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Entanglement Entropy in Warped Conformal Field Theories

Castro, A.; Hofman, D.M.; Iqbal, N. DOI 10.1007/JHEP02(2016)033 Publication date 2016 Document Version Final published version Published in The Journal of High Energy Physics License CC BY Link to publication

Citation for published version (APA): Castro, A., Hofman, D. M., & Iqbal, N. (2016). Entanglement Entropy in Warped Conformal Field Theories. The Journal of High Energy Physics, 2016(2), [33]. https://doi.org/10.1007/JHEP02(2016)033

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Download date:25 Sep 2021 JHEP02(2016)033 Springer February 4, 2016 January 13, 2016 December 6, 2015 : : : 10.1007/JHEP02(2016)033 Received Published Accepted doi: U(1) Chern-Simons theory in three × Published for SISSA by ) [email protected] , R , [email protected] , . 3 1511.00707 The Authors. c Field Theories in Lower Dimensions, AdS-CFT Correspondence, Conformal

We present a detailed discussion of entanglement entropy in (1+1)-dimensional , [email protected] Institute for Theoretical Physics, UniversityScience Park of 904, Amsterdam, Postbus 94485, 1090E-mail: GL Amsterdam, The Netherlands Open Access Article funded by SCOAP and W Symmetry ArXiv ePrint: in a WCFT viathe holography. concept of For geodesic theLower and geometrical Spin massive Gravity. description point In of particles theline in the Chern-Simons that the theory description captures warped we we the geometry evaluate introduce dynamics associated the of appropriate to a Wilson massive particle. Keywords: Warped Conformal Field Theories (WCFTs).ate We entanglement implement and the Renyi Rindler entropies methodour for to results evalu- a in single terms intervaldescribed and of in along twist terms the field of way correlation Lower wedimensions. Spin functions. interpret Gravity, a We Holographically SL(2 show a how WCFT can to be obtain the universal field theory results for entanglement Abstract: Alejandra Castro, Diego M. Hofman and Nabil Iqbal Entanglement entropy in warpedtheories conformal field JHEP02(2016)033 32 24 18 13 6 21 27 8 – 1 – ]. 5 ]. This states that in a quantum field theory with – 2 3 5 2 , 1 11 3 19 12 26 20 8 13 1 30 3.1.23.1.3 Renyi entropies Entanglement entropy at finite temperature 3.1.1 Entropy calculation 5.1 Wilson lines5.2 and cosets Equations of motion and on-shell action 4.1 A geometric4.2 background Worldline action 4.3 Two-point functions for holographic WCFTs 3.2 Twist fields 2.1 Modular properties2.2 of WCFT Quantum anomalies and preferred axes in WCFT 3.1 Rindler method geometric object in theThis bulk, remarkable e.g. prescription a relates minimal twoand area very geometry primitive in — objects the on — simplestof the quantum case the two entanglement of emergence sides Einstein of of gravity. aproperties the holographic of duality, spacetime its suggesting may field-theoretical that eventually dual a be [ refined through understanding the entanglement Entanglement entropy is a very usefultion tool structure for of organizing quantum our mechanicalfield-theoretical understanding systems. grounds, of one the In of correla- addition itsity to recent via being applications the is interesting Ryu-Takayanagi formula to on [ a the purely gravity study dual, of the holographic entanglement dual- entropy of a spatial subregion can be related to a simple 1 Introduction 6 Discussion A Entanglement entropy in CFT 5 Entanglement entropy: holography in Chern-Simons languange 4 Entanglement entropy: holography in geometric language 3 Entanglement entropy: field theory Contents 1 Introduction 2 Basic properties of WCFT JHEP02(2016)033 ]. ]. two 10 10 (1.2) (1.1) , U(1), 9 × any ) R ) invariance , class of sim- R , SL(2 . different × ]. Particular examples )  R 15 , , L π` 14 sin L , π   L π` log sin ]: vac 0 8 L – L π 4 6  − – 2 –  log ¯ ` ` c 3 − is enhanced to two copies of an infinite-dimensional are related to the identification pattern of the circle = ¯ L L 2 ], including the very simple theory of a complex free the length of the circle on which the theory is defined, ¯ L  EE 16 L ` S and vac 0 L ) (and, not unrelatedly, the similarly universal Cardy formula iP ] for recent work on the structure of entanglement entropy in 1.1 = 13 – EE S 11 are the separation of the endpoints of the interval in question in “space” and ), which we present here: ¯ ` 1.1 and the length of the interval, ` a UV cutoff. This formula exists because the global SL(2 `  One of our main results is the derivation, using WCFT techniques, of a formula anal- Given the infinite dimensional symmetry algebra enjoyed by WCFTs, one might then More recently, however, there has been a great deal of study of a However, entanglement entropy is also a notoriously difficult quantity to calculate in where in “time” respectively, and that defines the vacuum ofin the this theory. formula, The and non-relativisticclear nature the later. of precise the We meaning note theory of also is that evident space the and answer time is in naturally this expressed in context terms will of be the made charges of ogous to ( (massive) Weyl fermion. expect the existence ofthe a vacuum, universal similarly formula to for thein the well detail. entanglement studied entropy of CFT an case. interval In in this paper we study this question This is quite remarkable given thatories these (see types of however results [ other are scarce non-relativistic for field non-relativistic theories). the- densed As matter such, systems, WCFTs offer particularly a forof range Quantum WCFTs Hall of states applications were [ in constructed con- in [ ilarly constrained field theories, calledThese Warped Conformal WCFTs Field possess Theories (WCFTs) awhich [ is vacuum then that enhanced to is aries invariant single are only Virasoro non-relativistic, and under they a possessstandard Kac-Moody a a two-dimensional algebra. global similarly CFT Though SL(2 infinite-dimensional and these symmetry indeed theo- permit group there the as derivation exist of a notions a Cardy-type of formula modular for invariance the that the high-energy density of states [ of the conformal vacuumVirasoso in algebra, a greatly CFT constrainingversal the formulas dynamics such and as permitting ( for the the thermodynamic existence entropy of in uni- a high-temperature state). with and field theory alone, andquantum there field theory. are One fewa of exact single these results known interval available results in foris is the entanglement a that vacuum entropy for celebrated of in the universal adimensional entanglement formula two-dimensional Conformal entropy that conformal Field of applies Theory field (CFT) to theory, [ where the there vacuum on the cylinder of JHEP02(2016)033 ] to 20 – 17 ). R , and so should be dual 3 we describe the symmetry 2 ]; the reader familiar with these 22 , -gravity with no extra fields is related ]. This involves some novel geometric 3 . gravity, where the representation space 16 we use warped conformal mappings to 10 , 3 6 9 3 we explain how to couple massive particles to – 3 – U(1) rather than two copies of SL(2 4 ] for AdS ), as we will explain. × 1). In both cases we reproduce the field-theoretical 21 ) , ) for the entanglement entropy in the vacuum (and 1.1 R , 1.2 presents a novel covariant description of the Virasoro- SO(1 / ) 2.2 R , we study the same problem in the Chern-Simons description of Lower 5 — has been more recently understood in [ 2 We now present a brief summary of the paper. In section Note now that in three bulk (and two boundary) dimensions the Ryu-Takayanagi for- We turn then to a holographic description of warped CFTs and describe the appropriate 2 Basic properties ofWe WCFT start by gatheringfollowing some equations basic are properties basedresults of on can Warped the Conformal results skip Field in portions Theories. [ of The this section. We will also review and extend some results dimensions and U(1) charges.Lower In Spin section Gravity andentropy. re-interpret In section the resultingSpin geometric Gravity, where structures we evaluate the as appropriateand entanglement Wilson line. some directions We conclude for with future a discussion research in section Kac-Moody algebra which isare helpful part for of understanding the thederive definition preferred universal of coordinate formulas a axes such WCFT. that states as In related ( section to it by conformalalso transformations, interpret such our as the results finite in temperature terms state). of We twist fields, deriving expressions for their conformal tem living on the cosetresults SL(2 quoted above from a holographic analysis. structure of warped conformalliterature field before, theories. but Much section of this material has appeared in the first in a metric formulation ofof Lower a Spin massive Gravity, particle where moving we in constructentropy the the in bulk. worldline a action We Chern-Simons also describe formulationWilson the of line computation the of prescription theory. entanglement developed This inrequired requires [ a for generalization the of the Wilson line is now generated by an auxiliary quantum mechanical sys- mula relates entanglement to the lengthof of a a bulk massive geodesic, particle which moving isCFTs in equivalent we the to the bulk. will action then Tometric understand need its in to analog Lower for understand Spin holographic how Gravity warped to and couple study massive the particles resulting to geodesic a motion. background We perform this to a warped CFT,structure. these theories The also minimal holographic possessrelated to description a WCFT — in great the which deal sameto should way of CFT that be additional Einstein-AdS understood and as unnecessary ideas being (that we reviewinvolves below), the geometrization and of can SL(2 appropriately be called Lower Spin Gravity, as it secretly true of the canonical result ( generalization of the Ryu-Takayanagi formula.Einstein gravity While (supplemented there with other arethe fields bulk action), or solutions that a [ geometrically gravitational have Chern-Simons a term piece in that is warped AdS the vacuum of the theory, and not in terms of the central charge. The same is actually JHEP02(2016)033 de- (2.7) (2.3) (2.4) (2.5) (2.6) (2.1) (2.2) z , . The finite k 2  , 0 ) z ∂w ∂z (  P , ) 0 4 k z n and level . ( − − n ) c 0 n, ) z δ z ) as the right moving energy ( ( ) the operator that generates , . In terms of the plane charges z 1) g dz χ z ( . P n 2 ( 0 z − + T Z . 2 T 0 kγ 2 ∂w 0 ∂z = ! π w n 1 0 ) the operator that generates 0 2 − ( 0 n 2 z γz z ) n ( = ∂z 2 ∂z χ ∂z − ∂z ∂z z ∂ P c ( w 12 + 2 + =

