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Holographic entanglement entropy and gravitational anomalies

Castro, A.; Detournay, S.; Iqbal, N.; Perlmutter, E. DOI 10.1007/JHEP07(2014)114 Publication date 2014 Document Version Final published version Published in The Journal of High Energy Physics

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Citation for published version (APA): Castro, A., Detournay, S., Iqbal, N., & Perlmutter, E. (2014). Holographic entanglement entropy and gravitational anomalies. The Journal of High Energy Physics, 2014(7), [114]. https://doi.org/10.1007/JHEP07(2014)114

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Download date:26 Sep 2021 JHEP07(2014)114 Springer July 5, 2014 July 23, 2014 : June 22, 2014 d : : , Accepted Received Published 10.1007/JHEP07(2014)114 doi: and Eric Perlmutter c Published for SISSA by [email protected] , Nabil Iqbal b [email protected] , . 3 Stephane Detournay, a 1405.2792 The Authors. c AdS-CFT Correspondence, Anomalies in Field and String Theories

We study entanglement entropy in two-dimensional conformal field theories , [email protected] Cambridge, CB3 0WA, U.K. E-mail: [email protected] Universit´eLibre de Bruxelles and InternationalCampus Solvay Plaine Institutes, C.P. 231, B-1050Kavli Bruxelles, Institute for Belgium Theoretical Physics,Santa University Barbara CA of 93106, California, U.S.A. DAMTP, Centre for Mathematical Sciences, University of Cambridge, Institute for Theoretical Physics, UniversityScience Park of 904, Amsterdam, Postbus 94485,Physique et 1090 Th´eorique Math´ematique, GL Amsterdam, The Netherlands b c d a Open Access Article funded by SCOAP ArXiv ePrint: demonstrate that theDixon minimization equations of of motion this forsimple a action spinning examples results particle and in in demonstrate three agreement dimensions. the with We Mathisson-Papapetrou- CFT work out calculations. several Keywords: represented by a gravitationalanomaly Chern-Simons term broadens in the theanomalous Ryu-Takayanagi bulk minimal contribution action. to worldline We theribbon. into show CFT that a The entanglement the entanglement ribbon, entropy functionala and is may spinning also given that particle be by the — interpreted the that as twist is, the in worldline an this action anyon for — in three-dimensional curved spacetime. We Abstract: with a gravitational anomaly. In theories with gravity duals, this anomaly is holographically Alejandra Castro, Holographic entanglement entropy andanomalies gravitational JHEP07(2014)114 5 10 3 13 9 8 33 17 7 38 30 – 1 – 10 22 23 3 7 29 24 21 7 1 19 28 25 2.3.4 Holographic CFTs at large central charge 2.3.1 Ground2.3.2 state Ground2.3.3 state, boosted interval Finite temperature and angular potential 4.4 Thermal entropies 3.3 Spinning particles from an action principle 4.1 Poincar´eAdS 4.2 Poincar´eAdS, boosted4.3 interval Rotating BTZ black hole 3.1 Gravitational Chern-Simons3.2 term Holographic entanglement entropy: Coney Island 2.1 Gravitational anomalies 2.2 and R´enyi entanglement2.3 entropy in the presence Applications of a gravitational anomaly 1 Introduction Holography relates gravitational physics tolarge the number dynamics of of degrees quantuman field of effective theories freedom. geometry with from a The field-theoreticalHowever, degrees precise recent of mechanism work freedom governing has remains the somewhat shown emergence unclear. that of the concept of quantum entanglement is likely to E Chern Simons formulation of TMG B Details of cone action C Details of variation of probeD action Details of backreaction of spinning particle 5 Discussion A Conventions 4 Applications 3 Holographic entanglement entropy in topologically massive gravity Contents 1 Introduction 2 Quantum entanglement in 2d CFTs with gravitational anomalies JHEP07(2014)114  R (1.2) (1.1) log c 3 ]. These = 2 [ EE entanglement S define a normal n and ˜ n in AdS. Boundary intuition , ) ]: n anyon 1 gravitational anomalies · ∇ n (˜ , N ds min G C 4 A Z characterizing the spatial region. This is a = µ – 2 – 3 X 1 particle, i.e. an ρ G EE 4 S = spinning anom in the quantum field theory, defined as the von Neumann S X anomaly instead. Another motivation is holographic: as we will appears in the coefficient of the gravitational Chern-Simons term, 1 − µ . This equation relates two very primitive ideas on either side of the duality conformal X . The anomalous contribution to the entanglement entropy is now given by denotes the area of the bulk minimal surface anchored on the boundary of the is a curve in the three dimensional bulk and the vectors of a spatial region min ribbon C A Interestingly, just as a the length of a bulk spatial geodesic may be understood as the We find that the sensitivity of the field theory to its coordinate system manifests itself We have two main motivations in mind for studying entanglement entropy in such theo- In this paper we study holographic entanglement entropy in a different class of theories, and measures the anomaly coefficient in the dual two-dimensionalaction theory. of a massivestood point in particle terms of propagating the in actionfor of the this a bulk, result this comes new from term the may fact be that under- in two-dimensional conformal field theories with into a the twist in this ribbon: where frame to this curve. see, in theoriesbulk with geometry. anomalies, entanglement entropy probes other aspectsholographically in of an the elegantly geometric dual attached way: it to gives the physical meaning Ryu-Takayanagi to worldine, a essentially normal frame broadening the minimal worldline ries: from a field-theoretical pointbetween of anomalies view, and we entanglement entropy. would Indeed likefor the to celebrated the better formula understand entanglement the entropy interplay offield an theory interval in is the an vacuumtrolled of example by a the of two-dimensional conformal a precise relation between these two concepts, but is con- the manifold on which itand is right placed. central charges The appearing anomaly ina is the present gravity local in dual, conformal any algebras. the theoryChern-Simons holographic If with such term. three-dimensional unequal a left bulk theory admits action will contain a gravitational degrees of freedom implied by holography. namely, two-dimensional conformal fieldtheories theories suffer with from amay breakdown be of understood stress-energy as conservation a at sensitivity the of the quantum theory level: to this the coordinate system used to describe where spatial region — geometry and entanglement — and thus may provide insight into the reorganization of entropy entropy of thevery reduced complicated density and matrix nonlocalgravity duals observable. governed by However,the Einstein in entanglement gravity, field entropy, there due theories exists to with a Ryu and semiclassical simple Takayanagi and [ elegant formula for play a key role in an eventual understanding of this emergence. Consider the JHEP07(2014)114 . 3 2 and (2.2) (2.1) zz . Many 5 πT 2 ≡ will denote the ) z R ( ). In terms of the E T z z, and L E ...... , which acts as a source for the + + ij g (0) (0) z z we use two-dimensional conformal T ∂ ∂T 2 ]. We also discuss its interpretation in . 6 + + ij δW 2 δg 2 ) to various simple spacetimes of interest, (0) (0) z z g T T 2 1.2 √ – 3 – + ]. + = , the stress tensor is defined as 2 2 4 / 4 11 / ij Z , z z L R T c c 10 log , ∼ ∼ 2 − we apply ( = (0) (0) 4 ) run over the spacetime coordinates ( ) by a careful evaluation of the action of a cone in topologically T T W ) ) ] to instead compute entanglement entropy from bulk Wilson z z 7 i, j 1.2 ( ( T T , where ( presents a very different reformulation of the results of this paper, using ij E T , respectively. The algebras take the familiar forms z z ]: we believe that these works are incomplete in that they do not deal sufficiently 9 πT , 2 ]. Thus our results may also be viewed as the construction of the worldline action 8 5 – ≡ One can couple any QFT to a background metric A 2d CFT has Virasoro symmetries on the left and right that are generated by Previous discussion of entanglement entropy in topologically massive gravity in- Appendix A brief outline of the paper follows. In section 3 ) z ( stress tensor QFT generating functional Associated to each algebraleft is and an right independent Virasoro central zero charge. modes, respectively. theories (CFTs). More complete accountsdimensions, of can the subject, be including found generalization in to e.g. higher [ holomorphic and anti-holomorphicT components of the stress tensor, 2 Quantum entanglement in2.1 2d CFTs with Gravitational gravitational anomalies anomalies We begin by recalling some basics about gravitational anomalies in 2d conformal field the approach studied inlines [ in the Chern-Simons formulation of topologically massive gravity. cludes [ carefully with the subtle lack of diffeomorphism invariance expected in these theories. Dixon equations. Inholographically deriving section results that agree withWe the conclude field with theory a calculations brief fromimportant section discussion details and of summary the of derivations have future been directions relegated in to section various appendices. field theory techniques togravitational anomalies; study no and R´enyi use is entanglementwe made derive entropy the of bulk in holographic result duality field ( inmassive gravity, theories that following section. techniques with developed In in section terms [ of the action of a spinning particle and the relation to the Mathisson-Papapetrou- nonzero spin proportional toof the this anomaly action coefficient. inMathisson-Papapetrou-Dixon detail, We equations discuss demonstrating of motion the in for bulk particular aity that interpretation spinning [ particle its in variation general results relativ- of in an the anyon in usual curved space. We anticipate further applications of this formalism. a gravitational anomaly, the twist fields used to compute entanglement entropy acquire JHEP07(2014)114 , ω ∧ and (2.6) (2.7) (2.8) (2.9) (2.3) (2.4) (2.5) ω b,i a + 1 ω ]. dω 15 = , R 14 Γ or ∧ ]. The covariant cur- 15 , the metric transforms as i , spin connection ξ Γ + Γ i a d e . ] for a fuller description of the = , . ]. m jl 10 l R ab Γ R, R 13 k m dx , ? ∂ ∂ j ∧ k k 12  ∂ . i  k ij b R kl ) e c Y   j dx b π ξ j a R i − b +  ( . The choice of diffeomorphism or Lorentz anomaly c α 48 e 3 π i L R ij ∇ − a c c − T dω e π 96 – 4 – L  = = − c = 96 i ijkl , obeys ) = L ij a =  c ij ij R R b ˜ δg δe T ba T 1 2  = ∧ ˜ T R ij = = b − T ) are frame indices; see [ i ij ab ab T ∇ ˜ i R T a, b ) = Tr( ∇ R ( 4 (the “Bardeen-Zumino polynomials”) [ P ij , where ( Y ab R the rotation parameter. Non-invariance of the CFT generating functional ba α in a 2d CFT, one cannot consistently promote the background metric to a is the Hodge star of the curvature 2-form, − ) can be derived from a modified generating functional living in 2+1 dimen- R ab = c R 2.8 ? ab 6= α The anomaly presented thus far is the so-called “consistent” form, obtained from a The anomaly so defined can be presented in two ways. On the one hand, it is man- L This is succinctly captured in the language of the anomaly polynomial. For a gravitational anomaly in 1 c a 2d CFT, the polynomialcorresponds is to therespectively. choice of whether to express the curvature as for suitably chosen rents ( sions. Framed as ain diffeomorphism covariant form anomaly, becomes for instance, the non-conservation equation be done by adding a local counterterm to thegenerating CFT functional generating that functional satisfies [ a the “covariant” Wess-Zumino form consistency of conditions. theimprovement There anomaly, terms, in exists which covariance of the stress tensor is restored via where The presentation of the anomaly is a matter of choice: shifting from one to the other can the vielbein transforms as with implies that this stress tensor, call it On the other hand, theunder anomaly local can frame appear rotations asmetric. — a Working in Lorentz in anomaly which the — casecurvature tangent that 2-form the frame, is, stress we an have tensor anomaly passage the is vielbein between conserved metric but and not frame sym- formulations. Under an infinitesimal frame rotation, Non-invariance of the CFT generating functional implies dynamical field: that is,theories the with CFT chiral suffers matter from and a holomorphic gravitational CFTs anomaly. [ Examplesifest include as a diffeomorphismconserved. anomaly, Under in an infinitesimal which diffeomorphism case generated by the stress tensor is symmetric but not If JHEP07(2014)114 : + is M n and S i .Φ (2.10) (2.12) (2.13) (2.11) u n disjoint Z / Z n ∈ C , and have a n , for example, N ]. Z spatial intervals C 16 N = M -point correlator on at the endpoints . The branched covers ]. For C N + . 