between twoadjacent large but finite intervals [55]. The entanglement negativity may then be obtainedthrough a bipartite limit subsequent to the replica limit. Therefore in order to understandthe configuration described by a single interval at a finite temperature, we first consider afour-point twist correlator on the GCFT plane [55] (cf. eq. (2.9)): (cid:10) Φ n e ( x , t ) Φ − n e ( x , t ) Φ n e ( x , t ) Φ − n e ( x , t ) (cid:11) = k n e k n e / t ne t (2) ne F n e ( t, x/t ) t ∆ (2) ne × exp " − χ n e x t − χ (2) n e x t − χ (2) n e xt , (5.9)where k n e is a constant that depends on the full operator content of the theory. Thecorresponding weights of the twist fields Φ ± n e are given in eq. (3.2), from which one candetermine the weights of the composite twist fields Φ ± n e as [55]: ∆ (2) n e = 2∆ n e / = c L (cid:18) n e − n e (cid:19) , χ (2) n e = 2 χ n e / = c M (cid:18) n e − n e (cid:19) . (5.10)Equipped with eq. (2.7) for the two-point twist correlators, the universal part of the four-point function (which is dominant in the large central charge limit of the GCFT ) in eq.(5.9) can be factorized as (cid:10) Φ n e ( x , t ) Φ − n e ( x , t ) Φ n e ( x , t ) Φ − n e ( x , t ) (cid:11) = (cid:0)(cid:10) Φ n e / ( x , t )Φ − n e / ( x , t ) (cid:11)(cid:1) h Φ n e ( x , t )Φ − n e ( x , t ) i× (cid:10) Φ n e / ( x , t )Φ − n e / ( x , t ) (cid:11) (cid:10) Φ n e / ( x , t )Φ − n e / ( x , t ) (cid:11)(cid:10) Φ n e / ( x , t )Φ − n e / ( x , t ) (cid:11) (cid:10) Φ n e / ( x , t )Φ − n e / ( x , t ) (cid:11) + O (cid:18) c (cid:19) . (5.11)– 13 –ote that the arbitrary non-universal function of the GCFT cross ratios F n e ( t, x/t ) hasbeen neglected in the above factorization. We may justify this as follows. In the semi-classical limit ( G → ) of the bulk asymptotically flat gravity, the flat space holographicdictionary described in section 4 dictates that the dual GCFT theory has a large centralcharge c M → ∞ (cf. eq. (4.2)). Hence, we require a large central charge analysis of thetwist-correlator in eq. (5.9) for the entanglement negativity before giving its holographicdescription. In section 6 we will develop a monodromy technique to understand the largecentral charge behaviour of a specific four-point function of twist fields relevant to thecomputation of entanglement negativity for the mixed state configuration of two disjointintervals. There we will show that in the large central charge limit c M → ∞ the non-universal part of the four-point twist correlator is sub-leading in comparison to the universalpart. In the present context, we assume that the four-point twist correlator in (5.9) has asimilar large- c M structure and therefore the subleading contributions from the non-universalfunction F n e ( t, x/t ) in eq. (5.9) is neglected as shown by the O (1 /c ) contribution in eq.(5.11). Figure 1 : Schematics of the extremal geodesics anchored on different subsystems corresponding to thecomputation of entanglement negativity for a single interval in a finite temperature GCFT . The nullplanes descending from the boundary end-points are shown. The non-trivial contributions to the geodesiclengths land on the crossings of the corresponding null planes.
Now we utilize the flat space holographic dictionary in eqs. (4.5) and (4.6) to find thatthe four-point function in eq. (5.11) may be written in the following form (cid:10) Φ n e ( x , t ) Φ − n e ( x , t ) Φ n e ( x , t ) Φ − n e ( x , t ) (cid:11) = exp (cid:2) − χ n e L extr − χ n e / (cid:0) L extr + L extr + L extr − L extr − L extr (cid:1)(cid:3) , (5.12)where L extr ij denotes the length of the extremal geodesic in the bulk, which connects thepoints ( x i , t i ) and ( x j , t j ) on the boundary. Figure 1 shows the schematics for the config-– 14 –ration of a single interval A = [( x , t ) , ( x , t )] sandwiched between two large auxiliaryintervals B = [( x , t ) , ( x , t )] and B = [( x , t ) , ( x , t )] with B ∪ B ≡ B . As briefly al-luded to in section 4 the orientations of extremal geodesics anchored on different subsystemsfollow the construction in [60].From fig. 1 we identify thatL extr = L B , L extr = L A , L extr = L B , L extr = L A ∪ B , L extr = L A ∪ B , L extr = L A ∪ B . (5.13)In the replica limit n e → , we have from eq. (3.2) χ n e → and χ ne → − c M . Therefore,eq. (5.12) leads to the following expression for the holographic entanglement negativity E = lim B → A c G (2 L A + L B + L B − L A ∪ B − L A ∪ B ) . (5.14)In writing eq. (5.14) from eq. (5.12) we have first taken the replica limit n e → and subse-quently taken the bipartite limit B → A c in which the intervals B and B are extended toinfinity such that B ∪ B = A c [55]. We have also utilized the fact that for Einstein gravitythe asymptotic symmetry analysis following the Brown-Henneaux procedure [70] dictatesthat the central charges of the dual GCFT are given by (4.2). Therefore we concludethat the holographic formula for the entanglement negativity of a single interval in a finitetemperature dual GCFT relies on a specific linear combination of the lengths of bulkextremal surfaces homologous to the boundary intervals, as shown in fig. 1. Remarkablythe flat-holographic proposal for the entanglement negativity for asymptotically flat grav-ity in eq. (5.14) has exactly the same structure as in the AdS/CFT scenario obtained in[20]. Interestingly, implementing the flat-holographic counterpart of the HRT formula ineq. (4.7) we may rewrite our proposal in eq. (5.14) in the following form E = lim B → A c
34 (2 S A + S B + S B − S A ∪ B − S A ∪ B )= lim B → A c
34 ( I ( A ; B ) + I ( A ; B )) , (5.15)which shows a particular connection between two different entanglement measures, namelythe entanglement negativity and the mutual information, in holographic theories. Notehowever that these measures are quite distinct in the quantum information theory. It isimportant to mention here that this specific relation in eq. (5.15) seems to be uniqueto the configurations described by single intervals in holographic GCFT s at a finitetemperature.We now perform an explicit holographic computation of the entanglement negativityfor the finite temperature mixed state configuration described by a single Galilean boostedinterval in a thermal GCFT , using our proposal in eq. (5.14). The finite temperaturefield theory is dual to the Minkowski orbifold describing the locally flat geometry of FlatSpace Cosmologies (FSC). The length of the extremal geodesic in the FSC geometry withthe metric in eq. (4.19) is given in eq. (4.20). To relate with the field theory computations– 15 –n [55] we will consider the non-rotating geometry with J = 0 . In this non-rotating limit,we obtain another Minkowski orbifold, namely the boosted null orbifold. In this case,the expression for the length of the extremal geodesic homologous to the interval at theboundary in eq. (4.20) simplifies to eq. (4.21), namely L extr ij = √ M u coth √ M φ ij ! = 2 π u ij β coth (cid:18) πφ ij β (cid:19) , (5.16)where we have simply written β for β φ = 2 π M − / and u ij = u i − u j and φ ij = φ i − φ j are the differences in the coordinates of the endpoints of the interval at the boundary. Nowsubstituting for the extremal geodesic length in eq. (5.14) the holographic entanglementnegativity for a single interval in a GCFT at a finite temperature is obtained as E = c M (cid:20) π u β coth (cid:18) πφ β (cid:19) − π u β (cid:21) . (5.17)In obtaining eq. (5.17) we have used the understanding that B → A c corresponds to takingthe lengths of B and B to infinity. This matches exactly with the c L = 0 version of theuniversal part of the result obtained from the dual field theory in [55]. Although this standsas a strong consistency check for our proposal, it is important to mention that the analysisleading to eq. (5.14) relies on the large central charge behaviour of the dual GCFT anda bulk proof remains an open issue.Finally, it is interesting to note that using the flat space analogue of the HRT formula(4.7), the expression for the holographic entanglement negativity for a single interval in aGCFT at a finite temperature obtained in eq. (5.17) can be rewritten in the followingform E = 32 (cid:0) S A − S th (cid:1) , (5.18)where S A and S th are the entanglement entropy and the thermal entropy respectively, forthe single interval A in the holographic GCFT . Having computed the holographic entanglement negativity for various bipartite mixed statesinvolving a single interval in the dual GCFT , we now proceed to advance a similarholographic construction for the bipartite states described by two adjacent intervals in aholographic GCFT . As described before, the large central charge behaviour for theentanglement negativity in a GCFT indicates the plausibility of a holographic charac-terization for the entanglement negativity in a dual asymptotically flat spacetime throughflat space holography. To this end, we consider two Galilean boosted adjacent intervals A = [( x , t ) , ( x , t )] and B = [( x , t ) , ( x , t )] , as depicted in fig. 2, where the system A ∪ B is in a mixed state. We start with the following three-point twist correlator on the Note that the FSC geometry is defined for non-vanishing angular momentum J . Switching off theangular momentum leads to a Big-Bang like naked singularity [45]. The limit of J → has to be understoodin the sense of an analytic continuation. – 16 –CFT plane relevant to the computation of the entanglement negativity of two adjacentintervals [55] (cf. eq. (2.8)): (cid:10) Φ n e ( x , t )Φ − n e ( x , t )Φ n e ( x , t ) (cid:11) = k n e K Φ ne Φ − ne Φ ne t − ∆ (2) ne t − ∆ (2) ne t − (2∆ ne − ∆ (2) ne )13 exp " − χ (2) n e x t − χ (2) n e x t − (2 χ n e − χ (2) n e ) x t . (5.19)Utilizing equations (5.2) and (5.10) the three-point twist correlator in eq. (5.19) can berewritten in the following form (cid:10) Φ n e ( x , t )Φ − n e ( x , t )Φ n e ( x , t ) (cid:11) = K h Φ n e ( x , t )Φ − n e ( x , t ) i (cid:10) Φ n e ( x , t )Φ − n e ( x , t ) (cid:11) (cid:10) Φ n e ( x , t )Φ − n e ( x , t ) (cid:11)(cid:10) Φ n e ( x , t )Φ − n e ( x , t ) (cid:11) ! / , (5.20)where the constant K is given by K = k n e K Φ ne Φ − ne Φ ne k (1) p k (2) . (5.21)Now using the relation (cf. eq.