0 and n  + γP 3 2 w } → kγ , 0 2 and n +1 − = , z z − − + 0 n 0 , , ,P n ) 0 – 4 – z z ) 3 , z ) z n { 0 z 3 L ∂z ∂z  ( ∂z z ( n ∂ ) = + , w 0 0 ( c 0 . We can think of these transformations as finite − , w 12 P T ) n n n 0 0 T w n, = ζ ∂z ) P z z ∂w 0 δ − ( − z } 2 n k ( ) 2 n f ) = ) = = n 0 0 n z , z − k ( 0 z z z = ( ( z T 0 0 ) + { z dz ζ  z T P ] = ( ] = ] = ( 2 0 0 0 ) as a right moving U(1) Kac-Moody current. We can define n n n Z P  → z 0 (  π i ,P ,P ,L ). On this plane, we denote as 0 z 2 P ∂z n n n ∂z P − L ∂z L ∂z : the emphasis here is to explain and highlight some geometrical [  [ [ direction: z, w = w n 2.2 ) = ) = 0 0 L z z ( ( 0 0 T P ) the commutation relations are n ] in section Among these finite transformations, there is one that is rather interesting. Consider At the quantum level, we define (in CFT language) Consider a (1+1) dimensional theory defined on a plane which we describe in terms 16 ,P n L Under this tilt, the currents transform as doing a tilt of the where which is atransformation Virasoro-Kac-Moody properties algebra of the with currents central are charge where we choose the test functions( as momentum tensor and charges coordinate transformations Classical systems which areConformal invariant Field under Theories (WCFTs). these transformation are known as Warped properties of WCFTs. of two coordinates ( infinitesimal coordinate transformations in pendent infinitesimal translations in in [ JHEP02(2016)033 ) is a 2.3 (2.8) (2.9) (2.14) (2.12) (2.13) (2.10) (2.11) α , which π ) is ˆ v +2 u, x . ∼ 0 ) are the thermo- x n, δ τ, τ  , c ) is the U(1) axis; 24 ¯ τ . t π ) depend on how we Wick − . ¯ τ ,  k  0 a τ τ, + 2 2 | n, α cyl 0 ) coordinates and (ˆ a  ¯ aτ k δ πτ, t + 2 πiτL x, t 2 − 2 γ n αx , . x , ¯ − τ − 0 e ¯ a a +  x α P n, + 2 1 n ( cyl | 0 + t 0 t ¯ τP ∼ Z + 2 γ P = ) γ k δ ) πi = n ¯ a a 2 ¯ a 2 τ L ˆ e v + + 2 π −  – 5 – n n ¯ τ = a , w ( P | , L ¯ + 2 a ¯ a ) scaling coordinate and ix a x cyl n = = R − πik , ) ) e e = γ γ ) defined the coordinates on the plane, the transforma- ( πa, t ( n n ˆ u = 2 ,L ) = Tr L P ). The reality properties of ( 0 ) = z τ − | z, w τ n, cyl 0 | τ → → x (¯ τ ( a (¯ n n ,L | a ¯ a P | α kδ L ∼ ¯ a is the SL(2 cyl Z 0 ) + Z ) to allow for any choice of spatial cycle, and (¯ P x n x, t P ( a, a = cyl n ). In particular, if we do the change of coordinates P a, a Consider placing a WCFT on a torus. One way to proceed is to have For most of our manipulations, we will be interested in computing observables on the the relation between the partition functions using the ( rotate back to real time. For this parametrization of the torus, the partition function reads With this notation itchoices of is (¯ rather simple to relate partition functions labelled by different where we introduce (¯ dynamic potentials for ( properties of such functions. defines a particular spatialidentifying cycle, imaginary and time to appropriately, which introduceWCFT, defines temperature this the choice and is temporal angular not cycle. completely potentialthe equivalent However, way by to in to choosing a proceed other spatial is cycles. to Therefore, define a more general torus defined by the following identification 2.1 Modular propertiesModular of properties WCFT of partitionlowing functions sections (and when density evaluating matrices) entanglement entropy. will Here be we important review in the transformation fol- where on the cylinder, constant tilt that controls howto we operator define insertions a on space thethe quantization plane. slice planes The in as modes our of cylinder the relative cylinder are related to the modes on real time cylinder. Given thattion ( that takes us back to the cylinder is This is the usual spectral flowinvariant. transformation, which leaves the commutation relations ( which implies that the modes on the plane transform as JHEP02(2016)033 . ], is = ]). 22 S 29 – a, b (2.18) (2.15) (2.16) (2.17) 23 where ab η canonical circle ] it was explained 16 ). Therefore, WCFTs ) will not satisfy this . a, a 2.13 a ¯ aτ − . τ . ) and (  = ¯ 1 τ ]. Our goal is to explain in detail 1) 2.9 16 | − | − , z z τ (0 as it is calculated on the   ). The resulting notion of geometry was τ vx . | τ ˆ | ¯ τ Z + 2 τ Z a 2 z 2.18 ¯ aτ t – 6 – − πik ) = → e ]): ¯ τ τ t | is a spectral flow transformation. As shown in [ canonical  τ 30 (¯ 1 1 | ) = | 1 0 | 0 τ 0 is equivalent to | Z Z z Z 1) as ( S modular transformation properties. More concretely, , ≡ ˆ Z ) and ]. τ | a z 22 | ( ¯ a ) = (0 ˆ ]: it turns out that the most efficient way to describe this geometry Z Z 16 a, a canonical has ˆ Z ) that relates 2.13 Let us look at tangent space invariant tensors. In two dimensional Lorentz invariant As it was stressed before, WCFTs are non relativistic quantum field theories. As such Notice that partition functions defined for other choices (¯ by translations and thedescribed in boost detail in symmetry [ is ( in the Cartan formalism, where the symmetries acttheories explicitly the in first tangent invariant space. tensor of the geometry is the flat space metric This symmetry plays thecan same readily be role seen incouple from WCFTs to the that two manipulations Lorentz dimensional around boosts ( geometries play where in the CFTs; local this symmetry of space-time is given they do not naturally couplethat to the background natural Riemannian geometric geometry. structure Inwhich in [ is, this case in corresponds two to dimensions,The a a main form type point of is of warped that Newton-Cartan geometry WCFTs(sometimes geometry posses called (see a Carrollian for natural boosts symmetry example associated [ [ to generalized boosts This section is basedwhy WCFTs on have (and two extends)coordinates preferred results in axes space-time in in given [ system space-time; by in these this later axes will sections. which allow justifies us our to parametrizations pick of preferred the simple rule. For aimpose complete on treatment the of theory these see [ transformations and2.2 the restrictions they Quantum anomalies and preferred axes in WCFT This condition implies that The function the modular transformation that exchangesof the the spatial partition and function thermal under cycles, and invariance tion ( the modular properties of–the system partition is functions not aretition Lorentz function sensitive invariant after with to all. (¯ theWe From define torus this parametrization stand point, we denote the par- Note that we have kept track of the appropriate anomalies, since the coordinate transforma- JHEP02(2016)033 ) : -axis 2.18 t (2.25) (2.23) (2.24) (2.19) (2.20) (2.21) (2.22) and the U, V
0 n ab g + a,n J . b q ab − h b 0 n V ) given by + a a q U b,n J . ≡ a q .  V 0 0 n . n × 1 0 + b n 2  ¯ q + component of a vector. The second ). Notice that there is no canonical . ab n δ n = x . h b a ¯ q  − , q 2.18 a ≡ ,U n !
¯ q 0 a ! P ) 1 n 2 k a t x + n q 1 0 v a + L

a,n

0 V  – 7 – n J and = , q b = + )( q b = a n a b q δ q x a + a a ) by defining generators: q Λ 0 a,n 1) n U J ) and an invariant one-form ( ! ≡ a 2.3 + − 0 1 q 2 = ( ab b,n