20 , by tracing out degrees of , ρ n Z 17 ]. / n ρ . , in a state defined by some given C 19 – , + M ) log n i 17 ρ ρ . v ) through the scaling dimensions of the ( n − Tr ) n 1 2.13 − )Φ Z log Tr Z i twist operators Φ ( u = n ( N to positive real values, one can take the limit n = + – 5 – 1 − -dependent genus; when , and computes their 2 S n n Φ i n 1 1 v ρ → of twist operators inserted at the boundary of the N =1 = i Y n Tr n M * = lim S - and = N EE n S , one inserts ρ , with branch cuts along the entangling surface. The precise } i M Tr , v i u . The latter is equivalent to computing a QFT partition function { at the endpoints -dependence also enters ( n n − is the Ricci scalar; note the absence of explicit factors of Γ. In a ij g 1). Rather than computing partition functions on arbitrary genus Riemann ij − R n = 1)( act oppositely with respect to the cyclic permutation symmetry is a positive integer. In a 2d QFT, the spatial region is the union of − R n are Riemann surfaces of − anomaly as a correlation function on N 1 to obtain the entanglement, or von Neumann, entropy, We focus on 2d CFTs henceforth, calling a generic CFT Computing entanglement entropy via the replica trick amounts to computing the R´enyi n on a branched cover of M twist operators Φ ρ → = ( n As the subscript denotes,and Φ the correlator isnontrivial OPE. evaluated The in thetwist orbifold operators, theory to be derived momentarily. surfaces directly, we will employ thebounded twist by field endpoints perspective [ N Tr entangling region. of g Z relation is This technique isgral. applicable in Alternatively, any one state can that eschew can this be topological defined perspective, using and a instead functional compute inte- This procedure involves ambiguitiestinuation, in based principle; on we precedentdetails will in on employ similar this the construction, 2d naive we CFT direct analytic entanglement the con- calculations. reader toentropy For [ for further arbitrary where intervals. Upon analyticallyn continuing Consider a QFT living ondensity some matrix. Riemannian manifold Onefreedom exterior can to form somedefined a spatial as region reduced at density fixed Euclidean matrix, time. The entropy R´enyi holographic context, the usual AdS/CFT dictionaryin identifying boundary the data CFT with sources generating functional naturally leads2.2 to the consistent form R´enyi and [ entanglement entropy in the presence of a gravitational where JHEP07(2014)114 , ) C . z R ( c R and c T 6= 1 (2.15) (2.16) (2.17) (2.14) − n, L L c R c i ) ∝ . w , is a complex ( s 2 2 T w  h  ) ) — has genus zero v v 1 ] and elsewhere; the − − . n, v v 17 w w ). Before replication, R  − )( )( − . 1 n u u u u , we find spinning, anyonic u, v  − R − − 1 n h n ], it is sufficient to consider a w w (  − ( − 17 L R treated in [ n = 0, one finds h c      1 limit of entropy, R´enyi the twist R C = − c i 1 24 n 1 , n R ) s 24 c L z → = − c − ( /n 1 = L n n T n c = h   R  v u = , with endpoints ( R L 24 C 24 c c − − C , s and spin i , h – 6 – w ) w = = R z    ( h 1 n n n 1 n T Z Z + = h / / n n − L n n − z . The replica surface — call it Z Z C C . To compute the scaling weights of the twist operators, h n / / i i n z C n n ) )  C C  v v i i = R ) ) L c v, v, 24 c v v ( ( , these techniques still apply. The presence of a gravitational ). On the other, these expectation values can be traded for M + − − 24 R v, v, = ( ( c L )Φ )Φ − − L c . 2.14 6= u u h n )Φ )Φ L u, u, u u c ( ( ∆ = have nonzero spin controlled by the anomaly coefficient: + + , then u, u, 1 ( ( ± )Φ )Φ + + n, , coordinatized by Φ Φ w w R C h h ( ( T T h h 7→ . On one hand, one can compute the conformal transformation of the stress 1 1 n, , and can be conformally mapped back to the complex plane. If ) are defined through variations of the original CFT generating functional. Notice n, z n R ( i R ) T By deriving twist operator quantum numbers from the CFT stress tensor, this method To derive this fact, following the original method of [ For a CFT with w ( ∆ is independent of T anomaly. We are implicitly workingand with the consistent form ofthat the anomaly, in because computing entanglementoperators entropy are as analytically the continueds/ from anyons to spinless bosons. However, the ratio This derivation is a simplekey extension point of is the that case the calculation holomorphically factorizes. fully accounts for the dependence of and R´enyi entanglement entropy on the gravitational Defining conformal dimension ∆ = twist fields as advertised: These equations areand nothing anti-holomorphic stress but tensors,dimensions the acting conformal on Ward Virasoro identities primary operators for with the scaling holomorphic operators themselves. Recalling that maps consider the one-pointh functions of thetensor CFT under stress the tensor mapcorrelation components, ( functions with twist operators in the orbifold theory, by definition of the twist lives on the complexfor plane, all coordinate on anomaly does not spoil thetransformation use properties. of functional This integrals; brings onethe us need twist only to fields be the Φ careful central about novelty: their in a CFTsingle with interval in the ground state of a CFT JHEP07(2014)114 Z ) a as a z n (2.22) (2.18) (2.19) (2.20) (2.21) ρ ), this ↔ z 2.13 , we find an ). For holo- iθ Re 2.20 , unless otherwise = R z ). From ( R ∈ . , c R L . For a CFT in its ground c  − . ε z . iθ . n  Z R  / . c . n ε R R and Φ log 6 C − c This is familiar from the thermal i   + R L 6= 12 c c ε 2 R (0) L log − somewhere off the real axis. We find  + + c R ]. We will also comment on more general ), we note that only the imaginary part of log  C )Φ  c log ε z 21 ε R R ∈ + 12 ( 2.12 R  z  c + L c Φ 6 – 7 – + log h log  L ) yield L 1 c n R 12 c log c ], with n = ] in a CFT on a line, with 2.18 6 + 1 + , z 1   − L  ,R 1 n EE c 1 S = = = n ) and ( 1 + ]. The anomaly makes no contribution to ( n S  EE S 22 S 2.17 = n S , so ( see a contribution from the anomaly. If we define C do = ) shows, in the absence of Euclidean time reflection symmetry ( M 2.22 As ( We work with an interval [0 In fact, the anomalous piece had to be purely imaginary on general grounds. Thinking of Tr 2 Euclidean CFT partition function need not beCFT real. partition function oncan a ever Riemann receive surface, an asan anomalous in imaginary contribution. ( piece Thus, of the the Euclidean frame entropy. R´enyi rotation to a complex interval must lead to This setup is just aobservables rotation are of sensitive the previousdiffeomorphism to one, anomaly, the but observables in choice are adefine of theory a sensitive with frame. constant to time a slice. the Lorentz Equivalently, anomaly, choice in of the coordinates language used of to the In this case, we entanglement entropy graphic CFTs, this state is dual to2.3.2 Poincar´eAdS. Ground state,Now boosted consider interval a complex interval [0 and the entanglement entropy is easily obtained, This was first derived in [ We will reinstate the UV cutoff as2.3.1 needed for dimensional analysis. Ground state In this case, entanglement entropies in holographic CFTs with noted. Thus, we wish to compute With these results in andR´enyi hand entanglement we proceed entropy to inmerely easily 2d involves derive computing CFTs universal the with results two-pointstate, for function arbitrary of single on ( Φ interval a circle,the or correlator is at fixed finite by conformal temperature symmetry and [ angular potential, the functional form of 2.3 Applications JHEP07(2014)114 . , the 1 t (2.26) (2.24) (2.23) ). We 0 + R , whereas x L = 0 all the t R = ) yields a real z t, 2.22 , − : this immediately κ x R c Re = − 24 z ], as shown in figure L = c 23  R κ . . For a holographic CFT, this + ) is consistent with a sensible, R R ε R R L c R E × = 0. We see that at 6  2.22 − 1 and t − L S ) (2.25) c log J κ R = E − E R − 6 Ω c β . Plugging this into ( Re M =  − + iκ ε R =  βH  = − L L ,J ε e – 8 – θ R R log R c  as R c ] in a general Lorentzian CFT on a line at finite + 24 = Tr( κ log 6 L + ,R c L Z . Assuming factorization of left and right-movers, the , then 6 L c κ R c − R = L = E = 0 then should be EE t EE , suppose that we have quantized our theory with respect to the + . Flow of entanglement through boosted interval. S S 1 R E , and chemical potential for angular momentum, Ω. The CFT now 1 = − β H Figure 1 ; thus we understand how to compute entanglement entropy on the slice t ). = T 2.23 maps to a boost parameter θ There is a simple physicalTo understand argument figure for this due to Wall [ = 0. Consider now the entanglement through the boosted interval (ending at state is dual topartition the function rotating, is planar BTZ black hole. After Wickwhere rotation, the Euclidean 2.3.3 Finite temperatureConsider and now angular a potential temperature, spatial interval [0 lives on a cylinder with a compact thermal cycle, entanglement computed at If the hyperbolic boostreproduces ( angle is time coordinate t would like to relate thisleft-movers that to will entanglement computed eventually at passall through the the right-movers boosted are interval inside are inside contribution to the entanglement entropy due to an anomalous Lorentz boost: partition function of a non-anomalous CFTangular at momentum. finite temperature As and in chemical that potentialreal case, for Lorentzian the interpretation: complexity of underangle ( the analytic continuation JHEP07(2014)114 . ) R R c , β + (2.27) (2.28) (2.29) (2.30) L β L c ). For this R , , c L c . It is convenient  E . R Ω β πR i or multiple intervals  R M β πR sinh : . 0 R ) πε E β . sinh E  Ω R R i c = 0, there is a nonzero “Casimir πε β log − R −  24 R E L 12 c (1 c . This case sees no anomalous contri- log β = + iL = R 6 L = c  0 E R L + 7→ − β ; the respective inverse temperatures ( πR  ,J R R , β L – 9 – T do not scale exponentially with large β ]: aside from fundamental properties of unitarity, R ) β πR c sinh 6= E R = 24 c , L + 24 Ω L L i πε β T L + β 19 sinh c  L L − ). In analogy to the case of the complex interval on the c πε β (1 + . We can read off the and R´enyi entanglement entropies R log = β  L . L , β ) as 0 12 c = L E ] has shown that in such large central charge 2d CFTs with E log β L Ω ( as a boost parameter, or as parameterizing passage to a rotating 30 L β   6 c – β, E 1 n  = 1, 24 R , 1 + ), which yields EE → 19  S ) by simply taking to be real via the standard analytic continuation Ω = n in addition to the usual ground state energy, = 2.18 E 0 n 2.30 J S , there is an anomalous contribution that maps to a real entanglement entropy R )–( β 6= : this should be understood as the class of CFTs which are likely to have semiclassical 2.29 L R Recent work [ From the above results, we can also compute entanglement for a CFT at zero tem- To compute the entropy, R´enyi we plug the universal form of cylinder correlation c β + L In anticipation of ourrelevant class study of of theories holographic to consider. entanglement in TMG,or this without is gravitational an anomalies, andotherwise eminently R´enyi entanglement non-universal entropies cases, can involving be higher derived genus for manifolds (sub)subsection, we specialize to a classc of “holographic” CFTs at largegravity total duals, central charge although no usedefinition of will such CFTs be was given made inmodular of [ invariance holography and in compactness, this theirand section. essential density A property of is constructive that states the with OPE ∆ coefficients from ( bution. For holographic CFTs, this state is2.3.4 dual to global AdS. Holographic CFTsThe at results large in central this charge section universally apply to all CFTs with arbitrary ( in the thermal limit plane, we can think offrame. Ω perature on a cylinder of size For in Lorentzian signature. The formula neatly factorizes, and yields the correct Cardy entropy functions into ( and in the limit momentum” This effect stems from the imbalanced chiral contributions to the Casimir energy. to define left and right temperatures, are defined in terms of ( Notice that in the ground state on the cylinder, and we define Ω JHEP07(2014)114 , R = c 16 τ + inter- (2.31) L c ]. 31 multiple ). ) subject to a certain (TMG), which couples 2.30 1, these are non-universal 2.20 ,  , the ground state entangle- j R N > z c ε − + i z L c  , with complex modular parameter log 2 ) ]; for a more modern treatment see e.g. [ T i,j ( X . 34 = – N ) as the high temperature limit of entanglement 2 – 10 – 32 min M R 2.30 c topologically massive gravity temperatures above the Hawking-Page transition at ,..., 6 ], provided that the size of the interval is smaller than + L all = 1 28 c , i -point twist correlators with = 26 [ ]. This is true regardless of whether the CFT suffers from ]: N ) for particle, whose on-shell action yields the holographic entangle- R EE 24 c 30 S , + 2.30 L 28 c , This provides a modest consistency check of ( spinning 27 , 3 24 , 19 label the endpoints, } i z . Torus two-point correlators are non-universal. For a CFT satisfying the above { ]. The action is the sum of the Einstein-Hilbert term and a gravitational Chern- π 1. For example, one can think of ( 2 We begin this section by briefly recalling some salient features of TMG. As a second example, consider the and R´enyi entanglement entropy of For larger interval sizes there may be a competing saddle, holographically manifest as the disconnected / 36 3 , L 3.1 Gravitational Chern-Simons term The original work on TMG dates35 back to [ surface contributing to the Ryu-Takayanagi formula, i.e. the “entanglement plateaux” of [ our task is to constructwill an see, our action action for makes suchin satisfying a general contact particle relativity. with and previous understand literature its on dynamics. spinning As particles we computing entanglement entropy will requirejust modification as in in this ordinary case.tory 3d Roughly traced gravity speaking, out the by Ryu-Takayanagi a worldlineinvolves heavy a can bulk massive be particle, we viewed will asment show entropy. the that This trajec- in is TMG nothing the but analog the of extension this of idea the twist field spin into the bulk. Thus, We turn now tomentioned the holographic earlier, description the ofSimons bulk theories term action with in gravitational describing its anomalies.glement, such action. it As a In is theory trying sufficient tothe has to Chern-Simons understand work a term the with to gravitational anomalous ordinary contribution Chern- Einstein to gravity. The entan- Ryu-Takayanagi prescription for where 3 Holographic entanglement entropy in topologically massive gravity ment entropy is simply aminimization sum prescription over [ single interval contributions ( half of the circle. vals in the ground state.in As 2 general, but canexpansion be [ computed systematicallya in gravitational holographic anomaly. CFTs in At a leading large order in large entropy of a singleiβ interval on aproperties, torus, however, the entanglementargued entropy to of be a given singleleading by interval order on ( in large the torus has been N > JHEP07(2014)114 – 37 (3.5) (3.6) (3.1) (3.2) (3.3) (3.4) , 16 , and the solutions, 2 by [ 3 /` , 2 6  − τ νρ = Γ /CFT 3 R σ µτ Γ . 