(5.2)) (cid:10) Φ n e ( x , t )Φ − n e ( x , t ) i = (cid:0) h Φ n e / ( x , t )Φ − n e / ( x , t ) (cid:11)(cid:1) , (5.22)the universal part (which gives the dominant contribution to the entanglement negativityin the large- c M limit) of the 3-point twist correlator may be written as (cid:10) Φ n e ( x , t )Φ − n e ( x , t )Φ n e ( x , t ) (cid:11) = K h Φ n e ( x , t )Φ − n e ( x , t ) i (cid:10) Φ n e / ( x , t )Φ − n e / ( x , t ) (cid:11) (cid:10) Φ n e / ( x , t )Φ − n e / ( x , t ) (cid:11)(cid:10) Φ n e / ( x , t )Φ − n e / ( x , t ) (cid:11) . (5.23)Finally using the flat holographic dictionary in eqs. (4.5) and (4.6), we obtain the universalpart of the three-point twist correlator as (cid:10) Φ n e ( x , t )Φ − n e ( x , t )Φ n e ( x , t ) (cid:11) = exp (cid:2) − χ n e L extr − χ n e / (cid:0) L extr + L extr − L extr (cid:1)(cid:3) , (5.24)where L extr ij denotes the length of the extremal curve connecting the endpoints ( x i , t i ) and ( x j , t j ) of an interval on the boundary. In figure 2, we show the schematics of the extremalcurves anchored on the subsystems A , B and A ∪ B respectively, where we have identifiedL extr = L A , L extr = L B , L extr = L A ∪ B . (5.25)– 17 – igure 2 : Holographic construction for the computation of the entanglement negativity for two Galileanboosted adjacent intervals A = ( x , t ) and B = ( x , t ) . Extremal geodesics anchored on different subsys-tems are shown in: red - L extr ≡ L extr A , yellow - L extr ≡ L extr B , green- L extr ≡ L extr A ∪ B In the replica limit n e → , from eq. (3.2) we obtain χ n e → and χ ne → − c M . Notethat the large central charge limit has to be taken prior to the replica limit. This order oflimits is critical since the scaling dimension of the twist field Φ n e vanishes in the replicalimit and has to be understood in the sense of an analytic continuation. Hence, eq. (5.24)leads to the following expression for the holographic entanglement negativity for adjacentintervals E = 316 G (cid:0) L extr + L extr − L extr (cid:1) , (5.26)where we have again used the fact that for Einstein gravity the central charges of thedual GCFT are given by eq. (4.2). Therefore we conclude that the flat holographicentanglement negativity for two adjacent intervals in the class of holographic GCFT sthat we consider in the present article, is expressed in terms of a specific algebraic sum ofthe lengths of bulk extremal geodesics anchored on the endpoints of the intervals at theboundary. Remarkably the flat space holographic formula in eq. (5.26) has exactly thesame structure as its relativistic counterpart obtained in [21].It is interesting to note that the holographic entanglement negativity formula in eq.(5.26) may be recast, using the flat holographic HRT formula of [60] in eq. (4.6), in theform of another entanglement measure in such holographic GCFT s, namely the mutualinformation : E = 34 ( S A + S B − S A ∪ B ) = 34 I ( A : B ) . (5.27)– 18 –ote that this particular connection between the two different entanglement measures isspecial to the configuration of two adjacent intervals in holographic GCFT s. We start with the mixed state configuration of two adjacent intervals in the vacuum state ofthe boundary GCFT for which the bulk dual geometry is that of Minkowski spacetime.Substituting eq. (5.4) for the length of the extremal geodesic in pure Minkowski spacetimedual to the GCFT vacuum, in the expression (5.26) for the holographic entanglementnegativity for adjacent intervals, we obtain E = c M (cid:18) x t + x t − x t (cid:19) . (5.28)This matches exactly with the dual field theory result for c L = 0 in [55]. Next we turn our attention to the holographic computation of the entanglement negativityfor the bipartite mixed state configuration of two adjacent intervals in a thermal GCFT defined on an infinite cylinder compactified in the timelike direction. The correspondingbulk dual is the J = 0 FSC geometry described in section 4. Substituting eq. (5.16) for thelength of the extremal geodesic, in eq. (5.26), we obtain E = c M (cid:20) π u β coth (cid:18) πφ β (cid:19) + π u β coth (cid:18) πφ β (cid:19) − π u β coth (cid:18) πφ β (cid:19)(cid:21) . (5.29)Again this matches exactly with the dual field theory result for c L = 0 in [55]. Finally we compute the holographic entanglement negativity for the bipartite mixed stateconfiguration of two adjacent intervals in a finite-sized system described by a GCFT with periodic boundary conditions defined on a spatially compactified cylinder. The bulkdual is the global Minkowski orbifold in eq. (4.9) described in section 4. Utilizing the lengthfor extremal geodesics given in eq. (5.7), we obtain from eq. (5.26) E = c M (cid:20) π u L φ cot (cid:18) πφ L φ (cid:19) + π u L φ cot (cid:18) πφ L φ (cid:19) − π u L φ cot (cid:18) πφ L φ (cid:19)(cid:21) , (5.30)which is exactly the result in [55] obtained from the dual field theory computations, for c L = 0 . In this section we proceed to establish a holographic conjecture for computing the en-tanglement negativity in the context of flat space holography for the bipartite mixed stateconfiguration of two disjoint intervals in the dual GCFT . As briefly alluded to in subsec-tion 5.1.3, the computation of the entanglement negativity for such configurations involves– 19 –he large central charge analysis of a particular four-point twist correlator. From eq. (2.9),it is clear that the GCFT four-point function involves an arbitrary function of the crossratios which depends on the full operator content of the specific field theory under consid-eration. Also, for Einstein gravity in the bulk the semi-classical limit in the gravitationaltheory ( G → ) corresponds to the large central charge limit c M → ∞ in the dual GCFT .Motivated by these considerations, in the following we advance a holographic proposal forcomputing the entanglement negativity for two disjoint intervals in a GCFT .Before proceeding, we briefly review the computation of entanglement negativity fortwo disjoint intervals in the AdS /CFT scenario performed in [23]. In [25], the authorsdemonstrated that the entanglement negativity for two disjoint intervals in a CFT vanishesin the s-channel ( x → ) where the two intervals are far away, while remains non-trivialin the t-channel ( x → ) which corresponds to the two intervals being in close proximity.Inspired by these findings, the authors in [23] performed a monodromy analysis of thesemi-classical structure of the following four-point function in the vacuum state of a genericCFT : (cid:10) T n e ( z ) ¯ T n e ( z ) ¯ T n e ( z ) T n e ( z ) (cid:11) = z − ne z − ne x − ne G n e ( x ) , x = z z z z , (6.1)where T n e and ¯ T n e are respectively the twist and anti-twist fields inserted at the endpointsof the two disjoint intervals [ z , z ] and [ z , z ] . In eq. (6.1), x is the usual CFT cross ratioand G n e ( x ) is an arbitrary function of the cross ratio. Subsequently, it was found in [23] thatthe entanglement negativity for the two disjoint intervals in proximity obtained through thisprocedure has a holographic description in terms of a particular linear combination of thelengths of bulk spacelike geodesics homologous to specific subsystems.In the following we will utilize similar semi-classical techniques developed in [67] tocompute the entanglement negativity for two disjoint intervals A = [( x , t ) , ( x , t )] and A = [( x , t ) , ( x , t )] . This involves an analysis of the large-central charge behaviour ofthe following four-point twist-correlator in a GCFT vacuum : h Φ n e ( X ) Φ − n e ( X ) Φ − n e ( X ) Φ n e ( X ) i = t − ne t − ne t − ne exp (cid:20) − χ n e x t − χ n e x t − χ n e xt (cid:21) F ( t, xt ) . (6.2)In eq. (6.2), t , x/t are the non-relativistic cross ratios given in eq. (2.10) and F ( t, xt ) isa non-universal function of cross ratios that depends on the specific operator content ofthe field theory. In particular, we will focus only on the behaviour of the four-point twistcorrelator in eq. (6.2) in the t -channel defined as t → , x → , which renders the twodisjoint intervals in close proximity. We will be working with the GCFT s with only onenon-vanishing central charge c M for which the dual bulk geometry is described by Einsteingravity. We have employed a shorthand notation for