n g J ( ab a = g n q b to make another preferred vector. This means that classically, = 0. It is trivial to see that the loci of points on this axis a ab ¯ 0 q V a g x a q n , which are 2 c U ab − 12 ), we can write the Virasoro-Kac-Moody algebra in the compact form h ≡ n + V 2.21 · ] = 0 U b,n ). There is a quite transparent way to see this is the case. Let us covariantize ,J ’s and ( J a,n 2.14 J [ Quantum mechanically, the situation changes dramatically. The boost symmetry ( The existence of these invariant tensors is directly related to the existence of a preferred . In contrast, for warped geometry there are one index objects that are invariant under Using the This is already manifestboosts in ( the anomalous transformationthe of Virasoro-Kac-Moody the algebra partition ( function under correspond to fixed pointsway of to the raise transformation the ( indexthis in is the only preferred axis for a WCFT. becomes anomalous as a consequence of a non-zero level for the U(1) Kac-Moody algebra. metric means that the normof these is can sensitive only be to used the to define angles inaxis this in geometry. the classicaldefined geometry by associated to the WCFTs. equation This axis is just given by the These tensors permit two different notions of inner products between two vectors Clearly only the first of these can be used to define a norm. The degenerate nature of the From these objects we mayantisymmetric tensor also define tensor invariants: a degenerate metric There exists an invariant vector (¯ the boost symmetry, whichposition are vector defined as follows. Consider, in a covariant language, the and the boost transformation t, x JHEP02(2016)033 ) a q R , axis), axis). (2.28) (2.26) (2.27) t x as ). The term a q , we show that ]. While some 2 ], in analogy to is the classically 2.25 31 16 , . However, this is [ T 6 ab g ): a classical one (the a . ¯ q , a q = 0 a scaling structure ¯ q 1 ). a . , q b 2.24  q ], we could infer the existence of the second a q 16 0 1 − : here we must introduce a new one form ¯ 1, select which coordinate contains a scaling  k – 8 – − = is called a = b b a defined in ( a a ¯ q J J a,n is given by an invariant tensor J = 1 and c a q a q is the quantum anomaly selected axis with a scaling SL(2 , to geometry. There is an extra geometric structure needed, in λx X , which are 0 and b → a ). Following up on our previous discussion, here = 1) thus breaking the boost symmetry. This breaking is not severe, J x a ¯ q a T,X q . With this vector one can define the tensor a q Looking ahead, and to make contact with [ More explicitly, we choose the one-form as Not to be confused with the currents 1 . Then there are two anomalous terms given by the second line of ( a Our first task willsystem be is to on calculate its ground theby state. entanglement coordinates The entropy ( background of geometry a isU(1) single a preferred space-time interval axis cylinder when described and the entropy of an intervaltechnical as features the and thermal outcomes of entropythe this of applicability method of a differ the Rindler from method those observer is in [ equally a powerful3.1 CFT in a WCFT. Rindler method 3 Entanglement entropy: fieldIn theory this section wemethod,” will compute i.e. entanglement via entropy suitable in coordinate a WCFT maps by we using will the show “Rindler how to cast the entanglement The eigenvalues of symmetry and which onea does complex not. structure insimilar the role to usual that CFT of setup. the light As cone such, for a these CFT. preferred axes play a very preferred axis by demanding the possibilitya of scaling coupling symmetry a warped quantumorder field to theory have with this coupling, whichvector is nothing else but the existence of a covariantly constant Therefore, a WCFT has two preferredand axes another dictated one by selected ( This by will anomalies, be (which of in crucial our importance coordinates in is what given follows. by the and its existence allows us to unambiguously define a new preferred vector q accompanying the central charge not the case for(normalized the as Kac-Moody ¯ anomaly since it is governed by a well established anomaly; here is where the power of WCFTs reside. The classical part of the algebra can be easily written with the help of the invariant form JHEP02(2016)033 1 ` 2 X (3.6) (3.3) (3.4) (3.5) (3.1) (3.2) ). For a is related < X < 3.2 ` 2 D to a thermal − ) respects the D ρ 3.3 ), and for the task at , 2.1  ) to a set of coordinates x . . cyl 0 ¯ κ κ ) coordinates as: . This generalizes the results + κL  T,X . H t t, x ` 2 covers the region ρ − ) ) coordinates perceives a thermal , . = ¯ ` 2 ) iκ , where in an abuse of notation ∞ cyl 0 2 t, x L − X +  ¯ L L ¯ κP − ,X , 1  . + ¯  κ, x X i < x < ` 2 ¯ describing the vacuum state on L, X − = 0 − we will make use of warped conformal map- , t D + = exp x ( ¯ ρ ` 2 −∞ ¯ κ κ – 9 – D T H , in the above map; these scales are arbitrary, ,T ∼ ( κ  + ) κ ∼ t πx ∈ ) t, x ) and ¯ , ρ † will be made along the way. To relate κ ) coordinates the slice where the spatial identification is T,X :( U T,X 2 ( = tanh H t, x H L L :( Uρ π` 2 πX ). The real line ) coordinates. = D ) induces an identification in the ( 3.2 tan ) coordinates gets mapped to the line: tan D T,X ρ 3.3 . Moreover, the expected surprise that is a direct consequence of this then the segment is misaligned with the identification direction. For 1 ¯ L L T,X 6= ¯ ` ` ), we can interpret “inside” the interval as the causal domain of ( A ] to a case with symmetries different from that of a conformal theory, and appro- 31 , This transformation has several favorable features. First, the map ( To quantify entanglement entropy in 6 We interpret this result as the factdensity that matrix the observer induced in by ( this identification. More concretely cylinder identification ( and not the restis of depicted the in cylinder. figure Thefact domain is of that the causaility, which map turns ( out to be a strip, performed in the ( We have introduced twoand scales, the independence ofparticular the notice final that result in on the them ( will be used as a consistency check. In that cover only the “inside”appendix of the interval.warped In system comparison we with are ahand only relativistic the allowed system transformations appropriate (see of transformation is the form ( pings: we will show thatvia the a density matrix unitary transformationof to [ a thermal densitypriate matrix comparisons with aobserver CFT we first construct a mapping from the cylinder ( Notice that if later reference we denote theserefers two endpoints to by both of the ( circle is given by We will consider an interval inside this cylinder also oriented arbitrarily symmetry. In order to keep the discussion general, the identification that defines the spatial JHEP02(2016)033 ) ). t, x , we 3.2 (3.7) (3.8) (3.9) (3.10) H , expansion; in  .  in the domain of ) coordinates; we in the coordinate ζ x π κ  D 2 t, x  ) plane. , − ¯ ` 2 ` 2 ) T,X − 2 L . , in the small ¯ 2 L  , ¯ ) L L − ¯ X ` ` ( ζ ¯ ` L L 2 H + ( . + + ) ¯ ` 2  ζ ) ) on the ( ( ! 2 − ,T , π ¯ κ O ¯ ! 2 2 ); dotted line is the cylinder identifica- thermal  t, x − ( , S + , − ) 3.2 direction; this is necessary to guarantee     , ) = T L π` D +  and the domain covered by the coordinates ( ρ ζ ` 2 sin D π κ – 10 – − 2 log L π D −  , ρ  ¯ , ` ` ¯ ` 2 − Tr ( , ) + ¯ ` 2 ! 2 − = log − ¯ L L , ¯ ! ζ 2 =  ( ` 2 ! ) gives the image of this interval in the ( ∈ EE ) − ) 2 L S 3.3 ζ − , ¯ 2 L π ¯ κ ( 2 T,X  :( ∈ ) ). Straight segment is the interval ( D ) we kept terms that are subleading relative to t, x ( ). The naive answer gives an infinite range, so we introduce a cutoff parameter T,X 3.9 is a unitary transformation that implements the coordinate transformation ( . Diagram that depicts the interval t, x ); shaded region depicts the domain covered by ( U 3.1 For the above equality between entanglement and thermal entropy to hold, we need which defines the new regulated interval as where Notice in ( the following we will keep these terms since they could contribute to the final answer. the units are correctinterval. and Using that the the map regulatedobtain ( interval is actually contained in the original  Notice the factor in front of the cutoff in the to be ratherwe careful expect with the the entanglement divergentof entropy pieces the to have interval, of a requiring eachexpect UV the observable. the divergence introduction thermal arising entropy of On to from ainterest. be general the short IR To grounds, boundary divergent relate distance due these cutoff. tosystem divergences, the ( Whereas we infinite need size for of to obtain the length of where Thus Figure 1 relative to ( tion ( JHEP02(2016)033 ) is ) via ), we (3.15) (3.17) (3.11) (3.14) (3.16) (3.12) (3.13) 3.9 θ | ¯ θ ( 2.12 a | ¯ a S ¯ κ . i 2 − = , θ ) π 2 . πθ 2 ) . θ | . − ) ¯ θ τ ( | , a  z | ` ¯ θ, x ( ¯ a iκ , ) makes the torus non-degenerate  ˆ π Z Z ). In other words − − θ a have dropped from the computation since ` | = 3.13 + 2 3.9 κ log . Following the notation in ( L L ¯ a t ) log
θ a (  ¯ z θ πθ , in ( 2 and ¯ ζ ) − i ∼ ¯ z∂ θ∂ 2 θ κ ¯ | ) θ thermal ¯ ) and ( θ − − − ) we find S  ( πa θ τ kept constant. We can do this using Cardy-like → ∞ a ˆ | S = 2 3.5 is made arbitrarily small, all that is left is to – 11 – ¯ a z τ ` , ζ θ∂ τ∂ 3.14  Z − + − − ) = ` θ , z 1 | ¯ a, x L L ¯ θ ζ π 0 and ( π i i a H − | − ¯ a ) = (1 ) = ζ + 2 θ τ S | | ¯ = κ π t → − ¯ θ z ( ( ( τ τ = a ˆ | S ∼ ¯ a a ) S π ), the potentials relevant for the computation are 2 t, x ( 3.13 with the circle identification is what breaks the degeneracy of the torus. ); this is the frame with canonical modular properties. Using expres- D ζ , ) in the limit 2.15 κ π ), we now proceed to evaluate τ | = z ( 3.7 ) and taking derivatives as in ( ˆ Z is divergent as the UV cutoff is defined as πa )–( 2 ζ ). And from ( 2.14 ) does not dependent on them. 3.6 τ The problem has been reduced to that of calculating a thermal entropy. The entropy | τ Another comment is in order: as promised, both | defined in ( z 2 z ( ( thermal ˆ ˆ ˆ Since evaluate Z This illustrates that entropyMoreover, is we a can robust justZ observable pretend for to which be all in observers the agree canonical upon. circle and calculate where we have defined It is convenient to relateZ this definition to thesion entropy associated ( to the partition function and makes well-defined further derivations.yield It well is defined not results a evenmisalignment problem of in as the the degenerate formulae we case. will Still use it isS interesting to see that the There are two important pointsrather to emphasize large at we this expect stage. thethe identification First, edge of since effects the the interval, to yielding interval not the ( function. thermal be entropy Second, associated important keeping to and finite a we torus terms partition might as as well consider with Having established a relationvia between ( single intervaldenote entanglement the and partition thermal function entropy for where the data of the torus is built from ( 3.1.1 Entropy calculation JHEP02(2016)033 (3.22) (3.23) (3.24) (3.19) (3.20) (3.21) (3.18) ). Finally, U(1) isometries , 3.18 × ) R ··· 3 , + . is vac 0  L ) and using the fact that H 1 τ L π` πi , 2.17 ζ , +2 sin c 24 vac 0 vac 0 L we expect the minimum value of π . P L − is constant in the limit all we need . z 0 τ 4  . 2 P ) z τ πi k − 2 = 0 c q D Q 24 log e ρ  2 − τ ¯ ` ` z vac − 0 vac 0 2 k = 2 L − ) iπ 4 e ¯ L L log Tr( vac 0 k ,P vac − 0 q  = P c – 12 –  ` ( 24 due to subleading corrections in ( 1 − ¯  ` ` − ζ 1 1 τ vac = 0 − = iQ , L − = iP ¯ L L vac 0