2 3  + = 1, the so-called “chiral point,” 1 µ` σ ρν µ . is a real coupling with dimen- , Γ . µ µ 1 +  µν ∂ .  R C  = 0 3 1 µ βν ` µν µ ρ λσ g G 3 µ − C 2 Γ 1 4 C we review that approach to the theory. 1 . We briefly note that TMG can also be µ = 3 = −  λµν = E = 0. The theory also admits a wide class − 2 µν 2 R ` g βν g = µν µν 2 − R 1 + C C ` µν  µ , c √ g – 11 – R x − α ∇ 2 3   2 ` ] for TMG shows that the classical phase space of ∇ ] and references within. d R g = 1 µ` 42 43 + αβ µν − Z – µ g µν  − √ Specializing for the moment to locally AdS µν µ 40 1 2 C x 3 1 = R 3 4 −  d 1 αµν πG 3 µν ) that TMG admits all solutions of pure three dimensional  ` µν Z C G 3 32 R 2 3.5 3 + = 1 )–( πG L 16 c backgrounds is organized in two copies of the Virasoro algebra with 3.3 3 = TMG S is the Cotton tensor, µν C Many unusual properties of TMG are put into sharper focus in a holographic con- It is clear from ( Holography for TMG has been also the cradle of some controversy at 4 ], among many other authors. a gravitational anomaly inthe the gravitational Chern-Simons dual term CFT.action under This is bulk invariant is diffeomorphisms: up to consistent in a with particular, boundary the term the that bulk transformation captures of thewhich nonzero we divergence will of not the discuss CFT here. See e.g. [ asymptotically AdS central charges As detailed in the previous section, the unequal central charges indicate the presence of of non-Einstein metrics which arecast not in locally a AdS Chern-Simons formulation; in appendix text. This theory has39 been extensively studiedthe in application the of context Brown-Henneaux of [ AdS The Cotton tensor thus measures deviations from an Einsteingravity, namely, metric. Einstein metrics for which Consequently, all solutions ofequations TMG of have motion constant can scalar be curvature, written as where The Cotton tensor is symmetric, transverse and traceless, sions of mass. Even thoughequations the of action motion has are explicit covariant: dependence explicitly, they on Christoffel are symbols, the where we have included a negative cosmological constant. Simons term: JHEP07(2014)114 . – 3 51 (3.9) (3.7) (3.8) (3.10) ] and refer- , 58  – , τ νρ 54 R Γ c σ µτ − 24 Γ L : the ADM charges are 2 3 c 3 + = ”, were constructed in [ ]. 3 σ ρν µ Γ 50 3 . µ 1 – G ∂ 8 R .  solutions. One intriguing example 46 6= 0, which are not locally AdS , E − 6 3 R ρ λσ µν 37 c = Γ µν C C i r µ λµν π AdS − g = + 2 √  yields ,J x 2 L 2 µν 3 ` R g – 12 – it E d 6 c 2 ) is important: for a real metric, the left and right L + 2 ` c = Z + 24 R E + µ L r geometries is that they do not obey Brown-Henneaux 3.10 t  3 c π ) for ground state energy and Casimir momentum on 3 g µν i − → in ( √ πG R t = 2 i x = 2.28 3 32 S 3 d ) is not applicable; indeed, the nature and symmetries of a + ` G ] as applied to a gravitational anomaly. Z 8 3.6 44 3 ]. This is the usual elegant mechanism of holographic anomaly − 38 1 = ], πG )[ 45 16 AdS ) must vanish independently, hence restricting the space of real solutions 2.4 = ]. At finite temperature, the bulk admits BTZ black holes, which are dual `M 3.10 38 Euc S In the following sections, we will deal with the Euclidean theory for both technical and The novel solutions of TMG are those with One can also perform a precise matching of thermodynamic observables of the bulk ], along with warped black hole counterparts. Viewed holographically, the main fea- to the Einstein metrics offunctional pure for gravity. Our TMG presentationassumptions; relies of nevertheless, only the it holographic on is entanglement notset general of completely properties solutions obvious of how of TMG, toon for the apply which these the our theory, subtleties Euclidean result in modulo continuation to is the some the unclear. Discussion full mild section. We will comment and the equations of motion are The appearance of the factorhand of side of ( putative holographic dual CFTences are within). not very well understood (see, e.g., [ conceptual reasons. The gravitational Chern-Simonsreversal, term hence is a a Wick parity rotation odd term under time An interesting subset of such53 solutions, denoted “warped AdS ture of asymptotically warped AdS boundary conditions, so ( the Cardy formula [ Happily, this has beenany shown covariant to higher derivative equal modification the thereof Wald [ entropy of BTZ black holes in TMG or which match the CFTthe result cylinder ( [ to thermal states in the CFT. The asymptotic density of states of the CFT is governed by generation in AdS/CFT [ and boundary theories. Theaffects the presence observables of in the the bulk,is gravitational even the Chern-Simons for introduction term locally of AdS generically nonzero angular momentum into global AdS stress tensor in ( JHEP07(2014)114 , a ]. by 2 σ 25 n , (3.11) (3.13) (3.14) (3.12) M 19 ) we wrote 2.12 = 0. A suitable dy a a )–( σ 1) along a worldline dσ ) − 2.11 n σ, y ( ( π a . U . In ( . We use flat coordinates ) y n at conformal infinity [ ) σ 1 ( n Z φ e M . log ∂ + 2 n =1 ≡ n . are Taylor coefficients in the expansion

− dy n ) ) a 3 n R Z min K as the gravitational partition function for a G ··· cone 4 L n S + Z ( (log = n a ∂ n – 13 – σ 1 limit, one can understand the action of must be a bulk geodesic, and one eventually finds a − 1 − . As we will demonstrate in detail, this is the bulk (RT) C EE K → 1 = S n + 1 → EE yy n ribbon S g = lim + ( b EE S 1. The tip of this cone defines a one-dimensional worldline that dσ a − ], at least for the set of states that can be studied using a Euclidean n dσ ab 60 = δ , )  σ ( 59 asymptotic to a replica manifold , directions, and they measure the deviation of the trajectory from a geodesic. φ 1 n e a . gauge-invariant will play an interesting role in our analysis: the existence of σ n = . Furthermore one finds that consistency of the bulk equations requires (in the M 2 C M ] an efficient algorithm was developed for computing such actions in states that ), so that 6 ds  quite in the Consider a regularized cone in three-dimensional Euclidean space with opening angle We would now like to apply these techniques to TMG. The fact that the bulk action In [ yy extending into the bulk, i.e. (1 + g π metric ansatz for this cone is then In this parametrization the extrinsicof curvatures a statement we will verify in a number of2 examples. we take to extend along afor spacelike the direction two parametrized by perpendicular directions, and the tip of the cone is then at partition function. is not this anomaly will effectively introduceRyu-Takayanagi worldline new into data a along therepresentation worldline, of broadening the the anomalous usual contributions to CFT entanglement discussed in section This constitutes a proofment of entropy the [ celebrated Ryu-Takayanagi formula (RT) for the entangle- The Einstein-Hilbert action evaluatedlength on of this conical surpluscase simply of measures Einstein the gravity) that proper the curve admit a description involvinghere a but just Euclidean state partition thestudying result: function. the action in of the We a do bulkC geometry not with review a conical details surplus of 2 Using the AdS/CFT dictionary, we interpret 3-manifold In the semiclassical limitaction of we can approximate this partition function by computing the We now want to understandTMG how reflects the the computation gravitational of anomaly holographic ofwell-defined a entanglement procedure dual entropy CFT. for in In computing principle, entropythe R´enyi there for entanglement exists any entropy a as perfectly 3.2 Holographic entanglement entropy: Coney Island JHEP07(2014)114 a σ ), in σ (3.17) (3.18) (3.19) (3.20) (3.15) (3.16) ( ). The φ 3.9 ), it is clear ) makes this ) we actually ). To proceed y ( 3.14 a 3.14 , using standard U 3.16  . ) 2  ( . O ) 2 . +  ( b ν )) ˙ U O i X a y µ . ∂ ( + ˙ θ integral: ) X ab ) yy y − y ( g b 0 ) to first order in X ) σ θ ( √ dy  . Evaluating the integrals we find f b y a φ 3.9 µν (  dy 2 a Z g ) = 4 θ C ( 3 y ∇  ( q Z 3 ) θ 3 y i µG ds ): in writing down the metric ( − ( i  G µG – 14 – b C 16 4 4 = U Z a − − 3.15 − a 3 ∂  = G = = ab δσ 4 transforms inhomogenously:   CS a − CS , , δ U = cone Einstein cone , direction. To be more explicit, if we perform an infinitesimal S δS y cone Einstein S , cone ], although the generalization to TMG requires some care. We present S 62 , . 61 , B 6 . B ), we simply find s ). A very similar calculation yields ( µ Now we note that the explicit appearance of the metric functions We now attempt to repeat this calculation for the gravitational Chern-Simons part Let us recall the logic by first evaluating the Einstein part of the action ( We now compute the action of this cone ( 3.9 ) stores the information of the regularization of the cone: importantly, it can be picked X σ ( -dependent rotation: This is consistent with general expectationsmake for it Chern-Simons impossible terms. to construct However it a local appears covariant to expression analogous to ( then the curl of the vector field The Chern-Simons action is notshifts invariant by under a this coordinate boundary transformation. term from Instead the it endpoints of the expression qualitatively different from ( implicitly made a coordinaterotates choice as about we move how iny the the transverse plane parametrized by Obtaining this expression involves anrequires integration a by careful parts. understanding ofin The the appendix validity boundary of conditions this on procedure the integral, as we discuss of ( that it is simply measuringthere the is proper no distance obstruction along theby to cone-tip writing worldline. a In fully particular, covariant expression: parametrizing the cone tip Ricci scalar receives contributions of the form Although this expression was evaluated in the coordinate system given by ( to fall off exponentially outside a small core. techniques [ details of the calculation, includingappendix explicit expressions for the regulator function φ JHEP07(2014)114 ) is (3.21) (3.22) (3.23) define ). The 3.22 2 Still there is a . with no subscript 5 ]. It is also discussed C (see figure 64 ∇ 1 = 0. n 2 n · 1 piece of the action ( ..., n 6  . + 1 λ along the curve 2 via n 2 . The symbol V ∂ ,  2 1 ρ ∂σ , i.e: · ∇ n n a ds 2 dX := σ 2 µ λρ ) and n ds n z C z, + Γ ). In three flat Euclidean dimensions this expression is Z µ 3 – 15 – 1 3.22 ds ∂ i dV µG ∂σ to be covariant, but this is a delicate point, as in this 4 := − := µ 1 ): = V n ∇ CS appears , 3.17 ]. It is also known from the physics of anyons, where it appears along the . With this definition it is possible to write down a local y 63 ∂ cone S . Evolution of normal vectors yy ) also g were defined in terms of the base coordinate system. Essentially the these vectors have unit norm, and they are both orthogonal to the √ 3.22 2  = 0. Furthermore only the linear in n = of a curve [ a Figure 2 v σ and 1 torsion n ] in relation to the framing anomaly in Chern-Simons theory. The functional ( To that end, consider choosing a normal vector to the curve Up to an overallWe sign briefly related mention to the a history choice of of the handedness. term ( 65 5 6 derivation known as the worldline of a massive particlein with [ a coefficient that measures the fractional spin [ We emphasize that thiscoordinates expression at is evaluatedexpected at to the have a tip local of representation the along the cone, worldline. located in these where the dots denoteindicates subleading a terms covariant in derivative along ( the worldline: To lowest order in tangent vector expression which reduces to ( choice of this vectorlocal fixes SO(2) the freedom other ininformation this normal of choice: vector how for thethem our with local purposes, respect coordinates to we change the want coordinate along these choice vectors the of to curve, the store and the so we will we will need tosystem introduce as an we ingredient move along that the knows curve. about the twisting of the coordinate JHEP07(2014)114 = z (3.26) (3.24) (3.25) ) to be ) in terms ) via : certainly 3.24 ¯ z C . As it turns z, ). As empha- 3.24 C 3.24 , E   n 1 n derivative, we find the total ). Expressing ( · ∇  . n · ∇ to the TMG equations creates ˜ n x 2 1 µ ∂ non-covariance associated with a n i µ = + 2 ν + reminds us that this whole expression ˙ source n X ν µ E ˙ := ˙ X X µ ˜ boundary n ˙ µν X ) as a g ) and taking the µν q g – 16 – t 3.24  ) that we can obtain two real Lorentzian normal ∂ q 3.12  ds = 3.21 1 C ds Z in C ) and the other spacelike (˜ 3 Z n ), this analytic continuation cannot in general be justified, 1 G := 3 4 1 n G 4 3.24 = ) and this formula is equivalent to the relation between the = EE 3.24 S EE S ]. Nevertheless, we