describing the coordinates X i = ( x i , t i ) . This has to be contrasted with the t -channel x → , t → for the BMS field theory considered in[67]. We will use the methods developed in [67] to compute the Galilean conformal block utilizing theBMS /GCA correspondence briefly discussed in section 3 which essentially demonstrates the equivalenceof the two field theories under x ↔ t [44]. – 20 – .1 Four-point twist correlator at Large c M In this subsection we explicitly compute the large central charge limit c M → ∞ of theGalilean conformal block corresponding to the four-point function in eq. (6.2). To proceed,we recall some salient features of GCFT s relevant for the semiclassical large centralcharge analysis. There are two types of energy-momentum tensors in a GCFT andthe corresponding Galilean conformal Ward identities [67] look quite different from theirrelativistic counterparts. The finite GCA transformations t → f ( t ) , x → f ′ ( t ) x + g ( t ) , (6.3)are generated by the Nöether charges [67] M n = I dt T tx t n +1 , L n = I dt (cid:0) T tt t n +1 + ( n + 1) T tx t n x (cid:1) , (6.4)where T µν are the components of the GCFT energy-momentum tensor. Inverting theserelations, we obtain the components of the energy-momentum tensor as [67] M ≡ T tx = X n M n t − n − , L ≡ T tt = X n h L n + ( n + 2) xt M n i t − n − , (6.5)where L n and M n are the usual generators of GCA. Note that unlike the relativistic CFT sthe two independent components of the energy-momentum tensor L and M have distinctfunctional forms in a GCFT . This is a reflection of the fact that the GCA , unlikethe relativistic Virasoro algebra, does not decompose into two identical holomorphic andanti-holomorphic copies. The Galilean conformal Ward identities obeyed by these two ofenergy-momentum tensors are given by [67, 71]: hM ( x, t ) V ( x , t ) . . . V n ( x n , t n ) i = n X i =1 (cid:20) χ i ( t − t i ) + 1 t − t i ∂ x i (cid:21) h V ( x , t ) . . . V n ( x n , t n ) i , hL ( x, t ) V ( x , t ) . . . V n ( x n , t n ) i = n X i =1 " ∆ i ( t − t i ) − t − t i ∂ t i + 2 χ i ( x − x i )( t − t i ) + x − x i ( t − t i ) ∂ x i h V ( x , t ) . . . V n ( x n , t n ) i , (6.6)where V i are GCFT primaries, and χ i and ∆ i are the corresponding scaling dimensions.We wish to analyze the large- c M limit of the following four-point function of twist operatorsin the t -channel described by T → , X → h Φ n e ( X ) Φ − n e ( X ) Φ − n e ( X ) Φ n e ( X ) i = X α h Φ n e ( X ) Φ n e ( X ) | α i h α | Φ − n e ( X ) Φ − n e ( X ) i ≡ X α F α . (6.7)In eq. (6.7), F α are the GCA conformal blocks corresponding to the t -channel and we haveexpanded the the four-point function into a basis of GCFT primary operators denoted X , T are the usual cross ratios for the GCFT . – 21 –y the index α . Figure 3 shows this expansion of the four-point function (6.7) in terms ofGalilean partial waves. Figure 3 : Galilean conformal block expansion of a four-point twist correlator in the t -channel. The choiceof channel corresponds to two operators interchanging a GCA highest weight representation with the othertwo. The exchanged representation is labeled by α which denotes primary operators in the theory. In the large central charge limit c M → ∞ the blocks F α are expected to have anexponential structure similar to their relativistic counterparts [26, 72]. In the following,we are going to perform a geometric monodromy analysis in the semi-classical limit toobtain a large central charge expression for the Galilean conformal block F α . Recall thatunlike in the relativistic CFT s, the functional forms of the two energy-momentum tensorcomponents in eq. (6.5) for a GCFT are not identical and therefore we have to performa separate monodromy analysis corresponding to each of them. M In this subsection we will solve the differential equation for the expectation value of theenergy-momentum tensor component M . Subsequently we will utilize the monodromytechnique developed in [67] to obtain a partial expression for the Galilean conformal blockin eq. (6.7). Using the Ward identities in eq. (6.6) we obtain for the expectation value ofthe energy-momentum tensor M as M ( X i ; X ) ≡ hM ( X )Φ n e ( X ) Φ − n e ( X ) Φ − n e ( X ) Φ n e ( X ) ih Φ n e ( X ) Φ − n e ( X ) Φ − n e ( X ) Φ n e ( X ) i = X i =1 (cid:20) χ i ( t − t i ) + c M c i t − t i (cid:21) , (6.8)where the auxiliary parameters are given by c i = 6 c M ∂ x i log h Φ n e ( X ) Φ − n e ( X ) Φ − n e ( X ) Φ n e ( X ) i . (6.9) Note that the monodromy analysis can also be formulated using the GCA null vectors. The analysiswill be a bit more involved than the relativistic case due to the presence of the so called GCA multiplets[67]. Nevertheless the differential equations obtained via this technique will be the same as in the geometricmonodromy method. – 22 –he four-point function is not completely fixed by the conformal symmetry, and not all theauxiliary parameters c i are known. We will place the operators at t = 0 , t = 1 , t = ∞ and leave t = T free. Requiring that the expectation value M ( X i ; X ) vanishes as M ( T ; t ) ∼ t − as t → ∞ we obtain the conditions X i c i = 0 , X i (cid:16) c M c i t i + χ i (cid:17) = 0 , X i (cid:16) c M c i t i + 2 χ i t i (cid:17) = 0 . (6.10)Using the approximation that χ i ≡ χ Φ , being the conformal dimension of the so called’light’ operator Φ n e , vanishes when we take the replica limit n e → . This allows us todetermine three of the auxiliary functions in terms of the remaining one as c = c ( T − , c = − c T , c = 0 . (6.11)This leads to the following expression for the energy-momentum tensor expectation value c M M ( T ; t ) = c (cid:20) T − t + 1 t − T − Tt − (cid:21) . (6.12)The component M of the energy-momentum tensor transforms under a generic Galileanconformal transformation x → x ′ , t → t ′ in eq. (6.3) as [67] M ′ ( t ′ , x ′ ) = ( f ′ ) M ( t, x ) + c M S ( f, t ) , (6.13)where S ( f, t ) is the Schwarzian derivative for the coordinate transformation t → f ( t ) .Requiring the expectation value M ( X i ; X ) to vanish on the GCFT plane for the groundstate, this will lead to the condition S ( f, t ) = c (cid:20) T − t + 1 t − T − Tt − (cid:21) . (6.14)Eq. (6.14) is equivalent to the differential equation h ′′ ( t ) + 12 S ( f, t ) h ( t ) = h ′′ ( t ) + 6 c M M ( T, t ) h ( t ) , (6.15)with f = h /h , h and h being the two solutions of the above differential equation. Wewill solve this equation by the method of variation of parameters up to linear order in theparameter ǫ α = c M χ α . To zeroth order, setting M (0) = 0 , the solutions are given by h (0) ( t ) = 1 , t . (6.16)Therefore expanding up to linear order in ǫ α h i = h (0) i + ǫ α h (1) i , M = M (0) + ǫ α M (1) , (6.17)the differential equation to solve up to this order is given by h (1) ′′ i ( t ) = − c M M (1) ( T, t ) h (0) i ( t ) . (6.18)– 23 –fter solving eq. (6.18) we compute the monodromy of the solutions by going around thelight operators at t = 0 , as described in [67] which leads to the following monodromymatrix: M = πi c T ( T − πi c ( T −
1) 1 ! . (6.19)Now we utilize the monodromy condition for the three point twist correlator obtained in[67] r I − I πǫ α , (6.20)where I = tr M and I = tr M are invariant under global Galilean conformal transforma-tions. Using eq. (6.20) we can find the remaining auxiliary parameter c as c = ǫ α √ T ( T − . (6.21)Therefore the conformal block for the four-point function in eq. (6.7) may be obtained as: F α = exp (cid:20) c M Z c dX (cid:21) = exp (cid:20) χ α (cid:18) X √ T ( T − (cid:19)(cid:21) ˜ F ( T ) . (6.22)Expression (6.22) for the Galilean conformal block still has an unknown function ˜ F ( T ) .To determine ˜ F ( T ) we need to perform the monodromy analysis for the other energy-momentum tensor L , which we will do in the next subsection. For the particular four-pointfunction of twist correlators we consider in this section, we do not need to explore themonodromy for L . The reason is that, since the conformal dimensions ∆ Φ = ∆ n e ∝ c L ,they will vanish as long as we consider Einstein gravity for which eq. (4.2) gives c L = 0 . Therefore the monodromy problem for the energy-momentum tensor L becomes trivialand leads to ˜ F ( T ) = 1 . Nevertheless, in the next subsection we will explicitly solve thedifferential equation for L monodromy and show that this is indeed the case. L To get the full expression of the Galilean conformal block, we will next focus on the mon-odromy problem for the energy-momentum tensor L . We start with the expectation valueof the energy-momentum tensor L inside the four-point correlator [67] L ( X i ; X ) ≡ hL ( X )Φ n e ( X ) Φ − n e ( X ) Φ − n e ( X ) Φ n e ( X ) ih Φ n e ( X ) Φ − n e ( X ) Φ − n e ( X ) Φ n e ( X ) i . (6.23)Using the shorthands δ i = c M ∆ i and ǫ i = c M χ i , eq. (6.23) can be rewritten utilizing theWard identities in eq. (6.6) as c M L ( X i ; ( x, t )) = X i =1 " δ i ( t − t i ) − t − t i d i + 2 ǫ i ( x − x i )( t − t i ) + x − x i ( t − t i ) c i , (6.24)– 24 –here the auxiliary parameters c i are defined in eq. (6.9) and d i admit similar definitions[67]: d i = 6 c M ∂ t i log h Φ n e ( X ) Φ − n e ( X ) Φ − n e ( X ) Φ n e ( X ) i . (6.25)The smoothness of the expectation value L ( X i , X ) requires L ( T, t ) → t − as t → ∞ .Together with the freedom provided by global Galilean conformal transformations, this fixesall of the auxiliary parameters d i except one. Using the global Galilean conformal symmetry,we will place the operators at t = 0 , t = T , t = 1 , t = ∞ and x = 0 , x = X , x = 0 and x = 0 . This leads to the following values for three of the auxiliary parameters d i interms of the remaining one: d = c X + d ( T − − δ L ,d = c ( − X ) − d T + 2 δ L ,d = 0 , (6.26)where δ L = c M ∆ n e / and ǫ L = c M χ n e / denote the rescaled scaling dimensions of thetwist operator Φ n e . Substituting equations (6.26) and (6.9), into eq. (6.24) we obtain theexpectation value L ( X i , ( x, t )) as c M L ( X i ; ( x, t )) = − c X + d ( T − − δ L t + c X + d T − δ L t − c xt + c ( x − X )( t − T ) + c x ( t − − d t − T + 2 xǫ L t + δ L t + δ L ( t − + δ L ( t − T ) + 2 ǫ L ( x − X )( t − T ) + 2 xǫ L ( t − . (6.27)The transformation of the energy-momentum tensor L under the finite Galilean conformaltransformation in eq. (2.2), leads to the following differential equation c M L ( X i ; ( x, t )) = g ′ (cid:16) f ′ f ′′ − f ′′ ) (cid:17) + f ′ (3 g ′′ f ′′ − g ′′′ f ′ )2 ( f ′ ) − x (cid:16) f ′′ ) + f ′′′ ( f ′ ) − f ′′′ f ′ f ′′ (cid:17) f ′ ) . (6.28)As in [67], we now take the following combination of the expectation values c M ˜ L ( X i ; ( x, t )) = 6 c M (cid:2) L ( X i ; ( x, t )) + X M ′ ( X i ; ( x, t )) (cid:3) = c X (cid:18) − t − T ) − t + 1 t − (cid:19) − d ( T − T ( t − t ( t − T )+ δ L (cid:18) t + 1( t − T ) + 2 t − t − t − (cid:19) + 2 Xǫ L ( T − t ) . (6.29)Next we choose the ansatz g ( t ) = f ′ ( t ) Y ( t ) for the coordinate transformation to reduce thedifferential equation in (6.28) to the following form: c M ˜ L = − Y ′′′ − Y ′ c M M − Y c M M ′ . (6.30)– 25 –e can solve the above differential equation using the method described in [67] upto linearorder of ǫ α and δ α . The scaling dimensions of the light operator Φ n e vanishes when wetake the replica limit n e → . After computing the monodromy by going around the lightoperators at t = 0 , , we obtain the auxiliary parameter d as d = (1 − T ) Xǫ α − T − T δ α T − T / . (6.31)It is easy to check that the following is true from equations (6.9) and (6.25): ∂∂X d = ∂∂T c . (6.32)Finally, we obtain the full Galilean conformal block using eq. (6.25) as F α = exp (cid:20) χ α (cid:18) X √ T ( T − (cid:19)(cid:21) , (6.33)where we have used the fact that for c L = 0 , δ α vanishes. The complete Galilean conformalblock in eq. (6.33) exactly matches with the M monodromy result in eq. (6.22) for ˜ F ( T ) = 1 as anticipated before. c M limit In this subsection, we will use the large- c M limit of the t -channel Galilean conformal blockin eq. (6.33) to compute the entanglement negativity for the bipartite mixed state of twodisjoint intervals in proximity. Note from eq. (3.2) that, in the replica limit n e → thescaling dimension of the twist field Φ n e vanishes rendering it to be a light operator in thelarge- c M limit. Following [25] we now utilize the fact that in the t -channel T → , X → the dominant contribution to the four-point twist correlator in eq. (6.7) in the large- c M limit comes from the GCA conformal block corresponding to the primary field Φ n e . Notethat the twist operator Φ n e remains heavy in the replica limit, χ n e / → − c M . Therefore,the partial wave expansion for the four-point twist correlator in eq. (6.7) is dominated bythe exchange of Φ n e : F χ (2) ne = exp (cid:18) − c M X √ T ( T − (cid:19) . (6.34)Finally, using equations (3.6), (3.7) and (6.7), we obtain the negativity in the large c M -limitto be E = log (cid:16) F χ (2) ne (cid:17) ≈ c M X − T , (6.35)where, we have used the fact that in t − channel T → , and neglected the square-root inthe denominator. Note that this expression is in terms of the cross ratio in the t -channel, X/ (1 − T ) . In terms of the coordinates ( x i , t i ) of the end-points of the two disjoint intervalsunder consideration, the cross ratio is given by X − T = x t + x t − x t − x t . (6.36)– 26 –herefore the entanglement negativity for two disjoint intervals A = [( x , t ) , ( x , t )] and A = [( x , t ) , ( x , t )] in proximity is given by E = c M (cid:18) x t + x t − x t − x t (cid:19) . (6.37)We may now utilize the Galilean conformal transformations from the GCFT plane tothe spatially compactified cylinder to obtain the entanglement negativity in the finite-sizedsystem described by a GCFT defined on a cylinder with circumference L φ . The resultis E = c M π L φ (cid:20) u cot (cid:18) πφ L φ (cid:19) + u cot (cid:18) πφ L φ (cid:19) − u cot (cid:18) πφ L φ (cid:19) − u cot (cid:18) πφ L φ (cid:19)(cid:21) . (6.38)Finally we compute the entanglement negativity for the two disjoint intervals in a thermalGCFT living on a cylinder of circumference β , where β is the inverse temperature. Weobtain the following expression for the entanglement negativity E = c M π β (cid:20) u coth (cid:18) πφ β (cid:19) + u coth (cid:18) πφ β (cid:19) − u coth (cid:18) πφ β (cid:19) − u coth (cid:18) πφ β (cid:19)(cid:21) . (6.39)We will use these expressions for the entanglement negativity of two disjoint intervals inproximity to propose a holographic conjecture to obtain the same from the bulk computa-tions. In this subsection we will advance a holographic proposal for computing the entanglementnegativity of the bipartite mixed state configuration of two disjoint intervals in proximity ina holographic GCFT . According to the flat space holography, the GCFT is dual toa bulk asymptotically flat spacetime. As before, we consider two disjoint Galilean boostedintervals A = [( x , t ) , ( x , t )] and A = [( x , t ) , ( x , t )] in the ground state of a holo-graphic GCFT . The subsystem A = A ∪ A is in a mixed state, and the separationbetween A and A , denoted A s , belongs to the complementary subsystem B = A c . As theflat holographic proposals in equations (5.14) and (5.26) for a single and two disjoint inter-vals turned out to have exactly the same functional form as their relativistic counterpartsin [20, 21], we expect a similar holographic connection for the present configuration as well.We will make use of the monodromy computations in the previous subsection 6.1.3 tojustify our proposal. To this end we start with the following expression for the two pointtwist correlator in a holographic GCFT on the plane (cf. eq. (2.7)): h Φ n e ( x , t )Φ − n e ( x , t ) i ∼ exp (cid:18) − χ n e x t (cid:19) , (6.40)where we have used eq. (4.2) and eq. (3.2) to set ∆ n e = 0 . Now we utilize the holographic– 27 –ictionary in eqs. (4.5) and (4.6), to write eq. (6.34) as h Φ n e ( x , t ) Φ − n e ( x , t ) Φ − n e ( x , t ) Φ n e ( x , t ) i ≃ exp (cid:20) c M (cid:18) x t + x t − x t − x t (cid:19)(cid:21) = exp h c M (cid:0) L extr + L extr − L extr − L extr (cid:1)i , (6.41)where in the second equality we have made use of eq. (5.4). We now propose, based onthe monodromy computations in section 6.1.3, the following conjecture for the holographicentanglement negativity of two disjoint intervals in proximity located at the null infinity ofthe bulk asymptotically flat spacetime dual to a GCFT : E = 316 G (cid:0) L extr + L extr − L extr − L extr (cid:1) = 316 G (cid:0) L extr A ∪ A s + L extr A s ∪ A − L extr A ∪ A ∪ A s − L extr A s (cid:1) , (6.42)where c L = 0 and c M = G . Once again we observe that the holographic entanglement neg-ativity for the mixed state configuration of two disjoint intervals in a holographic GCFT involves a specific linear combination of the lengths of bulk extremal curves homologous tothe intervals as shown in figure 4. Remarkably our flat holographic conjecture in eq. (6.42)has exactly the same structure as its relativistic counterpart in the AdS /CFT scenarioobtained in [23, 29]. Figure 4 : Schematics of the holographic construction for the computation of entanglement negativity of twodisjoint intervals. The entanglement negativity is obtained via a specific linear combination of the lengthsof the bulk extremal curves situated at the crossings of the null planes descending from the endpoints of thetwo intervals. – 28 –t is interesting to note that, in the limit of adjacent intervals x → ǫ , where ǫ is theUV cut-off (L extr A s → in the bulk), we get back our formula for two adjacent intervals ineq. (5.26). This serves as a strong consistency check of our proposal. Now we make useof the flat version of the HRT formula in eq. (4.7) to recast our formula for holographicentanglement negativity in the following instructive form E = 34 ( S A ∪ A s + S A s ∪ A − S A ∪ A ∪ A s − S A s )= 34 ( I ( A ∪ A s ; A ) + I ( A s ; A )) . (6.43)Therefore we see that our holographic conjecture relates two very different entanglementmeasures, namely, entanglement negativity which is the upper bound of distillable entangle-ment, and the mutual information which measures entanglement correlation between twosubsystems. Again, this particular connection seems unique for the specific configurationof two disjoint intervals on the boundary field theory. Interestingly, in the limit of adjacentinterval A s → ∅ we get back the adjacent formula in eq. (5.27).In the following, we are going to employ our holographic conjecture to