− q L z τ 0 vac  S = ) = ` L  ): for a given value of to be complex, which occurs often in holographic duals to τ | ˆ 0 vac Z vac 0 z 0 ( 2 P τ ]: from modular transformation ( P z 3.18 ˆ iP S 2 k 22 = iπ , in ( e 0 are the cylinder values of the charges in the vacuum state in the 10 is real, as expected in a unitary WCFT we obtain EE ), we find L 0 S ) = P vac 0 τ | L 3.10 z ( ], the minimum value is ˆ Z and 20 , ) and ( 10 is a real vacuum charge. vac 0 3.7 P Q Gathering these results, the thermal entropy of the observer In other words, we assume that there is a state invariant under the global SL(2 is given by the unitarity bound 3 0 From these manipulations, itare is defined rather as straight forward to obtain Renyi entropies. These of the system. circle identification. Notice itwe is derive extensive this on same the answerinterpretation size using of of twist these the field contributions cylinder correlation will and functions become not and more periodic. holographically, clear. 3.1.2 the As Renyi entropies While one might imagine that theas first term a is response subleading it of might be the interesting leading to consider value to a misalignment of the segment with respect to the where we ignored subleadingusing terms ( in WCFTs [ where If the spectrum of If we allow the spectrum of to do is to minimize L the vacuum dominates the sum, the partition function is well approximated by where canonical circle. Notice that because the phase factor formulae available in [ JHEP02(2016)033 (3.28) (3.30) (3.25) (3.26) (3.27) (3.29) . With this ¯ β ) coordinates i → . T,X ¯ L  . . L π`
 and reduces to the thermal qθ x . β | sin π` iβ ¯ κ κ θ EE q L S π + → . a t sinh  | L a . ! = β Z π )
log q  X ) = iβ  qθ | θ ¯ β β |  θ − θ log q for which ( + 1 + cyl 0 a | q 1 a vac 0 a | ¯ a β, X  L Z i Z πiqθL 4 2 → ∞ + vac 0

− − ,T ` L yl – 13 – ). Note that the part of the entropy depending T c  2 0 ( log ¯ ` ` κ πx θP − q ∼ 3.23 − )  1 πiq − ` 2 ¯ β β ) coordinates is: 1 e −   T,X ` = a = tanh ( L | T,X q a ` L vac β 0 S β π` 2 πX  iP = Tr vac = 0 q D tanh tanh ρ iP EE ) system. Tr S = q S T,X 1 limit of this agrees with ( → q We turn now to afor slightly reproducing different these interpretation results of in theseNote holography, results; a first task one that that which we will our willwe be computations perform have useful been in up studying the until next the nowappropriate section. problem have reduced from been density a matrix Hilbert somewhat and space “canonical”, computing point in of its that view entropy. by There constructing is the a complimentary The thermal limit isentropy in obtained the by ( taking 3.2 Twist fields All the discussion goesidentifications on we as obtain before the with entanglement the entropy replacement to be: Now the identification in the ( 3.1.3 Entanglement entropy atAs in finite the temperature CFT case,entanglement a entropy small of tweaking of aneed the segment arguments to in above do can infinite is beis volume to used identified but change to along the at calculate its map the finite thermal such temperature. direction. that Concretely, the All consider original the we cylinder map in the ( The on the U(1) charge doescase, and not we depend leave on further the comments Renyi for index! the discussion. It is not evident why this is the Repeating again the modular manipulations as those above we find following partition function: Thus we want to compute Now the trace over (powers of) the un-normalized density matrix is computed by the JHEP02(2016)033 i H ) ρ X ( angle (3.33) (3.34) (3.32) (3.31) T h -th copy. i ), which we , 2 2.4  . ).
∂t ∂X q D 3.2 ρ  ) , 4 k in ( ) at the endpoints of the X C ( , X − D q implements the conformal belonging to the J ( C ) , then the precise statement q , we have C i q , x q O ) U Φ . In this section we will study q ( C 2 q Tr C R decoupled copies of the original P i C X ) q ≡ ( ) is implemented by considering a 2 of † q to the thermal density matrix q ∂t i ∂X X . As R ( )Φ i U † q 3.24 U 1 ], but will instead show that the above ) ∂x ∂X 8 X w )Φ , ( ( twist fields 1 7 q + J ). We will however keep track of the X h  ( , this is equal to the trace of the product of -fold copy by )Φ under q q } q . 3.1 X Φ R ( J h i . We will not review this method here, and refer  – 14 – , X, x D ). To make use of this result, we also need to ) { hO denotes the operator , i.e. q D ∂T ∂X and c located on sheet i D ρ and its 3.6 12 = 2 k ) ρ O C T C q . − in order to compute the Renyi entropy. The pattern of X ¯ R L L ( -th Renyi entropy ( , ) q ) + q C T x x i ) in ≡ ( R ( ) ) , and i ] for a detailed discussion of this replica method applied to ) T ( i α P ( C  define the endpoints of the domain 8 X , 2 X = Tr( ( ] it is shown that in a conventional 2d CFT, the effect of this 2 7 , (    q 1 O 8 -th power of , R X hO q i 7 ∂x ∂x ) ∂X ∂X ), this is given by the anomalous transformation law (   X ( . By the construction of q = = T 3.3 copies of the original space, where each replica is sewn to the consecutive in the spatial identification ( h R ] to determine the properties of twist “surfaces” in higher-dimensional CFTs q U U ) ) 32 on X X ) coordinate system by ( i ( ( → ∞ ) J is related by a unitary transformation by T † t, x † ¯ L X ( D U U ρ J L, h To identify the charges of the twist fields, we begin by computing the value of In this section the subregion of interest will be an interval on the plane, i.e. we will Twist fields defined in this manner have well-defined properties under conformal trans- In particular, in [ repeat here for completeness: Now in the ( understand the transformationtransformation of ( of this identification pattern and the operator with the We will not use the uniformizingresults map for studied the in Renyi [ entoropieswas can used be in re-casted [ inwith terms of holographic twist duals. fields. A similar method send the ordinary complex plane Recall that the points formations and can be consideredthe to properties of be such local twist fields operators in in WCFT, determining their dimensions and U(1) charges. is that for any operator where on the right hand side the expectation values are evaluated in the product theory on two-dimensional conformal field theory. non-trivival topology can be implementedfield by theory, considering with the additional insertioninterval that of local enforce the replicaIf boundary we conditions, denote coupling the together original the theory replica by copies. two-manifold that we will call traces in the construction ofmanifold the with one in a cyclic fashionthe along the unfamiliar interval reader to [ “path-integral” point of view, in which one considers the path integral over a branched JHEP02(2016)033 (3.38) (3.36) (3.37) (3.39) (3.35) . i ) ). Note also 2 ). We must . ). The oper- i x X ) . ( ( 2  3.3 ) , ) are the unique 3.38 † q aJ X τ | 2 ( vac 0 − τ † q )Φ  L ( 1 3.38 a = L L | ) in the thermal state X )Φ a 0 τ ( 1  x πia Z ( P q 2 X 4 k ) to find: J ( , )Φ + q log − ), Φ X τ 2.14  x ) we have: ), this should be evaluated on h ( vac 0 ∂  ( ) and a ∂ i L x J T 2 ( 2 h − vac 0 . πi 1 q a 2.12 3.25 τ 2 L τ = τ aT L L a + q + = −  − R i  c i 0 ), we have: = 2 24 τ ) vac P ). τ 0 0 a vac 0 h  q . X L (  2 3.31 iP a
J πiP 3.13 h
` 2 by vac 0 τ ` 2 vac 0 , 0 P – 15 – + + 2 ) P ) for the stress tensor (and in our case the U(1) . As shown in ( + τ ,J a +  | X 0 , + X τ a 2 a L 2 i ( 2 ` τ 3.38 2 2 ` ) a τ τ τ |