will use it for the rest of the paper and find physically , then we see from ( 66 t ], the justification of this statement is in principle a different question than + ) via 62 . We show that the consistency of the TMG equations near a spacetime with , ˜ n x coordinate system used above is ill-defined away from a vicinity of D 61 n, a = σ ¯ z Next, we note that this expression should be evaluated on a worldline This expression is one of the main results of this paper. It is manifestly real. However Let us attempt to interpret this expression in Lorentzian signature, with the path Now using the original expression ( t, spacelike. If we try to analytically continue the complex coordinates ( − appendix a conical defect constrainsextremized, the and worldline of we the showthe defect that desired in conical viewing defect. a ( way Weequations now that of move requires on motion. to ( discuss the physical content of the resulting generalization [ sensible results. out, the correct worldlinesized is in determined [ by extremizingthe the determination functional ( of the functional itself. We present two routes to its justification in unlike the steps leading toas ( the there is generally no U(1) isometrythe along which relation we can between continue the(justified) ( bulk spacetime. Ryu-Takayanagi formula Rather for static spacetimes and its (yet unproven) covariant where everything is now evaluated in Lorentzian signature. vectors ( We choose a notation thatas is one no of longer them symmetric isof with now these respect timelike vectors to ( we the now two find normal the vectors, following expression for the entanglement entropy: tion from the Chern-Simons term.is evaluated The in subscript Euclidean signature. C x entanglement entropy in topologically massive gravity to be where the first term is the usual Ryu-Takayanagi term and the second is the extra contribu- given it life, binding it todetail the in worldline the in remainder the form ofof of this freedom, a paper, but normal these vector. their normal existence Asis vectors we and precisely will are the explore what not associated in is true subtlegravitational dynamical needed non-covariance anomaly. degrees of to this account expression for the anomaly has taken a bulk gravitational degree of freedom that used to be pure gauge and JHEP07(2014)114 in with (3.31) (3.27) (3.33) (3.29) (3.30) (3.32) (3.28) anyon ) must be probe S ,  3.27 determined by 1) are normalized, s . − n , 2 n ) should capture the (˜ , and five constraints = 1 5 and ˜ ˜ n . n 2 λ 3.26 ˜ , n n · ∇ constraints .  and ˜ n S , n s 1 n + that we now have a dynamical + 1) + − = 0 ) as − · ∇ 2  µ ˜ n = = n n n is the velocity vector: ( s 4 3.26 2 4 4 µ λ . λ · ∇ λ + v 2 ). However, our construction requires µ appears ˜ ν ˜ n n + ˙ , n ˙ + 2 X s X v n, ): in other words, we expect an µ µ · ˜ n , = + v ), it ˙ = 0 · ˜ 2 n X s ν µ 3 ( λ v ˜ n ˙ 2.17 µν λ µ X · is a spacelike vector. These are the constraints s ds g + µ dX – 17 – X + ˙ n = and a continuously tunable spin µ q X v ˜ n 2 := m · 1 m ) should include variations with respect to both the , n µν λ g  µ n 2 v + q λ ds = 0 3.27 of this path to be the proper length along the path and µ m respectively results in v C s + ˜ n λ ˜ n  · Z µ ˜ n ∇ ˜ n n · · ∇ s = ds ) in the first place. Here n ˜ n C − 1 , is timelike and ˜ EE Z s and λ S  n 3.26 = µ = ) and the normal vectors ( = 0 v 1 ds ), leaving a single degree of freedom. However as it turns out this is s , v = 1. The total action for the spinning particle is then ( λ C µ · µ µ Z probe n v n 3.28 X S µ = v real constants. are five Lagrange multipliers that guarantee that i s : λ n ) remain normal to the curve , and so if we are to vary them then ( ˜ n and constraints n, m S In addition to the particle trajectory The variational principle for ( respect to Contracting with ˜ degree of freedom correspondingIndeed to we the have rotation three ofassociated the independent with normal components ( frame for along eachnot the a of worldline. true degree ofterms. freedom, as To the see action this, is only consider sensitive the to equation its variation of up motion to arising boundary from variation of We choose the affine parameter hence we have which also enforce that that allow us to write ( where the mutually orthogonal, and perpendicular to the worldline: particle position that ( supplemented with the following constraint action: in what follows we write the entanglement functional ( with Let us take athe step bulk. back As and articulatedphysics recapitulate at of what the a we start heavy expect particlethe of with CFT this the mass twist section, cone operator the actioncurved quantum action to space. numbers ( describe ( In in this section we will flesh out this interpretation, and to make it evident 3.3 Spinning particles from an action principle JHEP07(2014)114 (3.40) (3.38) (3.41) (3.34) (3.35) (3.36) (3.37) (3.39) ): In (2 + 1) 7 3.26 ]. 5 appears in both – 3 1 λ n . ). · ∇ ); details can be found in ˜ n s 3.31 s ( , , µ − inherited from ( , X s = νρσ = 0 = 1, both sides of this equation µ 5 = 0 , , µ 2 Note also that the normal vectors R λ ˜ n 3 n ) and 2 ) do not have a dynamical equation αµ 5 8 ρσ ν s ˜ there actually carries information about the n , s λ 1 n m = ˜ ∇ µG ν λ µ n . µ n, 4 µν v 2 v ˜ n v s n , + 2 1 2 n β · ∇ = v µ − − − µνλ · ∇ v s  n ν − 3 s v ˜ n λ = ] = µ s βµ − , s – 18 – n ˜ + n − ) with respect to 3 µρ ( = ∇ s µ µ 1 s G = v n · ∇ 4 ∇ µν α 1 3 3.31 = ρ s to be ˜ v n λ v = + ) on the resulting solution. µν + µν + s m s µ αβ µ = 0 gives s is insensitive to small variations of the normal frame along ) using the values of ˜ n v 3.26 n λ ∇ m ∇ [ s · ∇ 3.37 probe ∇ probe n S S ). ˜ s n δ ) are six equations, of which five may be satisfied by appropriate = ) we see that the spin tensor may be written in terms of the velocity as 3.39 spin tensor 1 ] for a more modern treatment and further references. λ 3.35 3.29 71 – . The equation that remains comes from the fact that i 67 λ . This variation, however, is not trivial and gives rise to ) and ( C 3.33 Thus to compute entanglement entropy in topologically massive gravity we should Next, using ( A more tedious task is to vary ( See as well [ In higher dimensions such a rewriting is not possible: 7 8 direction of the particle’sthe spin. spin In tensor, that case the MPD equations also includewhich a is relation an for identity the on evolution ( of solve the MPD equations ( and then evaluate the action ( indicating that the spin tensordoes is not just actually the carry volumethemselves any form have extra perpendicular vanished degrees to from of the thethe freedom. worldline equations action and principle. of motion: they are required only to set up These equations are knowndescribe as the motion the of Mathisson-Papapetrou-Dixondimensions spinning they (MPD) particles follow equations, from in the and classical simple general and relativity geometric [ action ( where we define the of motion: equivalently, the trajectory, up toimportant boundary to terms. us. As we will see, theseappendix boundary terms will be very choice of the sets of equations, and is: However this is notare a identically dynamical zero equation: and this as is an identity. Thus ( and again the appropriate contractions yield Eqs. ( The same analysis for JHEP07(2014)114 . i n , the (4.4) (4.5) (4.6) (4.1) (4.2) (4.3) to f ) for the 3.3 n The MPD unchanged: 9 3.37 f n ). A.9 and ) on a geodesic. We ) remains normal to the i vanishes. s ( n ρ 4.1 . n . This greatly simplifies v ). 3 µν  s . n 3.41 ) ρ · ∇ n , ˜ n . ∇ , 1 µ ( ) that leave f i ν s n = 0 n ( + n i n µ ν λ given by ( v ˙ ) = v ) = X ]) to guarantee that s i f , µ ∇ s s µνρ 72 ˙ ( ν ( X v n along the curve, i.e. consider the solution to . = 0 i µν i ds  to find g n n µ µνλ , , n C ρ  v – 19 – i q Z s n and spin n ∇ ν  µ − v 3 m ) = 0 1 in spacetime which is found by extremization of the µ ds G s ) = v µνρ i ( 4 C equation C  s m n Z ( h = relative to = 3 n ∇ 11 ∇ 1 f µ G 4 n n anom = S . Recall that the prescription is as follows: the entanglement ), which for the moment we parametrize as follows: s 2 EE ( using ˜ S n n also have other solutions, as it is a higher-derivative equation of motion: we believe ). Evidently anyons moving on maximally symmetric spaces follow . For this purpose it will be helpful to rewrite the action in terms of a are normal vectors defined in the CFT. As discussed in section 4.2 does ), the on-shell action itself depends on extra data, namely boundary data ) is f anom ) s n S ( 4.2 µ 3.26 X 10 and i n What does this actually mean in curved space? We require a way to compare So all we need to do is evaluate the second term in the action ( We will limit our discussion to spacetimes that are locally AdS This can be shownNote by that writing ( the Riemann tensor in terms ofThis the construction metric only using ( applies if the curve is a geodesic; in a more general case one should replace 9 10 11 (without proof) that they will have greater actions,parallel and transport in with this Fermi-Walker transport workcurve, (see we but e.g. will the [ not intuition investigate is them. still valid. it measures only the twist of To that end, consider parallelthe transporting first-order parallel-transport of the normal vectors where action is actually insensitive to smooth variations of single normal vector We note that although thetrajectory equations of motion can be entirely formulated in terms of the is a solution togeodesics. ( call this term the analysis, as here the contractionequations of then the Riemann reduce tensor to with and furthermore, it is clear that a geodesic It should be evaluatedfunctional on itself, a which curve results inmotion the Mathisson-Papapetrou-Dixon of equations a ( spinning particle of mass In this section weentanglement finally entropy apply on our various prescription backgroundsresults to of the found interest. actual in We computation will section functional of compare ( holographic with the CFT 4 Applications JHEP07(2014)114 . ) f s s ( , it and n anom n (4.9) (4.7) q S (4.10) (4.11) : q . However 3 . and ˜ η . q , )) i s  , ) we conclude that ( ) f ) (4.8) i η s : in fact, once we find s n ( 4.8 n ( 1) transformation: · q of this Lorentz boost Λ. This − · q , ) ) ) ))˜ η ))˜ f i i anom f s s s s s S ( ( ( ( ) to ( ( ) in terms of ˜ ˜ η with respect to variations of q q η η . s ( ( − − b f n 4.10 µ i ) that actually satisfies the boundary f n 3 s n b 1 n anom ( by a boost with angle a · G · S ) n via an SO(1 4 ) f ) η i ) + sinh( ) + sinh( n f f s s s s ( n ( ( ( q ) = q q q s = Λ( – 20 – )) ( ))  ˙ i s ) we have a η ( ) s ( η f log ds 4.6 η s ( µ n Z 3 ) measures the rapidity 1 µ G 3 ) (denoted in gray) along the curve by solving parallel transport 4 4.4 1 s = 0. We can expand G ( ) = cosh( = 4 ˜ q n ) will be related to ) = cosh( s ( f s ∇ = ( s n ( n into = n anom ) alone is sufficient to determine i q S n anom 4.8 ∇ S ), as it will in general not satisfy the right boundary condition at ), s µ ( ) now reduces to the total derivative ˜ q n , µ 4.4 ). Consider also a parallel transported normal frame along the path given by q ). In general 4.5 4.6 . Propagating -dependence comes entirely from the parallel transport of the normal frame and not equal to s To prove this assertion, consider a vector not we see that the total twist can be conveniently written as: ˜ Note that despite thedepends (through topological the character parallel of transportmetric equation) and continuously thus on on parameters the of state the of bulk the field theory. i.e. its from any rotation ofmeasures the the frame twist. itself. ( q Comparing ( The integral ( On the other hand, by its definition ( condition ( two vectors ( Instead at the endpoint it will be related to As we now show, theis integral the ( generalization of the idea of “twisting” to curved space (and Lorentzian signature). Figure 3 equation ( This has a uniqueis solution all along the curve, as illustrated in figure JHEP07(2014)114 . ) 2 CFT 3.25 ) (4.14) (4.18) (4.12) (4.13) (4.15) (4.17) x 1 with ∂ − = 2 )): n . , ) x, t, u , u 0 CFT : that is, we want to ) x, x = 0 (4.16) ( ∂ ˜ q u in the vacuum, stretching · 2 R` v R at rest − ) = ( as a vector that points in the . . f = = s ) is interpreted as the field theory q ( 2 µ ˜ · q ˜ = 0 q CFT . Recall from the original definition . v − ) du ˜ q f t CFT ∂ n = , ) + , ∇ t ˜ 2 q ) = 2 ∂ 0) i CFT 4 · ) s R , t dx ) in the interior follows from the fact that and ( ˜ ∂ s q , u, i = ( + n = 0 n , q (0 2 2 = ( ` x 1 q 1 dt – 21 – f − , ∇ + − n = ( 2 = µ 2 2 = u , ` 2 u i CFT ) n t = , q ) that the normal vectors should be defined with respect = 0 ∂ ) 2 , q v x . The geodesic equation admits the following solution: ds
∇ 3.21 − 0 ) = ( , = 1 , f 0 2 2 s R q ( u, q ( = 0. We first consider an interval = ˜ u to 2 in Poincar´ecoordinates, corresponding to the vacuum of the CFT u 2 R` v ) =