compute theentanglement negativities in various configurations described by two disjoint intervals inproximity in different mixed states of a holographic GCFT . Remarkably our formulareproduces the universal behaviour of the holographic entanglement negativity at the largecentral charge limit of the holographic GCFT . We start with the mixed state configuration of two disjoint intervals A = [( x , t ) , ( x , t )] and A = [( x , t ) , ( x , t )] in the ground state of a holographic GCFT . The dualbulk geometry is that of pure Minkowski spacetime. Utilizing eq. (5.4) for the lengthof the extremal geodesics in locally Minkowski geometry, one obtain for the holographicentanglement negativity from eq. (6.42) as E = 38 G (cid:18) x t + x t − x t − x t (cid:19) = c M (cid:18) l + l s t + t s + l + l s t + t s − l + l + l s t + t + t s − l s t s (cid:19) , (6.44)where we have denoted l = x − x , l s = x − x and l = x − x for the lengths of therespective intervals (cf. figure 6) and similarly for t , t and t s . remarkably this matchesexactly with the large central charge behaviour of the entanglement negativity in eq. (6.37)obtained using the monodromy method in subsection 6.1.3. Considering the adjacent limit l s → ǫ and t s → ǫ (where ǫ is the UV cut-off) and taking the leading order terms in ǫ , weget back the result for entanglement negativity for adjacent intervals in eq. (5.28). Next we will consider the mixed state configuration of two disjoint intervals in a thermalGCFT living on a cylinder compactified in the timelike direction with circumference β. – 29 –he dual spacetime is the locally FSC geometry described in subsection 5.1.3. Substitutingeq. (5.16) for the length of the extremal curve in FSC geometry in our holographic con-jecture in eq. (6.42) we obtain for the holographic entanglement negativity of two disjointintervals at a finite temperature E = 3 π Gβ (cid:18) u coth (cid:18) πφ β (cid:19) + u coth (cid:18) πφ β (cid:19) − u coth (cid:18) πφ β (cid:19) − u coth (cid:18) πφ β (cid:19)(cid:19) = c M πβ " ( t + t s ) coth (cid:18) π ( l + l s ) β (cid:19) + ( t + t s ) coth (cid:18) π ( l + l s ) β (cid:19) − ( t + t + t s ) coth (cid:18) π ( l + l + l s ) β (cid:19) − t s coth (cid:18) πl s β (cid:19) , (6.45)where the lengths of the respective intervals are denoted by l = u − u , l s = u − u and l = u − u , and the times are given by t , t and t s . Again this matches exactly withthe field theory computations at large central charge limit in eq. (6.38). We may take theadjacent limit l s → ǫ and t s → ǫ , to show that the leading order expression matches exactlywith the result for two adjacent intervals given in eq. (5.29). Finally we turn our attention to the holographic computation of the entanglement negativityfor two disjoint intervals in a finite-sized system obeying periodic boundary conditionsdescribed by a GCFT living on a cylinder of circumference L φ compactified along thespatial direction. The bulk dual is again asymptotically flat and is described by the globalMinkowski orbifold metric in eq. (4.9). We now employ the expression for the extremalgeodesic length in such spacetimes from eq. (5.7) to obtain the following expression for theentanglement negativity of the mixed state configuration described by two disjoint intervalsin a finite-sized system as E = 3 π GL φ (cid:18) u cot (cid:18) πφ L φ (cid:19) + u cot (cid:18) πφ L φ (cid:19) − u cot (cid:18) πφ L φ (cid:19) − u cot (cid:18) πφ L φ (cid:19)(cid:19) = c M πL φ " ( t + t s ) cot (cid:18) π ( l + l s ) L φ (cid:19) + ( t + t s ) cot (cid:18) π ( l + l s ) L φ (cid:19) − ( t + t + t s ) cot (cid:18) π ( l + l + l s ) L φ (cid:19) − t s cot (cid:18) πl s L φ (cid:19) . (6.46)Remarkably this again matches exactly with the field theory result in eq. (6.39) obtainedthrough large central charge computations in subsection 6.1.3. Again in the adjacent limitdescribed by l s → ǫ and t s → ǫ , we get back the adjacent intervals result in eq. (5.30). In the previous sections we have computed the holographic entanglement negativity in thecase of Einstein gravity in the bulk for which the dual GCFT at the boundary had only– 30 –ne non-vanishing central charge c M . At this point, we recall the fact that the represen-tations of the GCA algebra are labelled by the quantum numbers ∆ and χ . Thereforea vanishing c L would correspond to ∆ = 0 which describes a spinless massive particlepropagating in the asymptotically flat bulk spacetime.In this section we will incorporate the effects of a non-zero c L , and hence a non-zero ∆ , in the bulk in order to see the agreement with the field theory results in [55] moreclosely. We expect that a non-vanishing ∆ would introduce a spin for the massive particle.In this context we modify the bulk picture by introducing Topologically Massive Grav-ity (TMG) [59, 60, 63–66] which contains a gravitational Chern-Simons (CS) term. ThisChern-Simons term arises due to a gravitational anomaly present in the relativistic CFT whose İnönü-Wigner contraction leads to the GCFT s considered in the present article.From the perspective of the bulk, the dual operation to this parametric contraction on theboundary corresponds to taking the flat limit of the bulk AdS geometry. Therefore theflat-holographic connection between TMG in asymptotically flat spacetimes and GCFT swith non-vanishing c L and c M comes from two equivalent parametric contractions of eachsector in the original TMG-AdS /CFT correspondence [59, 60, 64–66].We start by briefly reviewing the salient features of TMG in AdS spacetimes. Theaction of TMG in AdS is the sum of the usual Einstein-Hilbert term, the cosmologicalconstant term and a gravitational Chern-Simons term [59, 66] : S TMG = S EH + 1 µ S CS = 116 πG Z d x √− g " R + 2 ℓ + 12 µ ε αβγ (cid:16) Γ ρασ ∂ β Γ σγρ + 23 Γ ρασ Γ σβη Γ ηγρ (cid:17) , (7.1)where µ has mass dimension one and describes the coupling of the CS-term, and ℓ is theAdS radius. In the limit µ → ∞ one recovers Einstein gravity. The asymptotic symmetryanalysis of TMG in AdS shows that the algebra of the modes of the asymptotic Killingvectors is isomorphic to two copies of Virasoro algebra with left and right moving centralcharges [59, 66]: c + TMG = 3 ℓ G (1 + 1 µℓ ) , c − TMG = 3 ℓ G (1 − µℓ ) . (7.2)Now we will go to asymptotically flat spacetime by taking the flat limit ℓ → ∞ leading to theflat space TMG. Remarkably the asymptotic symmetry group analysis at null infinity leadsto the Galilean conformal algebra, with both central charges non-vanishing [45, 48, 59, 60]: c L = 3 µG , c M = 3 G . (7.3)Alternatively, these central charges can be obtained from AdS by taking İnönü-Wignercontraction [59]: c L = c + TMG − c − TMG , c M = ( c + TMG + c − TMG ) /ℓ . From eq. (7.3) it is easy tosee that in the limit µ → ∞ we get back Einstein gravity in asymptotically flat spacetime. This should be contrasted with the Chern-Simons gauge theory of 3d gravity put forward by Witten[61]. – 31 – .1 Extrapolating the holographic dictionary
In [66], the authors computed the holographic entanglement entropy for a CFT with grav-itational anomaly using the theory of topologically massive gravity in AdS . It was foundthat the difference in the left and right moving central charges of the anomalous CFT gives rise to a non-trivial spin of the twist operators in the replica manifold, which in thecontext of AdS /CFT , corresponds to a massive spinning particle of mass m = χ and spin s = ∆ moving in the bulk geometry of TMG-AdS . As easily seen from the action in eq.(7.1), the Chern-Simons term is unaffected by the flat limit ℓ → ∞ and therefore the abovediscussion remains valid in the flat-holographic scenario as well [60]. The action of such aparticle was found to be [60, 66]: S flat-TMG = Z C ds (cid:18) χ q η µν ˙ X µ ˙ X ν + ∆ (˜ n. ∇ n ) (cid:19) + S constraints , (7.4)where ˜ n and n are unit space-like and time-like vectors respectively, both normal at thetrajectory of the particle X µ , and S constraints is an action imposing these constraints throughLangrange multipliers [60, 66]. In eq. (7.4) C denotes the worldline of the particle. Theaction (7.4) introduces two new vectors in the the 3-dimensional bulk, while the constraintaction S constraints imposes five constraints, leading to a single new degree of freedom. Thissets up a normal frame to each point in the bulk as shown in fig. 5, and particle worldlinesget broadened in the shape of ribbons [66]. Figure 5 : The topological Chern-Simons term in the TMG action introduces a normal frame defined bytwo auxiliary normal vectors n and ˜ n at each point on the worldline of a massive spinning particle. Figuremodified from [66]. – 32 –he equations of motion reveal that this is not a true degree of freedom in the sensethat the variations of the new vectors n and ˜ n along the worldline X µ does not affect theaction (7.4) [60, 66]. It is also interesting to note that straight lines governed by ¨ X µ = 0 in locally Minkowski spacetimes are still solutions of the equations of motion in the TMGbackground [60]. It is important to note that our holographic constructions for computingthe entanglement negativity in terms of bulk geodesics rely heavily on the straight-linenature of the geodesics. To proceed, we note that in order to