τ τa 2 ) on a general torus ( a ` ` 2 2 i τ  -number contributions and assembling all the terms, we 4 2 k k X ) | Z c ( 2 τ − − † q ( = = 2 X a log iπk ( | i i X X a 0 0 )Φ † q  -numbers that can directly be obtained from ( 1 ∂ Z P L . In a translationally invariant state the values of the currents ∂τ c h h † = = X )Φ given as before by ( ) in the Cardy limit by combining the Cardy result in the canonical ( 1 ], the form ( q q U πi 1 that have the correct singularity structure and approach a constant τ q θ 2 | R R X ) we find q H 8 i i ( , τ θ, ) ) − X )Φ ( 7 q (and similarly for the stress tensor). This determines the singular- ) = exp a Uρ | τ X X Φ X 3.35 = | a ( ( h ( y O i ≡ τ i Z q J T ( − 0 T h h x a h q D , with | iQ L a ρ h θ q = Z ) with the transformation to an arbitrary frame ( ∼ q = ) R i y τ 3.18 ) ( , q ). X qθ ( More explicitly, the OPE of the twist field with the U(1) current takes the form We now turn to its interpretation. From ( Finding also the anomalous )Φ X T ( x = h q ( J ity structure of the correlationanalytic function. functions of The functions appearingat in infinity. ( Thus the charges may be read off from the singularities in ( Now, as noticed incurrent) [ expectation value isΦ equivalent to the Ward identity for the conformal primary This is the desired result for the expectation values of the currents on the replica manifold. find after some algebra: As expected for ascale translationally like invariant the state, length the of totalτ the value spatial of cycle the energy and charge Putting this into ( We can now find frame ( ator part of thisdescribed expression by also requires usare to simply determine related to thethat zero from modes the definition of The anomalous terms are JHEP02(2016)033 . ` by ¯ ` (3.42) (3.43) (3.45) (3.40) (3.44) (3.41) and s is given by a : q X , leading to the q ) that is sensitive C ab . with ∆ on , h q i i ) a ab ) T 2 n 2 , g X . X a ). By inspection, it is clear ( ( † q † 1 X vac 0 ` , 2.22 )Φ P )Φ 1 = 1 , , and hence we don’t expect ): = | X X x ( L L 2 q ( ` q vX , 1 X 2.23 . Φ Φ − + h ab − h ¯ ` g T ! x q 1 b a = ,Q V 1 X − is a covariantized measure of the separation | L L → a V ab 1  s V = T h

b 2 ∼ the vector that corresponds to the identification vac 0 – 16 – p ab q to the full stress tensor ≡ X q a L g ) b i ≡ a q ∆ V 1 ) T ρ + a a ρ X V 1 limit of these charges is not obviously zero. We will n n Tr ∆ c 3.1 24 a (Tr → ≡  X q s q . As we are in flat space, we need not distinguish be- log ∆ a 1 = q ) and the metric is defined in ( p X q 1 − to go from ∆ 3.2 − 1 q a 2 = X q in the time direction. This is not surprising since the separation along S ≡ ¯ ` a X are given by ( of the identification pattern. Now, the cross product of 2 , 1 ). Correlation functions should now depend only on invariants associated with X a µ -axis is not invariant under the boost δ . One such invariant is its norm, as defined in ( Define ∆ a T angle ≡ X a -sheeted Riemann surface. This in turn determines the Renyi entropy, and we have: µ the and it is also ain boost the invariant. time We see direction. that At this point the correlator is an arbitrary function of We may now define a normalized vector which remains finite as we take the cylinder very large, carrying only the information of introducing another vector: denote by pattern associated to the cylinder ( where that there does notto appear the to separation be anthe invariant built fromitself (∆ to be a good measure. However, there is another invariant that we can be build by tween tangent-space and spacetime indicesτ (equivalently, there exists a∆ canonical vielbein We now need toexpect determine the the 2-point correlation two-pointsymmetries: function function while of of this primary is theof operators true, twist WCFT the on is precise field the somewhat implementation on plane novel. of the to these symmetries be plane. in fixed the by We case the two-point function ofq these twist operators determines the partition function on the following values for the conformal dimension and charge of the twist field Φ It is interesting to notereturn that to the this point, but first we proceed to compute the Renyi entropy itself. As usual, multiply by a factor of JHEP02(2016)033 T times (3.48) (3.46) (3.47) ): then q ] for the ) — then 22 direction, 3.20 . We might X 1 3.21 4 . ) q copies of Φ . − ). q ) U(1) — as in ( (1 q . × 3.41 vac 0 ) isQ vac 0 P U(1) invariant, as in ( R ) is neutral under the U(1) charge; this − P , = × = ) 3.46 ) we reproduce the previous value for 1 R exp ( , with respect to “translations” in the twist q q q 3.41 2∆ with respect to scalings of the Q − q ` ,Q ) comes with a few caveats. We are assuming implicitly ,Q – 17 – . Any computation performed on the plane will vac 0 i ∼ 3.46 1 )  L -fold theory we obtain 2 q q X + − ( † q c invariant under SL(2 q 1 24 , not by the central charge. As we have stressed above, if  )Φ = 1 = 0, and both of the charges above vanish, resulting in a vac not 0 1 vac 0 ) and evaluating ( X P ( L ∆ vac 0 . Thus the correlation function takes the form q that “creates the vacuum”, as well as a corresponding insertion s P = 3.48 Φ and h 1 = 1; in this case we have not traced anything out and are simply ), as expected. q vac 0 and twist q L 3.27 ∆ c 24 . It also has a U(1) charge ` − between the numerator and denominator of ( = vac 0 L to “annihilate the vacuum”. The freedom to move these operators around means † 1 A consequence of this is that the entanglement entropy is determined by the value of In our precise computation, these insertions of the vacuum operator have localized at However, if the vacuum is We now discuss some interesting features of the twist fields defined above. For example, The U(1) direction is anomalous, and hence ( 4 the vacuum is not invariant under the appropriate conformal group,that these vacuum are state not used directly implies to compute that the the expectationanalogous path value arguments integral in for the ( will partitionthe function. only anomaly If depend are the simple on vacuum to state invariant quantify is quantities. by charged, then keeping the track See extra of terms the section due anomalous 3.3 to transformation of of the [ path integral. spectrum) that allows usactual to computation add of conformal theautomatically dimensions. correctly It normalized is Renyi thus entropy, this reassuring subtraction that happens inthe vacuum the charges This subtraction — whilebe conceptually useful justified — unless is there is somewhat some heuristic, other and principle cannot (e.g. really a large central charge and gap in the that there is no translationally invariant quantization of thisthe theory endpoints on the of plane. theattempt interval. to separate In this the its vacuum contribution contribution to from obtain the the twist charges field of by the subtracting twist field itself: the vacuum on theoperator cylinder may maps be to understoodinvolve a as an being insertion non-trivial of Φ operator Φ of on Φ the plane. This vacuum answer with: In the usual case, wewe study have a vacuum thatregular is stress SL(2 tensor. the Renyi entropy ( consider first the case considering the expectation value ofthe the stress stress tensor tensor on the to plane. be One non-singular might then everywhere. expect Instead, however, we find a nontrivial as measured by direction, as measured by Putting in the values of ( Now the operator has conformal dimension ∆ JHEP02(2016)033 ]. ], and our discussion above can be 36 1. 35 – → entropy is also controlled by the vacuum 33 q ]. We will not attempt to do so here. Instead, – 18 – 37 thermodynamic ]. This prescription takes a special form for CFTs in two 2 , , we will assume that the line of reasoning described above is 1 . The preceding discussion is heuristic, essentially because one N 1 G 3.2 4 Under certain circumstances, however, it can be made precise through a careful im- It is well known that for standard CFTs the way to obtain entanglement entropies ticle trajectories in thethis means bulk, is relating that theminvariant under we at tangent will the space consider gauge endfest transformations. background the to comparison and This entanglement with point dynamic standard entropy.geometry of geometric fields versus view concepts What that warped like will that are geometry. make of mani- fully geodesics gauge in Riemannian plementation of this limitgiven in that the we have bulk aas [ good described understanding in of the section valid, properties and simply of calculate twist the operators appropriate for two-point WCFTs, functions through semi-classical par- Their quantum numbers are fixed completelyin by turned the determined charges by of the anomaliesgeodesic vacuum (under state, calculation some which in assumptions). are the Plugging holographicthe this bulk correct data yields into factors the the of Ryu-Takayanagican formula, really including understand the twistthe field as Renyi a index probe used of in a the fixed replica background only trick in is the taken limit that bulk describing the trajectorythe of field-theory a computation semi-classical involvingis particle. twist related fields: to This the the suggestsboundary entanglement two-point a entropy, theory function and connection two-point the of to functions geodesic twistonly for in fields necessary operators data the with to bulk perform large is the conformal known computation dimensions. is to the compute The quantum numbers of the twist fields. from holography is to performto a calculation the of edge the minimal ofRyu-Takayanagi area formula for the a [ entanglement bulk region surfacespace-time attached at dimensions. the In that boundary. case, the This minimal is surface nothing corresponds else to a than geodesic the in the identifications of a space-time cylinderany where our correct theory holographic is description defined. ofshow these As that such systems this we will is expect share the that in the case same the by property. using following We a section will the geometric we description holographic will of dual. derive holographic these duals of results WCFT; using a Chern-Simons formulation of 4 Entanglement entropy: holography inThe geometric results language of theenough previous to section determine show completelysystem clearly is the that in its entanglement the ground entropy state symmetries and of the of a interval has the an single problem arbitrary interval orientation are with when respect the to the known that the Cardy limit ofcharges the and not the centralviewed charge as of an the extension theorybeen of [ obtained those in results the to context the of entanglement non-unitary entropy. CFTs A in similar [ result has correlated. Another situation with a similar mismatch is Liouville theory, where it is well- JHEP02(2016)033 a q (4.3) (4.4) (4.5) (4.6) (4.1) (4.2) ) are a ¯ q , a q remains the t ) symmetry. ]. d 16 , . = 1 + 1) dimensions; the µ ¯ = 1 d A The vectors (¯ µ a ¯ ¯ q + 1)-dimensional warped A 5 coordinate, and a . . d ¯ q . x µ a IJ , a τ µ δ ν τ a , a ¯ q A q µ . , = ¯ ¯ = 0 selected by quantum anomalies. In = ¯ A µ a ! aI a µ ¯ ¯ q − ) vectors imply that A q q I ¯ ¯ t A q µ ρ x aI , δ → I