3 i 0 s = , ( 2 µ q R points in the time direction at the boundary and it is normalized such v − 1 with respect to the bulk metric; a similar definition applies to ( : as we saw in section 2, this is precisely what one expects from a theory with − CFT ) ) = t = ∂ is insensitive to smooth variations of 2 x, t q We turn now to some explicit computations. Finally, we discuss the explicit choice of boundary anom Furthermore, note that where ( that By construction, notice that and The tangent vector and a parallel transported normal frame is (using ( The boundary iscompute at the entanglement entropyfrom of ( an interval of length 4.1 Poincar´eAdS We first study AdS defined on a line. The metric is the physics is coordinate-invariantentanglement in entropy the now depends bulk.the in However a it subtle is waya on a gravitational anomaly. this nontrivial choice fact of that time the coordinate on Here (and in thetime following direction subsections) at we therespect define to boundary, ( the however bulk ittime metric. is coordinate The normalized that direction is toS used along ensure to ( that define the vacuum of the theory. The fact that the action of the normal frame backto in the ( coordinate systemthe used coordinate to system parameterize in the theinstructs bulk bulk. coincides us with to At that take: the used to endpoints define of the the CFT, and interval so ( JHEP07(2014)114 . : 3 ). π ) we 4.13 (4.23) (4.21) (4.19) (4.20) (4.22) 4.11 ) in ( points in the . x, t ) still κ, u ): sinh , ) 2.20 κ ) relative to ( 0 κ, x , t 0 sinh x . cosh . κ, x (  ε ) = 0 u expression ( R ) for an interval at rest in Poincar´eAdS s 2 R` (cosh s 2 κ , (  ( ) already satisfies the boundary condi- interval ( 2 η q − s R µ ( 3 q 1 log = G 2 µ ). Explicitly, we find R → ˜ ) and ˜ c q s = ( 6 boosted + – 22 – ) to q 4.15 , ) κ L s c . ( 0) anom q 4 = S sinh κ, ) = κ, EE s ): we see that still points purely in the time direction of the CFT, and ( S cosh n f 4.8 n (cosh κ, u 2 ) does not contribute, the entanglement entropy is simply given R = i − 4.1 n sinh u simply by boosting ( ( ) reduces to q ` . Much of the calculation can be taken over from above: we can obtain 1 , ˜ κ 4.8 q = µ q ) coordinates we have a spacelike line stretching from ). This means that a vector pointing in the time direction . Evolution of normal frame vectors x, t a boost parameter. In a Lorentz-invariant theory the entanglement entropy would 4.12 κ It is simple to evaluate ( Importantly, however, thus now has anow find component directed along the boundary interval. Evaluating ( with be insensitive to the appropriate 4.2 Poincar´eAdS, boosted interval Let us now consider the entanglementIn in the a ( Thus the spin term inby ( the proper distance, and we find the usual CFT tions ( time direction even after beingno parallel twisting. transported So to ( the other end of the curve; there is The normal vector perpendiculartransport, to the whereas boundary the interval vector doesthis that not is is change intuitively parallel under obvious to from parallel the figure boundary interval rotates through Figure 4 JHEP07(2014)114 ) to (4.28) (4.24) (4.25) (4.26) (4.27) (4.29) (4.30) 4.29 . 2  . dτ /` − ) direction at time 2 r − r + φ τr r + + + ) is: φr ( dφ e 4.13 to the Poincar´ecoordinates  2 − 2 − 2 κ, r . r r . along the R
− c − 2 − ) to compute this boost, we can 0 + R ) r , that is parallel to the boundary 2 2 + 6 1 2 ) (4.31) − r 2 t r R i 4.6 2 π` + . t L p 2 dr . c − , s 2 + ) − r 2
2 − t − 2 0 = dφ r ( , = CFT , t  2 + − ) R − 1 R ε τ 2 R 2 x 2 − β ∂ r ) ): the entanglement entropy depends on the  2 1 − dτ )( r  x , u 2 = ( − 2 + x 2.23 log – 23 – r ≡ − f ( Poincar´ecoordinates ( = R,L − 2 into the bulk, we expect it to be dragged by the 2 n R , − r P 3 c 1 x /β 1 2 ) ) ( n = R π` 6 r − τ + 2 CFT i ( 2 p ± + L n = φ φ, τ ds r c ( + i π := p 2 2 = = e P dτ L 2 2 + − R ) β r r EE 2 − coordinate is noncompact: thus technically this is not a rotating S r − − the length of the boundary interval in the Poincar´econformal φ 2 2 − P r r 2 R r s 2 )( r = 2 + r t − ± 2 x r ( − ): . The explicit mapping to AdS . Denote by 3 = 2 , 1 2 Rather than directly solve the differential equation ( ) 2.27 2 ` ds = 0. The boundary conditions on the normal vector are defined in terms of the time x, t It is now convenientinterval to in construct Poincar´ecoordinates, i.e a in 2d components normal vector We may now essentially takemap over the the endpoints of results the from interval( ( the previous section.frame, We use i.e. ( simplify our computation byto using AdS the fact that the BTZ black hole is locally equivalent coordinate appropriate to the black hole, In this case asboosted we black parallel hole horizon, transport picking up a nontrivial boost. We will assume that the black hole but ratherperature. a boosted We will black consider brane,τ a i.e. boundary a interval rotating of BTZ length black hole at high tem- This is dualcf. to ( a field theory state with unequal leftThe boundary and CFT right-moving lives temperatures, on a flat space with metric: 4.3 Rotating BTZWe black turn hole now to the rotating BTZ black hole, which has metric This matches the Lorentzian CFTboost result of ( the interval relative to the vacuum. and the total entanglement entropy is JHEP07(2014)114 at , ]. It P 2   R (4.34) (4.32) (4.33) (4.36) played 73 L R , β β x   ) = 49 , f L R s ). Note that β β πR πR ( 48   λ 2.30 2 P 4 R sinh sinh =   2 u log + L , c 2 , λ   remains perpendicular to the − 12 α L R v q R β β µ c ˜ n  
β + L R n β β πR πR , u i   ) can now be written (again in Poincar´e µ αβ  = 0 (4.35) Γ λp L ν β πR ˜ n 4.22 sinh sinh ds P u  ’s back to BTZ coordinates and compute the d ds H   2 q `R I = – 24 – ’s in ( − µ log sinh 3 q µ at the left endpoint and ending at 1 µ =  n G 3 4 P 1 µ R 2 ) simply states that d ] and we would like to discuss the connection with G ds ˜ q R β ). We find after some algebra πR 4 ) simply becomes = . The −  t 73 4.32 , = 4.1 4.28 49
anom ) = and sinh i 0 S s , anom x ( j 2 R S has a component directed along it that switches sign as we move ε p λ β 2 j L i q π  β ) using (  ` u 4.11 log = runs over i µ L c q + 12 ]. This agrees with the result computed from field theory in ( R c the usual affine connection. This is equivalent to the expression in [ 66 ), starting out at parametrizes movement along the curve, essentially playing the role that = µ αβ λ Note that if we are evaluating the action of a closed loop wrapping a black hole horizon, It is now straightforward to convert the EE 4.22 S with Γ is gauge-invariant under coordinate changes that are single-valued around the circle. From normal vectors willcircle. give the A simple same choicecoordinate answer, is system: provided just that to take it the is normal single-valued frame around to be the constantin in which any case convenient the spin term in ( The evaluation of thehas thermal been entropy studied ofour previously formalism. a [ black hole in topologicallythere massive is gravity no longer a choice of boundary conditions on the normal vectors. Any choice of cally in [ the structure of the secondsum term of is separate precisely left correct and to right allow moving the contributions. 4.4 answer to be written Thermal as entropies a where we have added back the ordinary proper distance piece, first computed holographi- and thus the total entanglement entropy is the right endpoint.boundary Note interval while that ˜ ( from one endpoint to the other. inner products in ( coordinates) as Here in ( where the index JHEP07(2014)114 ) ), is 2 R (5.1) 4.35 3.26 (4.37) (4.38) × 1 S ) does not ) and ( or ). Note that 4.35 C 3.8 4.25 . φ ∂ 1 `r -interval entanglement N = — in (2+1) dimensions. In in the presence of a gravita- 2 , v copies of our functional ( , r ) ∂ anyon N n rr /CFT , and hence are singular; but for the g 1 · ∇ 3 + √ . r n (˜ − . = `µ R − ds = 3 r c C πr ˜ G n 6= Z 2 L is a sum over µ , c = 3 – 25 – 1 R  G c ) for the BTZ black hole. Using ( φ 4 ∂ 6= anom = − constant; the distinction between these two notions is S L 4.36 2 r c r + r anom − S t ∂ ) this pathology drops out. We find  covariantly rr 4.36 2 g , i.e. a bulk worldline together with a normal vector. ` ). √ = 4.34 define a normal frame. This expression measures the net twist of this ribbon n n and limit of ( n Our prescription admits a natural extension to multiple intervals. We saw in section The resulting contribution of the anomaly to the entanglement entropy is pleasantly For concreteness, we evaluate ( → ∞ minimized over all pairings of boundary points allowed by the homology constraint of Ryu that for holographic CFTs,universal the to multiple leading interval order entanglement in entropyand large on total for central this charge. class Itable of is to CFTs, of conjecture course to the precisely whichentropy this following: in bulk regime, holographic the calculations CFTs bulk in with object TMG apply. that computes Thus it is reason- our work can also be viewedcurved as space. the construction In of the the contextof worldline of action the entanglement describing entropy, CFT anyons this in twist anyoninterval is on operator. the various simple holographic Finally, spacetimes avatar we towith illustrate computed a the the field formalism, theory entanglement demonstrating analysis agreement entropy of for CFTs a with single where ˜ normal frame along the worldline.a spinning In particle the with a bulk continuously this tunableparticular, spin action we — can showed an that be the interpretedPapapetrou-Dixon minimization as equations of for describing the the motion total of action a spinning reproduces particle the in general Mathisson- relativity. Thus worldline into a geometric: 5 Discussion We studied holographic entanglement entropytional in anomaly. AdS Our main result cana be easily non-trivial stated. dependence The on gravitationaltropy. anomaly the This introduces choice data is of transmitted coordinates into when the evaluating bulk entanglement by en- broadening the Ryu-Takayanagi minimal After adding the contribution fromthe entropy the density area extracted law, from thisR this exactly expression reproduces is ( consistent with that arising from the Strictly speaking these vectors arepurpose evaluated of at computing ( transported around the horizonmean and that the then vector compared is precisely to what itself. the spin Note term measures. that ( we have our point of view it is measuring the boost acquired by a normal vector if it is parallel JHEP07(2014)114 ] 12 82 , ]. It warped 81 83 (which This will 4 ], which is 13 80 ) for each geodesic. 5.1 chiral gravity story [ 3 ]. A challenge is to determine ) does not contribute (for spatial 79 AdS , 5.1 78 ]. The corresponding phase space contains the – 26 – 77 algebra, which can be obtained as an ultra-relativistic 3 ]. An interesting limit of the above set up is when the 79 , have not been generalized to warped CFTs, and there are 78 2 Virasoro algebra [ -invariant field theory living at null infinity could in some sense be 3 3 ]. Our construction opens the door to computing entanglement entropy 74 , 57 [ ] for a candidate example. 75 a gravitational Chern-Simons term. Holographic entanglement entropy was studied in that context further probe the nature of the putative dual theory. or thermal entropies [ bulk theory consiststhe only asymptotic of symmetry thesuggesting algebra gravitational the reduces Chern-Simons existence to piece. ofwould a a be chiral In flat interesting copy that version to of case, of a study the Virasoro entanglement entropy algebra, in these theories in order to whether a BMS dual to flat spacefield in theories (2+1)-dimensions. besides Again, specificto little examples flat is (such space known as about whatconstant) the these the and one putative Liouville general presented CFT properties in is [ such to as 3d the gravity with form a of negative correlation cosmological functions [ flat boundary conditions atconsist of null the infinity. so-called BMS limit In of that the AdS caseflat the limit asymptotic of symmetries thewith BTZ non-trivial black Bekenstein-Hawking holes, entropy [ which were recognized as cosmological solutions be slightly more involved thanheavily the used simple the applications fact presentedric that in space the section spinning admit particle geodesicdifferential equations equations. solutions) on and a will maximally likely symmet- actually require the solution of no known microscopic examples ofWe warped take CFTs, the away from view certainvide that limiting cases. an the application first, ofwith albeit our warped holographic, construction conformal to calculation symmetry. warpedseems of Regardless interesting AdS entanglement of in may entropy considerations its pro- own of in right warped a to CFTs, apply theory it our formalism to warped AdS. geometries are thought toCFTs be dual to somewhatin mysterious these field geometries. theories — tools Doing used this in in section subtleties the in field calculating theory entropies related itself to appears the challenging: choice of the ensemble. There are also In a similar spirit as the previous point, one could consider TMG with asymptotically There are other solutions to topologically massive gravity, such as warped AdS. These ] using a perturbative scheme that permitted application of the Ryu-Takayanagi formula. Their results There are some natural directions for future research: See [ It should be noted that there are also warped AdS solutions of Einstein gravity coupled to matter, 2. 1. 