compute the entanglemententropy from the bulk perspective in a AdS/CFT setting, one considers the notion of thegeneralized gravitational entropy [17]. The computation of generalized gravitational entropyinvolves a replication of the dual gravitational geometry in the replica index n followed bya quotienting through the replica symmetry Z n . In the quotient spacetime of the replicatedgeometry, there are conical defects along the entangling surfaces, namely at the endpointsof the boundary interval. We now propose, following [60, 66] that the two-point functionof the twist fields inserted at the endpoints of the interval on the boundary of the quotientgeometry is given by the exponential of the on-shell action of a massive spinning particlewith mass m n = χ n and spin s n = ∆ n . For such a particle propagating along an extremalworldline in the bulk geometry from a point x i with a normal vector n i to a point x f withnormal vector n f , the two-point twist correlator has the form: h Φ n e ( ∂ A )Φ − n e ( ∂ A ) i = e − χ ne S EHon-shell − ∆ ne S CSon-shell , (7.5)where S EHon-shell = q η µν ˙ X µ ˙ X ν = L extr ( x i , x f ) , (7.6)and S CSon-shell is the topological Chern-Simons contribution to the on-shell action. As de-scribed before, the effect of this topological action is to broaden the worldline in the shapeof a ribbon as the vectors n and ˜ n in eq. (7.4) define a normal frame to the curve C . Ineq. (7.5) the Chern-Simons contribution to the on-shell action in eq. (7.4) is given by thetwist in the ribbon-shaped worldline as the particle moves along it [59, 60, 66]: S CSon-shell = Z C ds (˜ n. ∇ n ) = cosh − ( − n i . n f ) . (7.7)Equation (7.7) essentially computes the boost ∆ η required to drag the orthonormal framegenerated by the vectors ( ˙ X, n i , n f ) from the point x i to x f . In the following subsection we will perform the computations of the spinning two-pointcorrelators for different bulk geometries in flat space-TMG using the modified holographicdictionary in eqs. (7.5) to (7.7). With this generalized expression for the two point twist-correlator in eq. (7.5) all our previous analysis in section 5 will simply follow and lead tomodified formulae for the holographic entanglement negativity in GCFT dual to bulkgeometries governed by TMG . All these results may be recast in the factorised Wilson line prescription in the Chern-Simons formula-tion of 3d gravity developed in [66]. – 33 – .2 Two-point correlator of twist fields with spin
We start with TMG in a pure Minkowski spacetime. A schematics of the bulk geometrycorresponding a single interval A = [( x , t ) , ( x , t )] in the boundary GCFT is shown infig. 6. We have two bulk normal vectors n ∂i erected at each of the bulk points y i ( i = 1 , descending from the endpoints ( u i , φ i ) of the interval on the boundary, which were chosenin [60] to be pointed along the directions of the corresponding null rays γ i : ˙ γ = ∂ r (cid:12)(cid:12)(cid:12) γ = ∂ t + cos φ ∂ x + sin φ ∂ y , ˙ γ = ∂ r (cid:12)(cid:12)(cid:12) γ = ∂ t + cos φ ∂ x + sin φ ∂ y . (7.8)Since these two vectors are null, the authors in [60] introduced two timelike vectors: n = 1 ǫ ˙ γ − ǫ γ . ˙ γ ˙ γ , n = 1 ǫ ˙ γ − ǫ γ . ˙ γ ˙ γ . (7.9)With these definitions we obtain from eq. (7.7) in the ǫ → limit S CSon-shell = ∆ η = cosh − ( − ˙ γ . ˙ γ ǫ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) − γ . ˙ γ ǫ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 2 log (cid:18) ǫ sin φ (cid:19) . (7.10) Figure 6 : Bulk setup for computing two-point correlator of twist fields with non-zero spin. There areboundary normal vectors n ∂i on each of the black points on the asymptotic boundary. The black pointsare on the null curves descending from these boundary points and they are equipped with normal vectors n i ∝ ∂ r (cid:12)(cid:12)(cid:12) γ i . Figure modified from [60]. In eq. (7.10) the boost ∆ η may be interpreted as the difference in the twist of thetwo endpoints of the ribbon-like geometries induced by the topological term in eq. (7.7).Therefore the two-point spinning twist correlator in eq. (7.5) in the case of pure Minkowskispacetime dual to a GCFT in its ground state is given by h Φ n e ( ∂ A )Φ − n e ( ∂ A ) i = (cid:18) ǫ sin φ (cid:19) − ne exp − χ n e u tan φ ! , (7.11) ( u i , φ i ) are the cylindrical coordinates related to the planar coordinates ( x i , t i ) via eq. (2.5). – 34 –here we have used eq. (4.4) for the extremal geodesic length and ∂ i A = ( u i , φ i ) denotesthe entangling surfaces, namely, the endpoints of the interval at the boundary.Next we proceed to compute the boost in the case of non-rotating FSC geometry. Inthat case the bulk null vectors in eq. (7.8) become (cf. eq. (4.10)) ˙ γ = β π cosh (cid:18) πφ β (cid:19) ∂ t + β π sinh (cid:18) πφ β (cid:19) ∂ x − β π ∂ y , ˙ γ = β π cosh (cid:18) πφ β (cid:19) ∂ t + β π sinh (cid:18) πφ β (cid:19) ∂ x − β π ∂ y . (7.12)Therefore using eqs. (7.7) and (7.10) we obtain ∆ η FSC = 2 log (cid:18) βπǫ sinh πφ β (cid:19) . (7.13)Similar computations in the case of TMG in global Minkowski orbifold geometries yields ∆ η GM = 2 log (cid:18) L φ πǫ sin πφ L φ (cid:19) . (7.14)In the following subsections we will utilize equations (7.10), (7.13) and (7.14) for the twistsin the ribbon to compute the topological CS contribution to the holographic entanglementnegativity for different sub-interval geometries in a holographic GCFT . In this subsection we will generalize the proposals (5.3) and (5.14) for computing entangle-ment negativity of various bipartite pure and mixed state configurations described by a sin-gle interval in a GCFT to incorporate the non-vanishing c L effects. To this end, we firstconsider the pure state configurations described by a single interval A = [( x , t ) , ( x , t )] in the ground state of a GCFT at zero temperature. To proceed, we replace eq. (5.3)for the two-point function of twist operators by the corresponding expression with non-zerospin in eq. (7.11). Now, using the modified holographic dictionary in equations (7.5), (7.6)and (7.7) we write the two-point function of the composite twist operators Φ n e inserted atthe endpoints of the single interval A = [( x , t ) , ( x , t )] as: (cid:10) Φ n e ( x , t )Φ − n e ( x , t ) (cid:11) = exp (cid:2) − χ n e / L extr − n e / ∆ η (cid:3) , (7.15)where L extr is the length of the extremal ribbon-shaped curve anchored on the entanglingsurfaces, and ∆ η denotes the difference in the twist at the enpoints of the ribbon. Nowthe entanglement negativity for the pure state configuration described by the single intervalin the GCFT vacuum may be obtained from eq. (5.1) as E = 38 G (cid:18) L extr + 1 µ ∆ η (cid:19) , (7.16)where we have used equations (3.2) and (7.3) and subsequently took the replica limit. Inthe following, we will make use of the holographic formula in eq. (7.16) to compute the– 35 –olographic entanglement negativity for the bipartite pure state configurations described bya single interval in the vacuum state of a holographic GCFT as well as for a GCFT describing a system of finite size. Later, we will consider the mixed state configuration of asingle interval at a finite temperature which involves an analysis of a particular four-pointtwist correlator in the large central charge limit in the spirit of subsection 5.1.3. We start with the simplest pure state configuration of a single interval in the vacuum state ofa holographic GCFT at zero temperature for which the dual bulk geometry correspondsto the pure Minkowski spacetime. Utilizing the transformations (2.5), the CS-contributionto the two-point function in eq. (7.10) may be written in the planner coordinates as: ∆ η = 2 log (cid:18) t ǫ (cid:19) . (7.17)We now substitute equations (4.4) and (7.17) in eq. (7.16) to obtain the holographicentanglement negativity as E = c L (cid:18) t ǫ (cid:19) + c M x t , (7.18)where we have used eq. (7.3) for the central charges of the holographic GCFT . Re-markably, we have reproduced the universal part of the complete result obtained in [55] viareplica technique. Next we move on to the computation of the holographic entanglement negativity for thebipartite pure state configuration of a single interval in a GCFT describing a finite-sizedsytem endowed with periodic boundary conditions. The corresponding bulk geometry isdescribed by the global Minkowski orbifold with metric (4.9). Using the expression for thecorresponding length of the extremal curve in eq. (5.7), and the twist in eq. (7.14), weobain the holographic entanglement entropy from eq. (7.16) as E = c L (cid:20) L φ πǫ sin (cid:18) πφ L φ (cid:19)(cid:21) + c M π u L φ cot (cid:18) πφ L φ (cid:19) . (7.19)This matches exactly with the universal part of the complete field theory result in [55]. Finally we focus on the mixed state configuration of a single interval at a finite temperature.The field theory is described by a thermal GCFT on a cylinder compactified along thetimelike direction with circumference β . As described in subsection 5.1.3, the definitionof the holographic entanglement negativity for this configuration involves two auxiliaryintervals B and B sandwiching the single interval A . This leads to a four-point twistcorrelator which admits a large central charge factorization of the form (5.11). For a thermalGCFT with unequal non-vanishing central charges (7.2), the dual gravitational theory– 36 –s described by topologically massive gravity in FSC geometries. For such GCFT s, usingthe modified flat-holographic dictionary in eq. (7.5), the four-point twist correlator in eq.