q and ¯ a = , , q a , ν b q I νρ J b ν τ δ τ G µ a a µ −→ ) invariant tensor – 19 – τ = τ , q µν d J b bJ I a q q q aJ I a q ! q aI t ,G → . Using these fields we can build spacetime tensors as aI x q a IJ µ ) are a δ IJ

τ q ) extended in the obvious way to ( δ = 0 , q = 2.27 µ = J b ¯ 2.26 q µν A I a µν metric is degenerate. G q µν G G IJ spans usual coordinates transforming under an SO( µν ) and ( δ G = ) and ( = µ 2.22 ¯ A ab 2.21 g , . . . , d µν G , a brief outline of the flat (i.e. tangent space) geometry that couples to WCFT = 1 I 2.2 as well as out of the preferred tensors a q Now we would like to extend these notions to curved space. In a nutshell, all we need To discuss warped geometry in dimensions larger than (1+1), the first step is to extend Here we consider a purely spatial (i.e. Euclidean) extension of the 5 which shows that the preferred U(1) axis. We define upper index tensors as Notice that the orthogonality properties of the ( to do is add vielbeinLet fields that us map call the these vector spacein invertible in standard fields the geometry. base Lower manifold index to tangent tensors space. built from these objects are generalizations of ( Therefore, the relevant tensors are extended as: As expected there nowstill also of exists the a form SO( ( the number of coordinates as where the was given. It wasand argued that it canthe be following constructed we out will of brieflygeometry; classically for elaborate invariant a tensors on complete this ¯ discussion formulation on for this ( subject we refer the reader to [ In order to understand how toportant describe to particle explain dynamics what in are warpedLet the geometry, us it necessary remind is structures the first to reader im- describenecessary that the to in background consider order geometry. a to dynamical describe geometry.In semi-classical section All particle we dynamics really it need is are fixed not background fields. 4.1 A geometric background JHEP02(2016)033 ) 2 G R , U(1) (4.9) (4.7) (4.10) × ) subspace R 2 , . AdS r dX of Euclidean AdS ): , γ τ µ ( ˙ x + µ µ to set the normalization x ¯ → ∞ A µ must describe a SL(2 r x dτ βdX . µν + Z G ¯ L L ], this deformation corresponds h − dT 17 ) (4.8) + L = = µ β − dτ e . parameterizes the flat U(1) connection ¯ dx A → setup [ µ Z β ¯ 3 ¯ A L, X 2 2 to make sure the action is invariant under m + = 0 e T + ( – 20 – , ν radius; ˙ 2 x holonomy at the center βL ∼ 2 ). If the geometry is regular (smooth) in the interior ) ¯ µν + A dX 2 G ¯ L 3.1 r µ + ) parameterize the boundary of our bulk geometry, and ˙ T,X x 2 = ( 1 4.7 ¯ − dr A 2 must be a flat U(1) connection deformed by a killing vector of is the AdS Z dτe R µ R ¯ ]. A consequence of this is that A = for our background field Z ], should include as well dynamics for the geometrical variables ν 1 2 16 β ) invariant geometry is nothing else than an euclidean 16 R dx = , µ ). It does not take too much work to write down the most general fully coordinates ( S µ dx ¯ T A µν , G µν and ) invariant metric. This freedom in the deformation is completely analogous to G R X , does the equivalent for the killing vector deformation. . For our immediate purpose of evaluating holographic entanglement entropy, we γ ¯ A The By implementing the above features, the (2 + 1)-dimensional background geometry for In (2 + 1)-dimensions, lower spin gravity admits a description as a SL(2 This is all the geometric structure needed (and available) to describe the trajectory of where we introducedworldline a reparameterizations. worldline Notice that einbein while we can redefine This fixes the value of 4.2 Worldline action We now have all ingredients togeometry describe ( the coupling ofcovariant action a point to particle lowest to non the trivial background order on the trajectory field as in our field theorywe computation must ( impose the vanishingover of this the cycle: in the following we willwe pick impose the the topology identification of the boundary to be a cylinder. In particular, Notice that the SL(2 of our warped geometry. while to the warping parameter.physical consequence We in will our see setup. below that thewarped value holography of is this deformation has no Chern-Simons theory [ invariant geometry while the SL(2 the freedom of selecting a particularsetups. vielbein from In a Chern-Simons relation connection to in higher the spin standard warped AdS a semi-classical particle. ALower complete Spin discussion Gravity of [ theand fully dynamical bulk theory,only called need the valuesthe of dual these WCFT. background fields which correspond to the vacuum state of JHEP02(2016)033 , µ for x 2 (4.16) (4.17) (4.13) (4.14) (4.15) (4.18) (4.11) (4.12) ), this and AdS e 4.7 . µ ˙ : x 1 µ , − ¯ ] A , m µν , ] [ dτ 2 T T = e ν ∆ 2 Z e ˙ . x ] X, , m h ih  µν [∆ [ = = + . We will see in the next section T αβ µ ν ν β shell ˙ ¯ ˙ ) we have picked euclidean G ˙ x x − x A , µ dτ e
on . E h ∂ m µν 4.7 ] α S ν ˙ G Z x − − ¯ = = 1 1 µ e A 2 µ ˙ x − µ ) 2 ν β [ e m β ˙ ˙ x ∂ x . The Euclidean action is obtained by Wick i ∼ α G ) 2 + µν ˙ = α x 2 – 21 – ( T ν ] G µαβ ˙ ∂ x µ ,T  µν − ˙ µαβ [ 2 x µν 1 2 ). This is directly related to the Chern-Simons origin 1 T + Γ X T G
+ Γ = µ ν ˙ 4.16 x ν ˙ = )Φ( . Because a particle has to transform in a representation x 1 ¨ 1 x 1 h − µαβ T − µν ,T e Γ U(1) symmetry we expect it to be defined by two quantum 1 G dτe and X × d 2 dτ ) Z Φ( and ∆ R m h 1 2 µν , 2 G = X ) Casimir, and a U(1) charge. E − , just as the geodesic equation for a normal charged particle depends on R S , h 1 m X = ) invariant metric, hence it makes sense to consider the direction singled out by R X , This is the geodesic equation in warped geometry. It is not universal, i.e. it depends We can obtain a standard looking geodesic equation (corrected by torsion) by picking The equations of motion for this action, obtained by varying with respect to to be the time direction. This amounts to considering the following euclidean action: . Notice the following peculiarity: this equation is first order for one of the components µ q ¯ m rotating the time component ofour SL(2 our geometry. In ( A priate boundary conditions fixing thegeometry trajectory at to the the point boundary where of the our operators three are dimensional inserted. More concretely we willwhere evaluate ∆ will satisfy the geodesic equation ( of our theory. 4.3 Two-point functionsNow for that holographic we have WCFTs the necessaryheavy particle operator action in we our can calculate WCFT the by two calculating point the function Euclidean of on a shell action with the appro- on a parameter of the trajectory, asthat can this be fact seencomponent has by becomes important multiplying arbitrary and by consequences. all paths that For have the our appropriate backgrounds boundary conditions of interest ( and a torsion field a preferred time parameterization given by the gauge choice where we have defined an affine connection constants, as expected: of the underlying SL(2 numbers: a SL(2 can be written in a compact form: of the first term to be canonical, the worldine action possess two physically meaningful JHEP02(2016)033 . ) r P and and 4.19 c (4.25) (4.24) (4.19) (4.22) (4.23) (4.20) (4.21) T r ) when the . r dr 4.18 ), however it is ) are conserved. . dr 2 X 2  ) r γ 4.16 r γ r γ ,P ih T ih ih 0 and with non-trivial direction. Using ( P + , − − r γ → X  2 ih r ihβ dr ,  ihβ ihβ r . The main appeal of writing − r + γ P e − − 2 c r ih X R X 0 ihβ 2 P ˙ Z − P , X r (  − µ 2 1 . . 2 2 r R − X r + 2 ihβ 2 e dx P γ µ − − ihγ − X 2 = P ihβ 2 2 h ∆ + X R E ˙ m − ihβ Z X P + 2 X  δ P 2 q δS X − m = 2 mR P R – 22 – r + r R X 2  = r c ), the canonical momenta ( P c T − r m shell r X = 0 2 0 ∆ − Z c p Z R r on 4.18 E ih 2 S ). We could manipulate explicitly ( = m ), which is not conserved, is given by + 2 = 2 = 2  τ 4.9 ( 2 X to the distance traveled along the 1 dr r r ih , P shell ∆ )–( ˙ ˙ r X − X = X = P . on c E 4.7 P 2 r r T E ˙ S 0 T P + Z δ δS ), and hence ( T and = ) we get ∆ 4.7 = 2 X T ih X P 4.18 = ∆ . X ) we find P shell is the critical turning point of the trajectory. Here we have made a choice: we − ) are constant, we have c 4.20 on r E X S Our action can now be written as There are two constants left to determine in terms of our boundary conditions: We are interested in finding solutions to the equations of motion in ( ,P . First, as in any projectile motion, the turning point is defined by the vanishing of dependence in ( T X P This fixes From ( Second, we need to relate gives where consider only trajectories thatseparation start along and end at the boundary P which is the( usual expression for systems satisfying a Hamiltonian constraint. Since where we used the constraintthese coming momenta from is that the the variation of on shell action is given by The canonical momentum to background is given byuseful ( to exploit certainX symmetries of the background.From varying ( Since there is no explicit JHEP02(2016)033 . ) k k we 0 we 4.26 (4.29) (4.26) (4.27) (4.30) (4.28) shell ]. This → − 39 [ on E ] or it can ) and ( S T 38 . 2 4.17 γ 2 , h + 2 X Λ m ∆ 2 1 R obtained for the WCFT and the U(1) anomaly ), which is obtained as a log √ Φ R 2 2 . ) 4.9 γ 2 2 to X γ 0, and to evaluate h c 2 . Implementing this cutoff gives = Λ, and as we take Λ . (∆ and ∆ h + T r 2 →  2 Φ
T, + T r µ 2 Q m ∆ ∆ ˙ x 2 or ∆ γ − m µ R 2 ¯ X A X → R ∆ is given by ( p ¯ L L 1 T p  β − + 2 ∆ ih = e dτe X Φ – 23 – = ∆ ∆ Z . 2 α γ h shell ihβ − ). , which are background parameters do not affect the that involve ∆ = + on E γ → ) agrees exactly with the expected result from field the- 1 S S h , T h − − 4.24 ∆ e ∆ = and 4.27 as an effective geometry. This term gives dynamics to the Φ ih ν i ∼ R Q ∼ ¯ ) A can be chosen to adjust them to any desired value. As part of the 2 ) and ( µ ¯ ,T m A shell 2 α ). 4.23 is real we can always re-normalize it away − coordinate. Evolution in this direction is not only undetermined, it is X γ + on E T = 1. This is the expected transformation rule for the U(1) anomaly S 3.40 )Φ( µν 1 γ G ,T 1 X ). Moreover we can relate the charge and scaling dimension of our WCFT field Φ( h 3.46 At this point it is important to point out that the equations of motion do not fix the The two point function ( From here we can estimate the (normalized) two point function; using ( . Moreover, if γ due to using time component. However,also it induce does a also scaling leads dimensions either of to the further field divergences that [ depends on the value of ∆ not important: the only importantexpect data from are a the U(1) endpointsfeature along Chern-Simons is this theory. axis. what This This makeslocal point is the what Riemannian could calculation we be description convergent. unsettling. ofterm In to the usual However, the this Warped geometries action AdS introduces of holography, in a the point our particle notation the following Finally, in order to reproduceevaluate the the results two point for functions entanglementtwist entropy above operators all at ( we the need values to do is to trajectory in the to and hence set obtain Notice that the valuesoperator properties of as holographic dictionary, we would relate the central charge In this expression weregularity used condition in that the the bulk. value of ory ( with the mass and U(1) charge of the bulk particle; by comparing both expressions we gives need to introduce aonly cutoff: keep the take leading the terms endpoints in Λ to lie at where we also used ( The integral diverges near the endpoints of the curve JHEP02(2016)033 . , 6 of B mn µν ˆ T G (5.6) (5.3) (5.4) (5.5) (5.1) (5.2) mn , then T − L along this 0. In what + , + B 1 L  . ! 1 B. − d 1 0 0 ∧