76 12 13 holographic entanglement calculations in warped AdS solutions of TMG is likely to be different. without in [ favor the hypothesis that those geometries describe states in an ordinary Virasoro CFT. The outcome of for multiple intervals, only now theIn functional Poincar´eAdS, is we extended showed to that include the ( anomalousintervals); however, term it ( does in the planar BTZ case. and Takayanagi. This is precisely analogous to the original Ryu-Takayanagi prescription JHEP07(2014)114 , = R ). d Tr should 4.1 F ) either spinning n Z ) suggests 4.2 ], and their 5.1 44 which occur in all = 6 QFTs. More with replica bound- 14 n d Z ]. ]. Thus some features of 30 71 ]. It would be interesting to . It is not clear if these addi- 3 84 ], and it would be nice to explicitly 25 manifolds at least, computing . Morever, action principles for n 3 and – 27 – n = 2, the would-be anomaly polynomial, Tr . Doing so would provide an orthogonal and comple- R d c ), so the only interesting case is 6= Z L c ∈ 2. (In k ]. This should yield the entropies R´enyi discussed in section 2 for d > 28 , . For this reason we expect that non-geodesic solutions to ( 25 E + 2 dimensions ( k in appendix don’t satisfy our boundary conditions or are subdominant when minimizing ( tional saddles imply ambiguitiesconstruct like these those solutions found in explicitlythe [ and Chern-Simons understand formulation their there roleequations is (if of no motion any). such are ambiguity: However,primarily first in sensitive for order to flat in boundary connections, conditions the of the momenta the of probe. the This particle is and carried the out explicitly action is mentary method to thosefor employed computing herein, bulk and loop comes corrections equipped to with the a classical prescription result [ for the trajectory of the particle even for empty AdS ary conditions at infinityvals. in a This semiclassical was approximation, donegeneralize for this for any number to Einstein of TMG. gravitybe inter- in For tractable locally [ [ AdS holographic CFTs with that there may bebeyond an holography. interplay between entanglement and anomalies that extends 4 interesting is the caseeven of dimensions mixed gauge-gravitational anomalies, vanishes.) The holographic description ofcontribution such anomalies to is well entanglement known entropy [ speculatively can still, be the calculated geometrical using our character methods. of expressions More such as ( 2 and admits twomembranes normal have vectors been ˜ constructedour (and probe studied) could in be e.g.role [ extended to to play higher in dimensions:with computing a gravitational entanglement spinning entropy anomalies. surface in might Of higher have dimensional course, a field-theories gravitational anomalies only occur in The Mathisson-Papapetrou-Dixon equations admit, in principle, multiple solutions In 3d gravity, one can actually compute the bulk path integral In any dimension, the bulk object that computes entanglement entropy is codimension We hope to return to some of these issues in the future. Recently holography has per- We thank A. Wall for discussions on this point. 5. 4. 3. 14 both: we hopefundamental that ideas it in may quantum havestep field something in theory, to that and direction. say that also our about work may the be interplay viewed of as these a two small mitted a refined understanding of the physical consequences of entanglement and anomalies JHEP07(2014)114 (A.8) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.1) we use t . ) λ µσ . νµ Γ , a ] ρ νλ + Γ = 0 . µν . g µν g − β  a ∂ µνλ − λ 1 ( b  to Lorentzian time λ νσ 1 2 e − √ Γ . ∇ E − ν µλ τ = L ρ µλ µβ | ) g , = ν β ν µν + Γ ∂ in Lorentzian signature with coordinates µνρ + Γ ( δ it . ν txz α i λ b + δ ρ µσ e = µ = Γ µνλ , a νβ +  ∂ ν – 28 – E g ) E ∂ τ  β λ µ

g ,  δ νµ ν ∂ [ − a − α ν a e µνρ δ √  ρ νσ αβ − − = g Γ = 1 2 µ µν = bµ ∂ a a ( txz = ω  = 1 2 µαβ  α µν = Γ σµν ] ρ µνλ  µν R [ a ) are: t, x, z The Riemann tensor is We define the Christoffel symbol and spin connection as This means that ifone, ever we they need are to related compare as a Euclidean epsilon tensor with aWe Lorentzian define symmetric and anti-symmetric tensors with factors When analytically continuing from Euclidean time ( Some other useful properties of the tensor are PHY11-25915 and by the DOE under Grant No. DE-FG02-91ER40618. A Conventions Our conventions for the epsilon tensor to thank J. Camps,Research Associate K. of Jensen the Fonds and dehas la G.-S. Recherche received Ng Scientifique F.R.S.-FNRS funding for (Belgium). from helpfulSeventh E.P. the comments Framework Programme on European a (FP7/2007-2013), Research“Strongly draft. Council ERC Coupled under Grant S.D. Systems”. the agreement is European STG a N.I. Union’s 279943, is supported in part by the NSF under Grant No. It is a pleasure to acknowledge helpfulJ. discussions Camps, with M. X. Ammon, J. Dong, Armas,Loganayagam, T. T. G.-S. Azeyanagi, Hartman, Ng M. and Headrick,for A. K. his Wall. Jensen, early contribution M. In tothe Kulaxizi, particular, this Solvay A. we collaboration. Workshop Lawrence, on are We R. “Holography are very for also grateful Black grateful to to Holes M. the and Ammon participants Cosmology”. of Finally, we wish Acknowledgments JHEP07(2014)114 (B.3) (B.1) (B.2) ) is a (A.14) (A.10) (A.11) (A.12) (A.13) σ ( φ .  dy . a dσ ) .  µ a σ, y τ νρ e ( a µ Γ V . U . ) σ µτ ρ σ = Γ ( v ) (A.9) 2 3 a bcµ φ , µσ ω e µνρ λ g +  + V abc s is defined as νρ ) to a spacelike curve with tangent ,V σ ρν  ρ 2 g ˜ n − a ˙ Γ , − X µν J µ ρ − dy , is . n, s a ∂ n ) µ λρ = ω ) is ) = ν V νσ  ), which we write as s µ µ v g ( ··· a πin ˜ n 2 = µ + Γ ρ λσ µρ ν + µνρ Γ , where we parametrize small deviations near 3.14 g µ X ze n  ( a – 29 – , ω 2 ds σ πn − 1 ∼ λµν = ` dV a µ ,V ν z ˙ µ ] g − K X ˜ n ˜ n := µ − a = + n J µ √ ( ω, V µ yy x a V s , and not in the metric components. If we go to complex 3 g ω a ∇ + [ d µνρσ = σ = R + ( Z = 1. The spin tensor ds µν b dV ω s 2 := v dσ , then we have the following identification pattern = a 2 1) and we will compute the action to first order in = CS V iσ dσ 2 S − ∇ n ab + δ n 1 ) and ) σ σ R ( , = ( φ spaces satisfy := 1 and ˜ e (2  are flat Cartesian coordinates on the space transverse to the cone. We denote 3 z − sl = a = ∈ 2 σ , we pick the following handedness: µ 2 a ds the AdS radius. v n J 1 with ` There is an important subtlety in this parametrization that is relevant for TMG: we The covariant derivative along a curve ∼ would like the informationperiodicity regarding of the the coordinates openingcoordinates angle of the cone to be stored in the Here the the total opening anglen of the coneregulatory by function 2 that smoothens out the tip of the cone. on a regularized cone, given by the metric ( B Details of coneIn action this section we compute the gravitational Chern-Simons action vector where where When we introduce a Lorentzian normal frame ( which, in terms of the one-form spin connection with Locally AdS JHEP07(2014)114 1 n z (B.6) (B.7) (B.8) (B.4) (B.5) := 0 limit w → ultimately . to find is regular at z a a  z ). In TMG this ) σ d D U dzd c 1 ∂ − 1 n cd ) with respect to the  ) . far from the tip of the ( z ) with the identification b z z  ∂ ( 3.31 b ab B.2 U ∼ δ a dzd . )  ) in appendix ∂ ¯ w φ .  ) falls off exponentially outside a : this is because there is no loss ab b  a ∂ σ ) D.2 ) U ( z ) + 1 U dwd a φ φ z z b + ( ∂ z . In a diffeomorphism-invariant bulk ∂ b ab a n √  U ∂ ( log ( a  f ab ∂  δ ) is taken to have compact support, which z  dy . From the point of view of computing a ab ( z 1 σ  a  ( ) √ Z − φ φ  b 1 this takes the form ( – 30 – ∂ π f σdy − a 2  2 1 n − ∂ ) and demonstrate that the resulting equations of d ) − s ab z ( δ z Z = µ ) = , and so the metric ( ) we now evaluate the integral over the  − 2 z X ( πi CS 2 ).  , −  z, B.5 ( z = we φ cone 3.17 = 0) = 0, and thus these terms vanish when evaluated at the σdy S dzd ∼ 2 CS , σ d ( w a Z cone U . To first order in S  4
r a = − variation actually involves a variation of the periodicity of the coordinates CS ,  cone S ). ) := exp r ) and then take a derivative with respect to ( B.3 f Using the explicit form of ( We now directly compute the action. The piece which survives in the We now discuss the explicit form of the regulator function. The coordinate B.3 C Details of variationIn of probe this action appendix weposition vary of the the particle action worldline of the spinning particle ( as in ( This is the result quoted in ( This integration by parts ismeans justified that the only if In evaluating this we havein neglected generality terms in nonlinear taking in tip of the cone. We now integrate by parts on the second term, to find takes the form cone: a suitable 2d metric that does this is with theory we are allowed to changeangle coordinates, instead storing in the the metric information components regardingis (for the example, not opening see actually ( physically equivalent, and such a procedure mayhas result the in usual different periodicity answers. the origin. We seek to interpolate between this metric and a small core thatholographic can partition function be the takendetermines asymptotic to the periodicity periodicity have of of size the thethe coordinate coordinates entropy system we describing should theas dual compute ( CFT. the To compute CFT partition function given an identification such Operationally, this means that the regulator function JHEP07(2014)114 . σ (C.5) (C.6) (C.7) (C.1) (C.2) (C.4) . δX i σ σ )] σ σ 1) ˜ n δX δX δX 3 − i i σ λ )] 2 ) µ n α σν µ ασ + (˜ ˜ n i δX Γ 5 σ ν α )Γ i α λ n ˜ n n ν µ ˜ n 2 µ ˜ ˜ n n λ µ ασ ˜ n µ α − ( 5 ˜ n n (Γ λ 5 ( µν s λ + 1) + g ∂ , (C.3) + , − ∇  σ 2 + ν ν ) ∂ − − ∇ n ˙ ν n µ ν ( = 0 X β β = 0 ˜ 4 µ n . n n µ v v measures proper distance and so (˜ µ λ n µ σ µ µ ˙ µ µ σ 4 ˜ n ν X n s ˜ n n ˜ n ˜ n + 5 λ 4 v 4 5 ) gives α constraints α δX 3 µν λ λ δX ˙ λ λ s n g n S + ) ) λ X ( ) σ + σ · + µ ν q + v ˜ n ˙ + 2 + 2 ν x µ αβ ν ˜ n ασβ 3 X ) +  µ 3 n ˙ ∇ Γ µ µ ), which were worked out to be: µ λ X ( µ λ σ v v R ˜ X α σν n ˜ n µ 2 3 n ∂ anom 3 + Γ ˜ n 4 + ds λ λ + n, S λ 3 α σ λ ˙ ds δ C µ ( λ + ( n – 31 – n + + X ) + + 15 Z n 2 C · µ µ , µ µ µν + − ν λ Z n ∇ m g n ˜ n n ν n ( ˙ ν ν ν 1 1 σ 2 X ˙ m − µν geod ˙ ∇ X n n ∂ µ λ λ λ d X g ν [ ds S )˜ )˜ µ n µ σ ˜ = = n ˙ + + 2 + ∂ n µν µν X = ds ds λ 2 µ µ µ g g µν ˜ n n , h to the explicit dependence on the coordinates. The contri- λ µν g σ σ ˜ n n n · C C ( h g ( geod ∂ ∂ Z Z ∇ ∇ n µν ( ( · ∇ ν X S 1 s s probe h h g q µν − + v ˜ n X λ S  only g 2 − δ h σ ds λ x ds δ ds ds [ ds C C C C ds C Z Z Z Z = 1. To simplify our results we will use the equations of motion from ds ds δ ds ∂ Z C s s s 2 m s Z C C C Z Z Z = = = , v = = = µ = = = ds dX geod anom anom S = S S µ X δ v constraints S constraints S For the terms involving the normal vectors we have We ignore total derivatives in this section. X δ 15 The variations of the constraints with respect to We vary first with respect bution from the geodesic is the usual one we have the variation of the action with respect to ( We will work work in a parametrization always where ning particle in general relativity. The action is with motion are equivalent to the Mathisson-Papapetrou-Dixon equations of motion for a spin- JHEP07(2014)114 = 0. (C.8) (C.9) ); the 1 (C.10) (C.11) (C.12) (C.13) σ λ ˜ n, v δX . σ ), as surely )] n, σ ˜ n α σν δX Γ n,
δX α i α σµ ˜ ) n σ Γ σ − ˜ n αν 3 g λ µν δX ) g − + σ µν ∂ σ g , α σν µ n σ σ , Γ n 2 ∂ (˜ β λ . βµν αµ ( ν v δX σ − )) that on-shell we have g v σ ] 3 R µν . − α µσ s λ µ ασ C.4 µν δX ) − ∇ Γ , s ν µν )Γ β )] β αν n g µ ) + ν v σµ σβµν v g µ σ ˜ n s ˜ µ n for linear combinations of ( 1 2 R α σν ∂ ˜ n α ˜ µ n ∇ 1 2 Γ ) and ( )(2 ˜ n n ˜ constraints − α n µ − µ ( ν α 5 S n v − ∇ ˜ n n ν n λ X C.3 ν µ ˜ n ] = δ = = n, v − ˜ + n µ ασβ − ∇ 2 µ – 32 – 5 σµ ∇ n σ β ν + µ ( s λ µν R v ˜ n n n g s s ν 3 µ ∇ + ds λ σ ˜ n λ ∇ µ n + ∂ = ν µ ν 4 v anom C µ + n σ n n λ Z S n )˜ v ( µ + µν σ ( X s + n σ n ∇ δ µν ν µν 4 2 v g g v νµσβ λ m λ + σ 2 σ ( m [ R ) to replace ∂ λ ∂ − [ [ s h = 0 gives us the final equation [( ds ∇ geod . cancel automatically and the remaining terms are C.4 C ds ds ds S ds Z 3 C C C X MPD ) there are some non-tensorial terms which we may be justified in λ C Z Z δ Z ) with the above definition, we find the following relations: = 0 = S Z 0 s + + X = = C.7 ’. Let us now gather together these terms and show that they add up δ ). ) and ( C.4 = ) we define 0 A.6 C.3 probe S 3.38 ) and ( ) and ( X δ ‘unwanted unwanted set of the equations of motion should be covariant even off-shell. In fact we have C.6 C.3 ‘unwanted As a final remark, one may be concerned that to demonstrate the covariance of the We thus have In ( full . As in ( µν which is the Mathisson-Papapetrou-Dixon equation. equations of motion wethe had to usebeen the a on-shell bit equations quick. of In motion reality for a variation ( of the particle path also induces a variation of the and hence setting s Using ( Our final step is to write the terms involving normal vectors as functions of the spin tensor To obtain the second equality weof used the the metric definition ( of the Levi-Civita connection in terms terms proportional to Next we use ( labeling ‘ to zero: Here we have used the fact (following from ( JHEP07(2014)114 ). (D.2) (D.3) 3.24 (C.14) ). The lies along D.2 C 2 , and thus the n dy ) z , the direction per- z y z ]: z ) are already satisfied, Q 62 A , 2 C.4 , e 61 ) (D.4) = 0 (D.1) z variation of ∂ ··· + 2 z µν 2 + K ) and ( z . z any = 0. From this we may construct . ≡ G ) z − dy as encoded in the metric ( 2 ) ρ C.3 β z Q a z µν ∂ C z C + + zd δX i 2 µ K 2 ] α z − 2 ρ n is of the form [ − ) + ( zz z . If we pick coordinates so that µ αβ C Q µν zdz log( yy Γ 1 C zd ]. In TMG, we will show that it instead requires g g  )( 2 + − 6 – 33 – 2 − 1 z ` z = z = z , evaluated at V 0; these divergences cannot be canceled by matching µ − µ ≡ − K zdz v yy y n + ( ) g ρ R ∂ , to compute entanglement entropy we are attempting → ρ X z T √ ∇ ( + 2 µν δ z ρ A ρ g A z V 2 v 3.2 ≡ z e 1 2 + v ≡ K + − U ) is a function which regulates the tip of the cone, i.e. µ is z ( ρ a µν + 2 ( A C 2 R A that can support such a singularity. In the case of pure Einstein dzd yy [ ie g A C 2 be a minimal surface [ : they should be parallel transported to the new location by demanding +2 e +( ˜ along n C , = µ µ v 2 n ) may be neglected. ), which follow from the extremization of the entanglement functional ( ds 3.37 C.14 = 0, i.e. satisfy (the Euclidean analytic continuation of) the Mathisson-Papapetrou-Dixon n ν C ∇ First we understand the geometric properties of We would now like to show that if we examine these equations of motion, they will ) = 0 then the most general metric near ν z z, equations ( velocity vector the acceleration vector on to the restconstrains of the the geometry set and of gravity, we this should procedure demand requires thatthus the their requires vanishing that coefficients of vanish. the This that traces of the extrinsic curvature and pendicular to the cone. generically contain divergences as where all of the functions appearing here are allowed to depend on to construct solutions to the Euclidean TMG equations of motion that contain a conical( defect along a curve Dixon equations and the construction ofa bulk solutions conical to defect. topologically massive gravity with 1. Singular termsAs in discussed TMG in equations detail of in section motion: boundary condition method. then the actionvariation is ( already stationary with respect to D Details of backreactionIn of this section spinning we spell particle out some details on the relation between the Mathisson-Papapetrou- δX By including this variationshell we can and find reduces a toequations set the is of MPD to equations equations note of that on-shell. motion if that However the is a equations covariant quicker of off- route motion to ( the MPD normal vectors JHEP07(2014)114 ) D.1 (D.9) (D.5) diver- (D.11) . /ρ  y ∂ ) (D.8) z 2  K ( yy z g O K , + − = 0  z ) is satisfied, provided ) ∂ z ) ) = 0 (D.7) V D.7 z = 0 (D.6) U µν probe yy V z 0 where the last equality is g  yy T 3 ρ iK g ] (D.10) v → 2 3 + − a −  − U βµν z z σ µν U log K g, X, n R z K [ C (  y K i µ ∂ − 2 µ ( µνρ  − . s + probe β − v 1 µ z + ( S µν . Note if the spin is taken to 0 this simply 1 2 + 2 g K z z (0) = =
y 2 ∂ + 1 z ] + ` A – 34 – ∂ ) s ρ m g ↔ z K 1 [ µ j − side of the equations of motion contain 1 y i µ z satisfies the MPD equations, then the singular part ∂ v z R − s 0 here with no loss of information as the geometric z iUK i TMG C Here we consider a different route. Consider sourcing µν σµρ S yy g  + g → + 2 1 2  z = and √  z s z K − i S K zz K yy y ) take the form + µν g ∂  σ R √ − a 3 2 iτ ( yy  3.37 e g m m  3 ) we find these equations become ρ − = 0, i.e. the geodesic equation.  2 2 1 / z 1 3 πG yy g 8 D.5 arise from the fact that the epsilon tensor used here is the Euclidean K do not depend on the strength of the singularity. ∼ − i = = . 0. We may set z µ z C z C a G K ρ ) and ( → ∇  ρ D.4 v ≡ Next, we simply directly evaluate the left-hand side of the equations of motion ( Next we note that in terms of the acceleration and the jerk the Euclidean continuation µ j This results in the Euclidean equations of motion of the bulk equationsdefect of along motion vanish and a geometry can2. be constructed Matching with delta athe functions. conical TMG action with a spinning particle by studying the combined action the ratio of the mass to the spin of the probe satisfies the relation Thus we conclude that if the curve gences, e.g. This divergence will vanish precisely when the MPD equation ( and the corresponding equationstates that with using the cone ansatz. The The factors of signature version and our previous definitionUsing of ( the spin tensor was in Lorentzian signature. properties of of the MPD equations ( These expressions require usvalid to as evaluate and its covariant derivative, sometimes rather unfairly called the “jerk”: JHEP07(2014)114 µ X probe S (D.15) (D.16) (D.17) (D.18) (D.19) (D.20) (D.12) (D.14) . i 1) − 2 2 n relative to other ( 5 i λ , ); the geometric terms .  . 1)+ µν   − ) D.11 . , δg α µν 2 1 µν α Γ ( n Thus we need only verify that  δ ( α 4 α L µν 16 λ − ∇ µν + δg + L ] ˙ µβ constraints X + αν S · δg [ µν probe ν µ 2 µν + T L n ∇ 1 2 3 δg = 0 (D.13) + λ αβ  and the metric µν ] g anom t g + S 1 2 αµ α µν µν [ + ˙ √ probe ν X – 35 –  x T + takes the form · L 3 νβ µ g d 1  ∇ δg √ n α geod µ y 2 Z , probe 3 S λ ∇ ν d S = have been redefined with factors of ˙ − ∇ + = αβ X , i Z 2 g µ 1 λ µν n ˙  n t = X · probe 2 1 probe 1 = µν · ∇ S δS n g = 1 2 probe λ q h µν α µν probe δS Γ ds n ds T δ ds C C C Z Z Z s m i i = = = geod anom S S . To understand this consider taking the divergence of ( i n We will do this in two steps: first we compute the Euclidean stress tensor arising from constraints . Thus consider first variations that treat them separately While we have checked it explicitly in this case, the fact that the stress-tensor of a brane is conserved S 16 α µν if the action of thephism brane invariance. is In extremized this withvectors respect context as to it well all helps as dynamical us with variables to respect follows understand from to bulk why the diffeomor- we particle must trajectory. vary with respect to the normal Taking into account the relation between Γ we find that the full stress tensor is depends on the metric bothΓ explicitly and through the definition of the Christoffel symbol In this appendix somesections of of the this paper. This metric variation can be divided into two parts, as with This conservation law is equivalent tothe the backreaction MPD of equations. the probe creates the desired singularitythe in probe. the In spacetime. Euclidean signature and vanish as an identity, and so we find In this section weis will the show desired that conicalconsistent the defect only resulting studied if spacetime extensively we sourced above. minimize by The the the resulting action spinning equations of probe are the then probe over the dynamical variables where we have defined the stress tensor of the probe by JHEP07(2014)114 ). i ν 2 D.20 (D.29) (D.24) (D.25) (D.26) (D.27) (D.28) (D.21) (D.22) (D.23) n ) arising µ 2 n , 5 C.4 λ  )) s + 2 ( g ν 1 1 x √ . ) and ( n − µ 1 ):  . n y C.3 )) 4 ( αµβ s λ ! ( L (3) D.18 x δ )) ν αβ s + 2  − ( ) α x i ν y ) ( + Γ ∂ g v , ν − ∂x µ v ( 2 √ (3) )) . . y µ . − n δ . s ( 2 ( αβν ) ( 3 α  n ) ν 2 L 2 λ = 3 (3) ν 1 X να n n δ ∂ g λ i s µ αβ n  µ 1 ∂s ν 1 ; from the second to the third line we α µ − µ 2 i + 2 2 ν n √

) n )) + v ( n n ν y ) n s µ ) α coming directly from explicit dependence ( s + Γ ν ( 1 µ v ν d ∂ − ds v ds x v ∇ ∂y (3) to the total stress tensor as given by ( µ ν 2 µ µ ( 1 ds δ ( 1 ν 2 − βµν ( Z n α ds ds n n v n n probe µ 1 L y 2 i 2 Z 2 µν µ 1 α – 36 – ( ds S n as defined in equation ( λ Z Z λ v s ( n L α αβ i C s s = (3) α s i i Z µν ] δ + 2 t = ) + ) +  + Γ = + − ) ) ) να ≡ [ α ν 1 ds v ν 2 ν 2 α µ n n n µν µν ∂ L ds αµν s αµν αµν Z µ µ ∂y L ∇ ( 1 ( 1 s L L L µ n n i ( 2 Z α ( ( n , β αβ β αβ ∂ = s ∇ ∇ Γ Γ s s 2 g = β αβ − −    √ Γ i i i αµν = = L αµν − + + + α L ν ν ν ∂ = α v v v αµν µ µ µ ∇  L v v v g α 1 m m m ∂ ) simplifies to √ h h h  ds ds ds α ) but with the spin tensor defined as D.27 ∂ Z Z Z ∂y C.12 = = = The contribution to the variation of µν t on the Christoffel symbol is In particular we find that From the first to thefrom second the line variation we of usedused the the ( Euclidean action analogs with of respect ( to equation ( Using the following identities The first term is We want to evaluate theTo contribution start, of consider the term and begin by constructing the tensor We now need to actually construct this variation. For convenience we define the notation JHEP07(2014)114 .  ) and (D.34) (D.35) (D.36) (D.30) (D.31) (D.32) D.34 .  . α ) ∗ ν ) ∗ s )  µ α ( zy ) v ), i.e. we have ν zy probe = 0 (D.33) C s ( T µ ds ( − 2 D.3 v = ( µν Z probe = dy )  ds T z zy α z . Z zy + C probe ∇ K ) i  T )  ) (D.37) to 0 at that point. Computing its ν 2 + α z − n µν z ∇ ) U µ z i z, ) ( 1 C ) defined in ( ( ν 2 n i sources a delta function in the Einstein K ··· z  µ + n ( (2) ): µ A i + z, + m ( 1 ∇ + z 2 α ( n ) ∂  ( yy πδ ν A z, µν ds ∂ ) to set g s A and the spin tensor following from ( g ∇ − 2 yy 2 ∇ v Z ∂ 1 g ` = µ + ( – 37 – ds s ∂ ( 3.19 i z v + − yy Z α g ∂A = s R iv ∂ ) dzd i ∂A + ) 0 µν ∂ z µν + = z g ( → z, ), we find after considerable algebra lim ) gives ν ( a α 1 2 ∂A  K ˙ v A ) L ∂  2 µ − z α ˙ e v µν D.32 ( 3 2 D.26 s yy ∇ K m i α µν =  h 2 L πg R 2 ) gives us for the total stress tensor: 2 3 yy  4 ds + − ds g ] 3 = Z D.31 αν = 1 [ πG µ = 8 zy L probe C ) + ] x µν probe ) and ( ( T αµ ), expanded only to first order in ( [ zy ν D.22 L T D.2  α Furthermore, constructing the velocity We would now like to show that the backreaction of this stress tensor on the geometry ∇ , and we can use the gauge freedom ( y Here for ease ofdimensional comparison delta with function the using Cotton the tensor function we have chosen to represent the two- where we have neglected terms that are regular at the tip ofevaluating the cone the or stress are tensor higher order ( in of Cotton tensor we find that the singular terms take the form delta function structure in theansatz Cotton ( tensor. To that end we consider the regulated cone In this calculation we will construct local equations of motion evaluated at a fixed value As it is well knowntensor, that the we part must proportional show to that the new spin-induced delta function sources the corresponding The conservation of this stresson tensor general is grounds. equivalent Note toanswer that the can MPD the be equations, normal written as vectors in one themselves terms have expects of vanished, the and spin(as the tensor. dictated full by the TMG equations) creates a conical singularity: Adding ( After the dust settles, we find Combining this result with ( JHEP07(2014)114 ]. (E.1) 87 (E.3) (E.4) (E.5) (E.6) (E.2) , (D.38) conical 34 radius via , 3 )) R F spinning ] − 88 L , F ( . ,  . a ∧ 1 # ¯ µ` J  β  0 1 0 0 A a " 1 + e Tr( ∧ ). ` 1  = A Z k − 3.6 − In doing so we will show explicitly ∧ a = k ω A πµ` 17 ,J R 6 2 3 4  ) we see that the delta function struc- ]. c # 7 + − = , = . ] 1 0 3 ) can be rewritten as [ R D.11 0 0 R 3 R − dA 1 ` A G " 3.1 [ µG and in the fundamental representation ∧ 4 4 c = ,A , k CS A J – 38 – = + a S =  )  J abc k s  1 1  Tr µ` µ` ,J a ] = e # b Z − ` 1 2 1 / ,J π (1 + k 1 a 4 +  J − − a k ] 2 0 ω / 0 ] = L = 1  A A ) are [ is related to the gravitational Planck length and AdS [ " [ L 6 = R . The TMG action ( c k , = CS CS L 0 (2 S = S , and the Chern-Simons action is L,R J A ) sl ) L A ]. R 1 k µ` , ∧ 85 (2 − A sl + ) are the Brown-Henneaux central charges ( ] for an different (but compatible) approach to this subject. ∈ R 86 = (1 dA , c L L,R = c A 18 TMG F S We start by writing the TMG as a Chern-Simons theory with a constraint [ It is interesting to note that a spinning particle in TMG creates a pure (i.e. non- See also [ Our conventions for 17 18 It is convenient to define left and right levels, where ( The Chern-Simons level with Define where In this appendix weSimons language will using construct the techniques the developedhow in effective [ the action Chern-Simons fordifferent and the perspective spinning gravitational on probe prescriptions how inclarify are to certain Chern- equivalent. construct properties a of This spinning the metric-like particle will construction. and, give for a some readers, may appropriately. Conversely, a spinlessdefect particle geometry in [ TMG actually creates a E Chern Simons formulation of TMG ture agrees provided that we have in agreement with the previous results. spinning) conical defect, provided that its spin is tuned to the Chern-Simons coupling Comparing these with the equations of motion ( JHEP07(2014)114 = ∼ R . In L (E.8) (E.9) (E.7) 2) along F , (E.10) ] . These U (2 , 91 particle in i so , ¯ h 88 h, | ¯ h A ∈ : we will pick the = G i for the gauge group ¯ h R when and an extended conformal h, infinite-dimensional | 0 3 R , ¯ c J ] C 6= ) , ), defined in the via a highest- L . A c R ; h . massive and spinning ,  = 0 U − ( i A ] it was argued that the Wilson line ¯ h S 7 ¯ h C − Z = h, | s 1 is a local symmetry of the probe . There is an infinite tower of descendants ¯ ], in [ J exp[ exp R G ) 94 U P – , R ¯ h ,  , D i . More concretely, we replace the trace over 92 + ¯ h R – 39 – (2 R Z h sl h, | = h ∈ ) = = 0. One of the purposes of this appendix is to show 1 ) = tr m = C ) of ordinary TMG, only with fields now valued in a higher spin ` ± s ( C , i ( 0 which have a global symmetry group is design that R ¯ h ¯ E.3 J R C W U h, . In this notation, the mass and spin of the probe are geometries. For present purposes we highlight the following ) | W ¯ n 0 3 A ] took ) and 19 ; 1 7 as the Hilbert space of an auxiliary quantum mechanical system L ]. − U ) is a one-form that enforces the “torsion free” condition ¯ satisfies J ( ,J R R ( 90 S i , β n , so that upon quantization the Hilbert space of the system will be ¯ h ) (2 1 , with an appropriate choice of representation = 0 Following the work of [ 89 h, − R U i sl | ) J ): . ¯ h ). The analysis in this appendix in principle extends in a straightforward manner to ∈ R C , h, 1 | E.7 ,R 1 ± (2 is the massive degree of freedom that characterizes TMG. Without it, the , (2 J 0 sl sl . The natural unitary representations for the probe are 2). J β 3 , × L G ⊃ ) the curve dynamics of precisely the desired representation by a path integral, where the action how to account for non-trivial spin in the representation. that lives on theintegral Wilson over some line. fields This auxiliary system can be constructed as a path Here created by ( The construction in [ AdS particular the highest-weight representation ofweight SL(2 state R One can interpret We need a representation that carries the data of a Our goal is to construct a massive and spinning probe in the Chern-Simons language In this notation, , There exist proposals for theories of topologically massive higher spin gravity, see e.g. [ . The addition of this constraint is not just a matter of aesthetics: the Lagrange 2. 1. = SO(2 19 (2 R compute holographic entanglement entropy for these theories. furnish a dual descriptionsymmetry. of Higher the spin TMG currents actionsare in are identical holographic known to in CFTs closed the with formalgebra first-order only action in ( first-order formulation: namely, they and show that itconnections, captures i.e. the locally same AdS physicsproperties as of the ( MPD equations. We specialize to flat correctly captures thesl dynamics of aG massive particle in AdS F multiplier theory remains adegrees topological, of parity-odd freedom [ Chern-Simons theory of1. gravity Wilson with lines. no local JHEP07(2014)114 . µ ˙ x µ (E.22) (E.16) (E.17) (E.18) (E.21) (E.13) (E.14) (E.15) A , L = , U R A L A , A , R ) ) ] (E.12) , i.e. U s + s C ( C ( ) L 1 − R L R − ) U R s d A U L (E.19) (E.20) U ∂ ; ) s + L d ) is ∂ = U R and + C ( ] = 0 ] = 0 . L = ( , A L ]. L L . L R U R 7 . S R )( L A U ) s ,P U ,P W 1 − ( )( 1) L R R 1 − ] = 0 ] = 0 s ) it is useful to treat the connec- ( − A P L R A ) (E.11) − 1 ,D exp[ L R C ¯ h P ) E.8 ,A + [ ,A ( L ( 20 + [ 2 ,D 1 → ¯ R h R c U → L + ) R − ˙ L 2 U P L 1 ) is D W P R − ¯ c 1 U s ) s P = 2 C L −