(5.11) has the large-central charge structure: (cid:10) Φ n e ( x , t ) Φ − n e ( x , t ) Φ n e ( x , t ) Φ − n e ( x , t ) (cid:11) = exp h − χ n e L extr − χ n e / (cid:16) L extr + L extr + L extr − L extr − L extr (cid:17) − ∆ n e ∆ η − ∆ n e / (cid:16) η + ∆ η + ∆ η − ∆ η − ∆ η (cid:17)i , (7.20)where L extr ij are the lengths of the extremal ribbon-shaped curves anchored on various sub-systems constituted by the single interval A and the auxiliary intervals B , B , and η ij arethe corresponding twists in the ribbons. Taking the replica limit n e → followed by thebipartite limit B ∪ B → A c , and utilizing the definitions of the central charges in eq.(7.2), we obtain the following modified formula for computing the holographic entangle-ment negativity for the bipartite mixed state configuration of a single interval in a thermalGCFT with both central charges non-vanishing: E = lim B → A c G (2 L A + L B + L B − L A ∪ B − L A ∪ B ) , (7.21)where we have defined L X = L extr X + 1 µ ∆ η X . (7.22)where X is the specific subsystem under consideration. We now compute the holographicentanglement negativity for a single interval located at the asymptotic null infinity of thegeometry described by TMG in FSC. The holographic computations are identical to thosein subsection (5.1.3) for the extremal geodesic lengths L extr ij and the remaining contributionto the holographic entanglement negativity comes from the Chern-Simons term as E CS = c L (cid:18) log (cid:20) βπǫ sinh (cid:18) πφ β (cid:19)(cid:21) − πφ β (cid:19) . (7.23)Together with the Einstein gravity result eq. (5.17), the total holographic negativity be-comes E = c L (cid:20) log (cid:20) βπǫ sinh (cid:18) πφ β (cid:19)(cid:21) − πφ β (cid:21) + c M (cid:20) π u β coth (cid:18) πφ β (cid:19) − π u β (cid:21) . (7.24)The above expression for the holographic entanglement negativity exactly matches with theuniversal part of the complete field theory result obtained in [55] using the replica technique.We may also rewrite eq. (7.24) in the instructive form eq. (5.18). Next we turn our attention to the bipartite mixed state configuration of two adjacent inter-vals in a GCFT with unequal non-vanishing central charges. The holographic entangle-ment negativity for the case of Einstein gravity in the bulk was discussed in subsection 5.2.In this subsection we utilize the modified dictionary in eq. (7.5) to advance a holographic– 37 –roposal for computing the entanglement negativity for the mixed state configuration of twoadjacent intervals living at the null infinity of the geometries described by flat space TMG.From equations (7.21) and (7.22), it is easy to see that the expression for the holographicnegativity in such scenarios is simply obtained by replacing L extr X by L X , for each subsystem X . Therefore our proposal for the entanglement negativity for two disjoint intervals reads(cf. eq. (5.26)): E = 316 G ( L A + L A − L A ∪ A ) , (7.25)with L X given in eq. (7.22). We start with the bipartite mixed state configuration of two adjacent intervals in the vacuumstate of the boundary GCFT . To compute the holographic entanglement negativity, weuse eq. (7.10) in our modified holographic entanglement negativity formula eq. (7.25) toobtain to topological contribution to the entanglement negativity as E CS = 316 G µ (∆ η A + ∆ η A − ∆ η A ∪ A )= c L (cid:20) t t ǫ ( t + t ) (cid:21) , (7.26)where ǫ is identified as the UV cut-off. The expression for the total holographic entangle-ment negativity, after including the Einstein gravity result eq. (5.28), becomes E = c L (cid:20) t t ǫ ( t + t ) (cid:21) + c M (cid:18) x t + x t − x t (cid:19) , (7.27)which matches exactly with the universal part of the field theory result in [55]. We next compute the holographic entanglement negativity for the bipartite mixed stateconfiguration of two adjacent intervals in a finite temperature GCFT . Here, the bound-ary theory is defined on an infinite cylinder compactified in the timelike direction leadingto a finite temperature GCFT and the dual gravitational theory is decribed by TMG inFSC geometry. The Chern-Simons contribution to the holographic entanglement negativity,using eq. (7.25) and eq. (7.13), is given by E CS = c L βπǫ sinh (cid:16) πφ β (cid:17) sinh (cid:16) πφ β (cid:17) sinh (cid:16) π ( φ + φ ) β (cid:17) . (7.28)The total holographic entanglement negativity after including the Einstein gravity resulteq. (5.29) becomes E = c L βπǫ sinh (cid:16) πφ β (cid:17) sinh (cid:16) πφ β (cid:17) sinh (cid:16) π ( φ + φ ) β (cid:17) + c M " π u β coth (cid:18) πφ β (cid:19) + π u β coth (cid:18) πφ β (cid:19) − π u β coth (cid:18) πφ β (cid:19) . (7.29)– 38 –q. (7.29) correctly reproduces the universal part of the result obtained in [55] using fieldtheoretic methods. Finally, we focus on the computation of the holographic entanglement negativity for abipartite mixed state configuration of two adjacent intervals in a GCFT describing asystem with finite size. The boundary theory is described by a GCFT on an infinitecylinder compactified in the spatial direction with circumference L φ . The Chern-Simonscontribution to the holographic entanglement negativity is obtained using eq. (7.14) andeq. (7.25) as E CS = c L βπǫ sin (cid:16) πφ L φ (cid:17) sin (cid:16) πφ L φ (cid:17) sin (cid:16) π ( φ + φ ) L φ (cid:17) . (7.30)The expression for the total holographic entanglement negativity after including the Ein-stein gravity result eq. (5.29) becomes E = c L L φ πǫ sin (cid:16) πφ L φ (cid:17) sin (cid:16) πφ L φ (cid:17) sin (cid:16) π ( φ + φ ) L φ (cid:17) + c M " π u L φ cot (cid:18) πφ L φ (cid:19) + π u L φ cot (cid:18) πφ L φ (cid:19) − π u L φ cot (cid:18) πφ L φ (cid:19) . (7.31)This matches exactly with the universal part of the complete field theory result obtainedin [55] using replica technique. Finally we compute the holographic entanglement negativity for the bipartite mixed stateconfiguration of two disjoint intervals in a GCFT with both central charges non-vanishing.The entanglement negativity for such configurations involves a four-point function of spin-ning twist operators. Therefore to obtain the entanglement negativity via field theoreticmethods, we need a semi-classical monodromy analysis of the four-point twist correlator fornon-trivial central charges c L and c M . In this paper, instead of developing a correspondinggeometric monodromy technique, we propose a generalization of the holographic conjecturein eq. (6.42). For two disjoint intervals living at the null infinity of the geometries describedby TMG in asymptotically flat spacetimes, we replace L extr by L in eq. (6.42) to obtain E = 316 G (cid:0) L extr + L extr − L extr − L extr (cid:1) , (7.32)where L is defined in eq. (7.22). In the following, we will apply our conjecture to differentbipartite mixed state configurations involving two disjoint intervals in a GCFT and findagreement with the İnönü-Wigner contractions of the corresponding CFT results in [29].– 39 – .5.1 Two disjoint intervals at zero temperature We start with bipartite mixed state configuration of two disjoint intervals in proximity A = ( x , t ) and A = ( x , t ) in the ground state of a holographic GCFT which isdual to TMG in pure Minkowski spacetime. Using eq. (7.10) and eq. (4.4), we obtain theholographic entanglement negativity from eq. (7.32) as E = c L (cid:18) t t t t (cid:19) + c M (cid:18) x t + x t − x t − x t (cid:19) . (7.33)For comparison, we reproduce the corresponding expression in the context of AdS /CFT from [29]: E = c (cid:18) z z z z (cid:19) + ¯ c (cid:18) ¯ z ¯ z ¯ z ¯ z (cid:19) , (7.34)where we have allowed for unequal central charges c and ¯ c for the left and right movingsectors, respectively. Now following eq. (2.1) we take the İnönü-Wigner contractions [40–42] z = t + ǫx , ¯ z = t − ǫx, (7.35)to obtain, up to first order in ǫ , E = ( c + ¯ c )8 log (cid:18) t t t t (cid:19) + ǫ ( c − ¯ c )8 (cid:18) x t + x t − x t − x t (cid:19) . (7.36)Using eq. (2.11), we see that the GCFT result in eq. (7.33) is exactly reproduced. Thisserves as a strong consistency check of our proposal. Next, we consider the bipartite mixed state configuration of two disjoint interval in a finitetemperature GCFT . The boundary theory is living on a cylinder compactified in thetimelike direction with circumference β and the dual bulk theory is described by globalFSC geometry. Using eq. (4.20) and eq. (7.14) in the holographic entanglement negativityformula in eq. (7.32) gives E = c L βπ sinh (cid:16) πφ β (cid:17) sinh (cid:16) πφ β (cid:17) sinh (cid:16) πφ β (cid:17) sinh (cid:16) πφ β (cid:17) + c M " π u β coth (cid:18) πφ β (cid:19) + π u β coth (cid:18) πφ β (cid:19) − π u β coth (cid:18) πφ β (cid:19) − π u β coth (cid:18) πφ β (cid:19) . (7.37)It is easily verified that the above expression matches perfectly with the İnönü-Wignercontractions of the corresponding result obtained in the context of AdS /CFT in [29]. Finally, we consider the bipartite