B and a U(1) gauge field 1 2 ) to associate with the two ) algebra, which is given by Z B βdX . R R . ξ = , , ) + 0 − 2  dT , dX , B ρ n = e n ν ζ ∧ ,L B B is a parameter whose value (but not its m + B ζ m ! µ 2 ξ holography. These are the usual problems ) gauge field ∧ − B dρ R 2 ( 0 1 0 0 B , , and thus can be set to one of mn 1 2 U(1) Chern-Simons theory in the bulk. The mn 2 3 ˆ

T B T – 24 – × = dX , + = . = = ) + ν ¯ L L 1 R L µν − dB , ρ dx − mn e G µ ˆ ∧ T = + dx B β  µν dρ ,L 0 G Tr L ! Z = ). If this configuration is smooth, the holonomy of 1 0 0 0 B L CS − k −

) the Euclidean signature metric = = S 5.3 1 ¯ L, X L + ) corresponds to the flat connections T is the Killing vector of choice that was projected out. This degenerate Killing can be related to the central charge. ( 4.7 m ) down onto this two-dimensional subspace to obtain a degenerate Killing form ∼ ζ CS R ] it was shown that the minimal way to describe the holography of warped conformal ) k , To connect this Chern-Simons formalism to the geometric language of the previous The equations of motion simply tell us that both gauge connections are flat. The Of course, our results don’t rely on this choice of representation. 16 6 T,X For example, if we take thewe subspace obtain orthogonal from to ( the hyperbolic generator where form is used to find the geometric degenerate metric defined in the previous section: The conjugacy class of the omitted generator determines the signature of the metric section, we must pickscaling a directions two-dimensional in subspace theSL(2 of bulk. SL(2 We can then project the normal Killing form of The topology of the( 3 manifold hascycle a must contractible be trivial cycle which described sets by the identifications vacuum ( Here sign) can be changed byfollows real we will rescalings use of an explicit matrix realization of the SL(2 field theory was inrelevant bulk terms degrees of ofand a freedom the SL(2 are bulk a action SL(2 is simply with holographic renormalization in warped spaces.elegantly by Lower spin suppressing gravity this avoids interaction this problem term from the action. 5 Entanglement entropy: holographyIn in [ Chern-Simons languange is the situation in the standard setup in AdS JHEP02(2016)033 ) ) U R R , , (5.7) (5.8) (5.9) SL(2 × ) would make ) capture the R , R − , , L  ]. Note that the
2 − 40 c SL(2 + − L × )’s [ ) ) 2 ] for SL(2 R R , P , 21 Tr( . ) λ , R s ) act naturally from the left and , + ¯ A U(1) gravity. We first briefly review R  , (which can be related to the mass of U × 2 SL(2 ) c − ds R DU ∈ , 1 U , and thus we are simply re-asserting the s transform as highest-weight representations 3 − A 3 is a Lagrange multiplier that guarantees that ) gauge fields (as is appropriate for standard λ PU + R – 25 – ,  , U ds dU Tr ]).  = 41 denote the pullback of the bulk gauge field to the path ds LUR L, R s ds C DU ¯ → Z A , ). Two copies of SL(2 s U R ] = A , ¯ A . We may now compute the Wilson line by performing the path ) is actually itself AdS A, µ SL(2 ; ], referring the reader to that work for a more detailed discussion. R ds , dX ∈ 21 µ U A U, P, λ [ ≡ S s A is the momentum conjugate to P gravity), the correct quantum mechanical system is a particle living on the SL(2 ) via Now the worldline action describing the system was shown to be In the case where we have two SL(2 To compute a Wilson line in an infinite-dimensional representation of the gauge group, A version of this problem has been studied previously in [ We would now like to obtain the results described in the previous sections — regarding 3 s ( into a timelike coordinate. These considerations will turn out to be important when µ the particle) and the covariant derivative is where the external sources X where the representation has quadratic Casimir equal to It can betransforms shown as that a upon highestgroup quantization weight manifold representation the SL(2 under Hilbertfamiliar both fact space SL(2 that single-particle of states on aunder AdS its particle isometry moving group (see on e.g. [ AdS group manifold, right as quantum mechanical system is picked to havespace a furnishes global precisely symmetry group the such infinite-dimensionalple that representation its this in Hilbert auxiliary question. system to Wesymmetry then the along cou- bulk the gauge worldline) fields inthen (viewed as the computes external standard the sources way. Wilson for Integrating line the out in global this question. auxiliary system correlation function the Wilsonwe lines will are adapt picked that to discussionthe intersect to Lower the prescription Spin of boundary. SL(2 [ In this section we first construct an auxiliary quantum mechanical system living on the worldline. This variables. gravity (as well as a higher-spinin generalization), an where infinite-dimensional it highest-weight was representationphysics shown of that of SL(2 bulk heavy Wilson particles lines inrelated the to bulk. the The mass Casimirs characterizing and the other representation charges are of the particle. To compute a boundary theory X determining boundary conditions on our probe. entanglement entropy and correlationThis boils functions down — to from coupling massive the particles Chern-Simons to Lower description. Spin Gravity using Chern-Simons Taking instead the subspace orthogonal to the elliptic generator JHEP02(2016)033 . ) x ), )- 1 R 1). R R , ]. and , , , = ). We (5.11) (5.10) f B at the 43 , R U , SO(1 U ) we may 42 ) changes / = ) R R . The isom- i , , R 3 7 U . , ¯ A in this section has SL(2 SL(2 ¯ A, A ∈ ) gauge field × invariant under this U(1). The U(1) part ) R , U , R × U ]) is acted on by SL(2 , ] to compute this action ) fixed, and which should ¯ . The action of SL(2 A 2 2 R is a coset representative. 