A R 1 ( Z − ˙ C 2 2 L U − U ( L [ ¯ c
[ P ,A L = ) P , ∂ 2 , ∂ R W ( ) s − − ( s W P  , (  − – 40 –  L R 1 = 0 (Tr L = 0 are elements of the algebra and correspond to the 2 1) R 0 R A − L L ) = A ), the equations of motion are L P R ). The effective actions for each Wilson line are as A P (Tr L λ − C P ) C P P L R C + Z s ( L − h + ( R ( λ P ( 1 λ R 1 ) R
λ E.15 R h , respectively. The information about the representation − L ) = 2 s ˙ 1 + − exp and U ( W L 2 W U − 1 L
+ 2 R L U + 2 P L R P = 2 − R U R 1 ˙ P U  R 2 − U U → L L → c  gives the following equations for Tr( U ) and ( R  R U U and R L R L L d R D ds d R 1 ds U D P D 1 − U E.13 L R L are understood to be the pullback of the connections along ,P − R ,P ). Here ) = tr U P D ) U ). The action for the “right movers” in L R R C and s , ( L,R ( , we have P R U separately, i.e. a L ) L Tr , A R s P J R ( R W a ds Tr SL(2 A L U P C SL(2 ds ∈ Z = → → ) ∈ C and s = L ) R Z P ( s L U U L ( R = S A L From the actions ( To construct a Wilson line in the representation ( R In what follows S is encoded in the quadratic Casimirs via 20 Integrating out And for where conjugate momenta to R and the local symmetries are given by which is invariant under with and an analogous definitionfollows. for The action for our “left movers” in tions where For more details on this construction, we refer the reader to [ JHEP07(2014)114 ) E.19 (E.28) (E.29) (E.30) (E.31) (E.32) (E.23) (E.24) (E.25) (E.26) (E.27) we get are equivalent c J L,R abc U  ), we can write these ) with . Next, define A.11 , . We find . µρ 3 . s ] = 0 (E.33) , µ νρσ E.24 ∇ µ ˙ L ν x , ρ µ a µ R c ] = 0 v s e ) are equivalent to s ν , e, P a c ρσ ) using tangent space indices is + v . s J J e, P . ν µ + [ = , v E.24 , = = 0 := L E.29 1 2 + [ = 0 b a mv P p v c νρσ S a a µ s = = 0 ] = 0 = ∇ ] = 0 v J ∇ bc R c µ µ α ,S a ) and ( p p p  J , e, S ρσ To see that the dynamics of b := ab β , e, P s ∇ v – 41 – s v ν abc + [  v E.23 + [ + − abc . Then contracting ( 1 2 ] = 0  P + a , e S β ] = 0 1 2 R p − µν a p ∇ S ∇ s p α ∇ b ν a = ∇ v e, S := e e, P J [ µ a µ c + p e s + [ − := ∇ αβ = P R s P P gives ∇ ab ) structure. We start by rewriting the MPD equations in terms ∇ s c ∇ ) for spacetimes that are locally AdS R J , abc E.23  ). An equivalent rewriting of ( ) and we get a ] = ,R A.9 b J ,J . a SL(2 3 J ). In terms of covariant derivatives, as defined by ( ∈ a J E.20 With this notation it is straight forward to compare with the Chern-Simons formula- Summarizing, the MPD equations ( Next, consider ( tion. The equations ofand motion ( for the leftequations and as right moving momenta are given by ( Emphasis should belocally placed AdS on the fact that this is only valid for spacetimes that are Multiplying by where we used ( where we introduced the notation where and using [ where the canonical momentum is given by to those of themakes MPD explicit particle, the it SL(2 isof convenient the to momenta write of the the MPD particle, equations in a way that A. Wilson lines and MPD equations. JHEP07(2014)114 ) R for A ) are z E.13 s (E.36) (E.37) (E.38) (E.39) (E.40) (E.34) (E.35) ( and R , in ( and ¯ P L C ) z A R L,R , A S , 0 ; ) and z . s ρJ R 1 , ( J − U R L L e ( − ¯ z P P P e R 1 0 )) )) S − s s ρJ J ( ( + − − x x e e 21 by . C , ) 0 − − . Thus = = 2 L J s . x x c 2 . ( ( A c E ; source 2¯ L L (3) (3) it ) = P R, √ δ δ s U )) = ¯ a ( ( − s − ρ ( L µνρ µνρ φ 2 R ) = R   S ) we identify s P ρ ρ and P = ( ds ds R ¯ ] + z = dx dx Tr( R E.32 ds ds A source [ L, Z Z ,S ,L CS a ) with respect to the gauge fields i i – 42 – , ,P L 1 S , 2 0 dR , R P − To fix the Casimirs relevant for computing entan- c ) with ( J 1 E  2 − ) = ) = E.35 + it c 1 by x x µ` 2 R R ( ( LdL + E.33 )) = φ √ P + s + φ ( 1 + = 2 1 L,µν R,µν R L = −  ) = F F and P P s i z L ), and the probe is described by the actions t (   as the Fefferman-Graham radial cordinate. Our probe should ) we introduce L − Tr( source 1 1 ] µ` P µ` ρ E.4 R, L source a − A E.36 1 L, [ 1 1 + −  ). Comparing (  CS La R S π which have to satisfy π k k = = 2 2 E.2  s L R 1 µ` A A is given by ( − ). If we vary the action ( 1 CS  S i E.15 are flat. In order to solve ( In Euclidean signature it is convenient to introduce complex coordinates = Note that this identification is not unique; the normalizations and relative signs are not fixed. This is 21 L,R S simply because we have not specified how the Chern-Simons variables map to spacetime variables. which is, for example, achieved by In this appendix webe use parameterized (at leastindependent close of to the boundary) by the field theory coordinates where and ( we obtain B. Backreaction of Wilsonglement entropy, lines. we consider the backreactionswitch of to our Euclidean probe signature on the and gravity consider background. the We action This proves that the conservationated equations to of the the Wilson leftA lines and right are moving equivalent momenta to associ- the MPD equations, provided the connections where we used ( JHEP07(2014)114 ) 3.41 which (E.45) (E.46) (E.47) (E.41) (E.42) (E.43) (E.44) α : this sets α = ¯ . α ) is normalised . ¯ z 2 , R 6 1) c )) 1 we find ρ, z, α − C ( . n ) → , = ¯ R ( φ . In the following we will . n ( 0 0 W O  3 δ J J )
1 µ` 2   E t log ( ¯ ¯ z z 1) + ( ¯ ¯ dθ z z d d 2 n 1 + − πδ 1 1) − − −  n ( 1 − z z ) = dz dz R αk 1 6 ¯ α z c (   → ) on our Euclidean CFT spacetime by = ¯ 2 n z, . =   r ( 2 Having constructed independent left and ) because i 2 c 2 1 1 r, θ µ` µ` + c 2¯ (2) )) 2¯ )) + lim 2 = √ 1 1 − πδ E.36 C ¯ z √ C dr ( 1 1 + ( – 43 – ρz =   R L  ρ , k k 2  W W , 2 e 2 2 1 z 2 2 ¯ c c L 1) 6 are at the boundary of AdS + c  log ( log ( r r 2 ¯ z − α C ) generate the asymptotics, whereas the coupling to the ∂ ) in the main text. In the limit n n n dρ = =  = ( 1 1 = − − = O  L, R 1 1  (or 2 1 2 1 1 ¯ 1 z µ` ` source source The strength of the conical singularity is governed by n ds → →  1) + ) in a semiclassical approximation. . L, n n R, − z a a iθ 1 − ∂ = lim = lim −  n E.47 and the corresponding conical metric reads ( re EE L αk 6 c S = ). = = z source 2 E.9 2 c R, c 2 a 2 and ¯ √ √ = iθ re ] we postulate that 7 source = L, when the endpoints ofexplicitly the evaluate curve ( upon using ( 2. Holographicright entanglement Wilson lines, entropy. we canin compute [ holographic entanglement entropy. Following the logic z should be identified with This fixes the Casimirs. As expected, they reproduce the values of mass and spin in ( a Here we have introduced polar coordinates ( For the purpose ofreplica computing geometry entanglement at entropyconical the defect (and boundary), on hence the the boundary mimicking effect metric. the of This correct the is simply Wilson achieved line by setting should be to create a Moreover, the antisymmetricsuch that epsilon tensor in the coordinates ( Let us now calculate the metric associated to the solution. It is convenient to write This is indeed a solution to the equations ( source is taken into account by where the gauge functions ( JHEP07(2014)114 ) C ]. 7 E.52 (E.56) (E.52) (E.54) (E.55) (E.48) (E.49) (E.50) (E.51) ). The = 0 is L E.22 A , α . ± . 6= 0, so the task . x ∆ , and the boundary . 2 2 L ) ∆ -dependence. Un- c  c L i 0 A
ρ . a 1 ≡ A = ¯ i 6= 0 we can gauge-up f ρJ − s ≡ − 0 )) that characterizes − dx L s 2 R, (0) ( = x 0 A = 0) (E.53) P L x ± s (0)) Z s U exp ( Tr ( ) α (0) , x − f ) ±  ). L s s − i x  1 , ( ( , in terms of ¯ a ) − ρ 0 i L ). The solution with ) f − α ); in particular, combining ( = 0 to u 0 U s , E.51 ) ) gives ( dx ) 1 s (0) ) f R, ) exp 0 − α x 0 s L s 0 P E.13 ) ( ( x L, ) = ( E.52 U ) 2 0 f Z s ) = s P E.53 c ρJ s ( ( ) L, f (  in ( s µ − s s − ¯ α U ( L ( ( x ) L ± − L = α s U ( (0) U 0 − ) is 1 L L 2 L, – 44 – − ) determines ∆ exp ( ) ) = exp P f f 0 ) = exp ( ) = , x E.21 s , ρ u s ( 0 ± ( ) is still a valid expression when , ρ ) Tr E.53 ρ The path L = x L s ± = Tr ) = exp ( ( ( 0 U x ≡ s ˙ = α ( ( 22  E.50 R, 0 0 2 2 c ds L, L, = 0) U C ,L αP r ,U Z s 1 for the set of backgrounds ( ∆ α )), which is a solution to the equations of motion ( dR , R ( ) − s 1 − − s ρ ∆ ( α e ( −  = R , ρ R ) 0 (0). Equation ( LdL ) = background. s α f = R, ( Consider first evaluating the action of the probe for an open interval s = 3 shell − ± U R ) L − x 2 cosh = A f ( A s on s L is a flat connection, to construct solutions with ) = ( ( L, s ρ α ( L S ≡ . More explicitly, taking the trace of ( R ) we find A ≡ L ) = 0, and the on-shell action is given by U ) our probe will transform as s = 0 to U 0 α ( is the parameter along the path, varying from L L L, E.49 ˙ A s P E.51 For the right invariant action the logic goes exactly the same with the appropriate To do so we impose boundary conditions on ( Since We start by solving for the dynamics of We remind the reader that we will be rather brief here. Details of the technique can be found in [ 22 a solution to the equations of motion ( replacements: provided a background connection of the form where ∆ data for where appeal of this approachat is hand that is ( to compute ∆ with ( We have made ader ( particular and conventional gauge choice for the from with in any locally AdS satisfies the following boundary conditions Here A. Open path. JHEP07(2014)114 )), C ) can ( (E.59) (E.60) (E.62) (E.57) (E.58) shell ) we have E.58 − ) has to be ), one com- s on E.52 ) for a closed L,R, ]. C E.55 ) and ( ( 7 S . . The total action − ), ( φ
L,R a ) and ( E.54 W (0) (E.63) exp ( R E.51 ) ]. ). ) is the holonomy around E.49 0 U α . ∼ f 1 J 95 s ) φ ∆¯ − ( ) 2 ¯ λ E.47 C ), these define the on-shell ac- ¯ c L f ( ( 1 s f − ( − , tr R L,R . (0) ) (E.61) ) = E.46 0 R U 6 φ L (0) ) 0 c W ) J 1 0 . L shell f φ 2 π R, J − πa s − λ 2 ) φ φ ( P as the diagonal matrices whose entries ( f . Then using ( ¯ λ on f R s − ( π φ πλ ( 1 R, f ) tr λ from the equation ) Tr − 0 L S 2 tr s 2 J c = 2 ( = 2 α 2 φ + 2 = exp(2 ˙ (0) ¯ c ¯ α φ λ L and r ) = R ( – 45 – π 0 r f ds L 2 shell αλ π 0, this prescription reproduces our earlier results. L, f λ tr C − , − ∆ Z L f 6 ) the closed Wilson loop will reproduce the thermal on αP c s = 2 = − − L, π = Tr α = α S = we find E.47 s ! ∆¯ ∆ = = 2 2 φ 2 ), ¯ ) is simply c exp (∆ a at shell gives . Defining ), we solve for ∆¯ , and we take ∆ r − π 2 f 2 α To end this appendix, we will compute ). In the semiclassical limit E.57 c E.15 on L,R ) and ( and U . Combined with the Casimirs ( thermal ∆¯ U R, E.54 + 2 p L α S = S

P φ E.57 i E.46 )–( U ) = ∼ ) and ( 0 J φ and ∆¯ E.53 2 cosh L α λ ) and ( E.50 ( f this yields the formula for the thermal entropy in [ R E.50 is the diagonal matrix whose entries are the eigenvalues of ¯ c φ = ¯ λ L The main difference with respect to the open interval derivation is that U( It may be verified that when applied to the backgrounds treated in the main text To summarize, given a pair of flat connections in the gauge ( c For is the sum of ( On the right, one proceeds analogously; the result is where And using tr where we used that forthe a spatial closed loop cycle the combination are the eigenvalues of spatial curve; using ( entropy of the solution. Again, most of the derivationsa here periodic follow function, from [ the on-shell actions in turn yield the entanglement entropy(Poincar´eAdS ( and the rotating,boundary planar conditions BTZ on black hole), with theB. appropriate Thermal choice entropy. of putes the single interval entanglement entropybe as solved follows. for ∆ Equations ( tions via ( Analogously to ( The right moving action ( JHEP07(2014)114 , M ), one , (E.64) (E.65) JHEP A 209 (1984) , ]. E.65 81 (1984) 269 (2013) 090 )–( SPIRE IN 08 , with the mass ][ E.63 (1937) 163 − B 234 R 6 dx E ). JHEP  R , 1 , c J and L R R c L ]. Proc. Roy. Soc. Lond. k E , E Nucl. Phys. − J , ]. 1 + SPIRE arXiv:1310.4294 2 − [ ]. IN J . M Entanglement entropy for logarithmic Acta Phys. Polon.  ][ SPIRE , Proc. Natl. Acad. Sci. U.S.A. ]. IN − = , ][ SPIRE R = IN ) for arbitrary ( ¯ a Anomaly inflow and thermal equilibrium SPIRE ]. 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