mixed state configuration described by two disjoint in-tervals in the proximity in a finite-sized system. Using eq. (7.14) and eq. (5.7) we get the– 40 –olographic entanglement negativity from eq. (7.32) as E = c L L φ π sin (cid:16) πφ L φ (cid:17) sin (cid:16) πφ L φ (cid:17) sin (cid:16) πφ L φ (cid:17) sin (cid:16) πφ L φ (cid:17) + c M " π u L φ cot (cid:18) πφ L φ (cid:19) + π u L φ cot (cid:18) πφ L φ (cid:19) − π u L φ cot (cid:18) πφ L φ (cid:19) − π u L φ cot (cid:18) πφ L φ (cid:19) . (7.38)This may also be seen to match with the İnönü-Wigner contractions of the correspondingresult obtained in the context of AdS /CFT in [29]. To summarize, we have established a holographic construction to obtain the entanglementnegativity for bipartite states in GCFT s dual to bulk (2 + 1) dimensional asymptoticallyflat Einstein gravity and topologically massive gravity (TMG). For the former the bulkasymptotic symmetry analysis leads to dual GCFT s with central charges c L = 0 , c M =0 . In this context, we have obtained the holographic entanglement negativity for variousbipartite pure and mixed states in a GCFT utilizing our construction. These include thepure state of a single interval dual to a bulk (2 + 1) -dimensional Minkowski spacetime andthat in a finite-sized system dual to a bulk global Minkowski orbifold. The correspondingmixed state of a single interval at a finite temperature is dual to a bulk non rotating flatspace cosmology described by a null orbifold. Subsequently, the holographic entanglementnegativity for the mixed state configuration of two adjacent intervals in a GCFT wascomputed utilizing our construction. Our results for these bipartite states exactly reproducethe corresponding replica technique results in the large central charge limit.Following the above computations, we used the geometric monodromy method [67] inthe BMS field theory to find the large central charge behaviour of the entanglement nega-tivity for the mixed state configuration of two disjoint intervals in the GCFT . Utilizingthe M and L monodromy for each of the two distinct components of the energy-momentumtensor leads to second and third-order differential equations for the four-point twist corre-lator. Solving these equations, it was possible to obtain the dominant conformal block forthe four-point twist correlator in the t-channel describing the intervals in proximity witheach other. This leads us to the entanglement negativity for the mixed state configurationunder consideration for zero and finite temperature and also finite-sized system describedby a GCFT at its large central charge limit. Subsequently we advance a construction tocompute the holographic entanglement negativity for this mixed state configuration in zeroand finite temperature and also finite-sized system described by a GCFT dual to appro-priate bulk gravitational configurations. Interestingly our results exactly match with thecorresponding replica technique results in the large central charge limit obtained throughthe geometric monodromy analysis described above. This constitutes a strong consistencycheck of our holographic construction for the mixed state configuration in question and mayalso be extended to the other configurations discussed here in a straightforward fashion.– 41 –urthermore we demonstrate that in the limit of the two disjoint intervals being adjacent weretrieve the corresponding holographic entanglement negativity for two adjacent intervalswhich further demonstrates the consistency of our holographic construction.Subsequently we have extended our construction to obtain the holographic entangle-ment negativity for the bipartite states described earlier, in a GCFT with non zero c L dual to a bulk flat space topologically massive gravity. This describes massive particleswith spin propagating in the bulk and also renders both the scaling dimensions for thetwist fields to be non zero. Our results for the adjacent and the single intervals matchexactly with the corresponding replica technique results in the dual GCFT with boththe central charges being non zero. For the mixed state configuration of disjoint intervalsour results for the holographic entanglement negativity obtained through our constructionis identical to the İnönü-Wigner limit for the corresponding replica techniques results fora relativistic CFT which constitutes a consistency check. This is because a completereplica technique analysis and a large central charge limit for the above configuration is anontrivial outstanding issue.It is well known that flat space chiral gravity is a limit of the flat space topologicallymassive gravity for which the Newton constant G is taken to be infinity and such that theproduct of G with the coupling constant µ of the topological term in the action is heldfixed. The corresponding dual GCFT in this case has the other central charge c L = 0 and the GCA is identical to the chiral part of a (relativistic) Virasoro algebra. In theAppendix utilizing our proposal, we have computed the holographic entanglement negativityfor the bipartite pure and mixed state configurations described by single, adjacent, anddisjoint intervals in the dual GCFT mentioned above and the results are similar to thoseobtained earlier for a generic TMG. We would like to emphasize here that our constructiondescribed in this work addresses the significant issue of the characterization of mixed stateentanglement for a class of dual GCFT in flat space holography. It is to be mentionedhere that the replica technique computation of the entanglement negativity and the largecentral charge monodromy analysis for the mixed state configuration of two disjoint intervalsin a GCFT dual to a bulk TMG is an outstanding nontrivial issue that requires attention.Furthermore it has been shown in the literature that the GCFT dual to a bulk flatspace chiral gravity is related to a conformal quantum mechanics (CFT ). This is anextremely interesting open avenue to explore in the future as described by the progress inthe corresponding AdS /CFT correspondence. We hope to return to these exciting issuesin the near future. A Holographic entanglement negativity in flat chiral gravity
In this appendix, we will discuss a special case of flat-space TMG, namely the conformalChern-Simons gravity (also called flat space chiral gravity) [73]. The dual boundary theoryis described by GCA with central charges c L = 24 k, c M = 0 , where k is the Chern-Simonslevel. The action for conformal Chern-Simons gravity is given by S CSG = k π Z d x √− g " ε αβγ Γ ρασ (cid:16) ∂ β Γ σγρ + 23 Γ σβη Γ ηγρ (cid:17) , (A.1)– 42 –ith G → ∞ , keeping µG = k fixed (cf. eq. (7.1)).Note that in this case the two-point correlator in (7.5) only gets a contribution fromthe Chern-Simon term. In this context, we are looking at a massless spinning particle inthe bulk. All the previous analysis in flat-space TMG will now follow with c M = 0 andholographic entanglement negativity formula for a single interval A becomes E = lim B → A c kµ X A + X B + X B − X A ∪ B − X A ∪ B ) , (A.2)where we have defined X = 1 µ ∆ η . (A.3)Therefore, using eqs. (7.10) and (A.2), the holographic entanglement negativity for a singleinterval in the ground state of a chiral GCFT is obtained as E = 38 G X A = c L (cid:18) t ǫ (cid:19) . (A.4)Similarly, we may compute the holographic entanglement negativity for a single interval A = [( x , t ) , ( x , t )] at a finite temperature or for finite-sized systems using eq. (7.13)and eq. (7.14). The results match exactly with those in the flat-space TMG case as wellas the field theory results in [55] with c M = 0 which strongly substantiates our holographicproposals.Next, we modify our holographic entanglement negativity proposal for two adjacentintervals A = [( x , t ) , ( x , t )] and A = [( x , t ) , ( x , t )] at the boundary of a manifoldaccommodating flat chiral gravity: E = 3 kµ X A + X A − X A ∪ A ) . (A.5)Using eq. (7.10) for the spinning contribution in pure Minkowski spacetime, eq. (A.5)yields the following expression for the holograpic entanglement negativity for two disjointintervals in the chiral GCFT vacuum: E CS = 3 k η A + ∆ η A − ∆ η A ∪ A )= c L (cid:20) t t ǫ ( t + t ) (cid:21) , (A.6)which matches exactly with the c M = 0 version of the dual field theory result in [55].Similarly, we can obtain the holographic entanglement negativity for adjacent intervals at afinite temperature and for finite-sized systems in the present scenario using eq. (7.13) andeq. (7.14).Finally, for two disjoint intervals A = [( x , t ) , ( x , t )] and B = [( x , t ) , ( x , t )] inproximity in the chiral GCFT with c M = 0 , we write E = 3 k X + X − X − X ) , (A.7)– 43 –ith X given in eq. (A.3). Using eq. (7.10) and eq. (A.7), we obtain the holographicentanglement negativity in the ground state to be E = c L (cid:20) t t t t (cid:21) (A.8)Similarly, we can obtain negativity for two disjoint intervals at a finite temperature and forfinite-sized systems using eq. (7.13) and eq. (7.14). Once again the results match with theflat-space TMG results with c M = 0 as well as the corresponding İnönü-Wigner limits ofthe relativistic field theory results [29]. References [1] G. Vidal and R. F. Werner,
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