21 0 , . One way to understand x A, and label all other points is as a coset of SL(2 g 1 ; rather than AdS 2 2 − as the coset SL(2 2 R 2 ]. In particular ]). AdS 21 = 44 U, P, λ [ in AdS . This is overcounting, as there is a S L x 0 − . x to LU 0 ). The element ), label different points on a two-dimensional x (see e.g. [ gh , ] exp( 0 0 → ). It has been shown that single particle states λ , i.e. given any element 8 ) where 0 = R γL γL D U – 26 – L , 5.7 U P ). D 4.4 , i.e. U = exp( U D [ h Z ) connections as defined in [ R , ) that leaves the reference point ) as the appropriate highest weight representation [ ) = defined in ( ) and (2 R R ¯ 1 A , , ¯ ) required to move sl A − R A, ). We can then couple this system to a SL(2 , ( βL R SL(2 ) are , W + ¯ A is a single copy of SL(2 1) to be generated by ⊂ 1 , 2 A, 1) αL , ’s, modulo the action of exp( g = exp ( copy of SL(2 g Different Such a system is given by a single particle living on AdS For our purposes, the most efficient way to represent AdS It is important that appropriate boundary conditions must be placed on In this section ( Here we immediately specialize to the case of interest, but it should be clear that the discussion applies 7 8 single invariant metric on this manifold,is simply which via is left thus multiplication seen on to be AdS nothing to do with the tensor to any homogenous space. where manifold, and there is a canonical way (which we do not review here) to find a SL(2 and thus one isby tempted to the pick element a ofsubgroup reference SL(2 point SO(1 not be used to labelWe points. take So the we SO(1 instead understanddecompose AdS it as etry group of AdS transform under this SL(2 first present a brief review of coset geometry We now want to adaptfactors the out discussion and above will to be thequantum (easily) case mechanical dealt system of with that SL(2 at transforms thea as end. a highest-weight The representation non-trivial under partfollow only then the is algorithm to above find to a compute the Wilson line. Chern-Simons gauge fields invariant:the the metric, remaining acting part onprivileged subgroup of it is SL(2 (on-shell) the identity, and as so diffeomorphisms. we impose the The5.1 boundary conditions only Wilson lines and cosets beginning and endare of invariant the under tangent-space path. Lorentzsubgroup rotations, These which of correspond boundary gauge towhy conditions the transformations 3-parameter this are in subgroup chosen ( is such privileged that is they that it leaves the geometric metric associated to the In the semi-classicalTechniques limit — which this we will amounts reviewpurely to below algebraically — simply in were terms developed computing of in the [ data characterizing bulk the on-shell flat connections action. integral over all worldline fields, JHEP02(2016)033 , s 0 ), 0 L B L i g 5.11 (5.15) (5.12) (5.13) (5.16) (5.17) (5.14) ), this . This ∝ g i 5.6 should be g ) ) invariant. − as in ( − L . This is the L 5.5 , 0 gh . As mentioned + as in (  + g . γL +
= e + 2 B L X c U L − ) case. , , , ) R 2 ), leaving only , P will then gauge away the = 0 = 0 and the external source s ) = 0 5.11 0 Tr( 0 a SL(2 , L λP ds  DP × λ ) PL ), and as above we couple that , ) − + 2 + R s R L should satisfy ( , a , Tr(   s − U a ≡ i,f , and may be viewed as tangent space g + ds − 0 − DU − L L 1 ( U − s 4 π ds + dh i B 1 + − − PU ,P + in the SL(2 L h  as left-multiplication. As the physical degree of 2  – 27 – is a number times . This can easily be done by promoting the c 1  g to a dynamical degree of freedom (which we will s ds ) gauge field which we now call U dU Tr , which changes the coset representative but not the a + = in the decomposition ( s 0 R  ) = , ¯ h = L A = exp 2 h i,f ds  f P U at the two ends of the path. Writing g C ds g DU ds Z Dg Tr( = U(1), that which leaves the geometric metric ( i  g ] = 1 × rather than ) − B g ; R g gauge field 1 s , − a ) algebra. h R , right -multiplication by (2 corresponding to sl U, P, λ, [ U S right ) along the worldline. The quantum mechanics along the worldline is now a s a . For the choice appropriate to a Euclidean bulk coordinate ab ˆ T Finally, we turn now to the choice of boundary conditions on the field Thus the action is still We now generalize the construction above to make the dynamical degree of freedom 5.2 Equations ofTo motion compute the and Wilson on-shell line action suitable we boundary now conditions need on only computewe the find on-shell the action equations after of supplying motion to be coset element itself. Thuswhich our we boundary can condition solve to find where the solution isanalog of ambiguous the up boundary to condition further right-multiplication by form is a one-parameter subgroup generatedSO(2) rotations. by This operationfreedom acts is on a coset, “invariance” reallyequivalent to means that left-multiplication by is valued in the above, the key requirementleged” is subgroup that of the SL(2 boundaryThis conditions is are equivalent invariant under to a demanding “privi- that the subgroup leave invariant the reduced Killing but the covariant derivative is now where the worldline degree of freedom sort of construction isalthough familiar here in we the aresymmetry context doing associated of it two-dimensional with along conformalglobal left-multiplication a field symmetry by one-dimensional to theory, SL(2 worldline. an external There SL(2 is still a global the coset representative component of the now call dynamical gauge theory incomponent its of own right. Integration over JHEP02(2016)033 . B B then gauge (5.21) (5.18) (5.20) (5.22) (5.23) i,f h field. We . B . . i,f ) . . i g s ) worldline ( respectively: we s ( α ] = 0 = 0 s ? a s − P ,P a ) s f a s ) = . and ): ( = 0, and will show that s must vanish. Note that if ] x α ( + [ s λ ( ]. We stress that different a P L ,P λ , = 21 s ds dP a α = ) (5.19) ∆ , and we have s ,P + [ ? ( ds dα ) P , s )) can be parametrized by a constant ( ds s dP dsλ ? ( , ? i U = 0 = Z , )) , then we obtain eventually the following 2 ,P )) and so does not change under this gauge ? component of s c ) f P Mh ( λP s 2 ) 0 h ds 1 µ s ( DP f ( L − − f ? 5.18 g α X h + 2 U 2 ( )): – 28 – s − L  ] = ≡ ( e s ) = i ? B a f [ h u g , ,P s i ) = . Actually the physics depends only on the difference s ) ( − g 0 s s varies. 1 B ( ˙ L U ( shell h ) = − 1 , − λ U s ) + i,f i − i . We work in a gauge where ( γ on h s s ? h e i ) contains the information of the gravitational background ( = 0 these equations are: S a ds dg  g x means that the L ( ) ,U B + s = L f 1 a from the left and take the trace, we find that the on-shell action s ) = h − i ( ) follow from varying with respect to and an element of the algebra ,U s = 0 and then perform a gauge transformation on all quantities P ds  L Dg ( ? ? 1 U P u ds dg B − f LdL 5.17 and the gauge transformation parameter g  1 ? 1 − f ) = ) with P ) = − s h x ( g ( ? 1 = 5.15 B P − — even those related by gauge transformations — are physically inequivalent ? will always be flat, so the most efficient way to find a solution is to start in h B αP B 2∆ − e In this gauge the most general solution ( The main complication in solving these equations arises from the external source is nonzero, as it is still a fluctuating degree of freedom along the worldline, not an is not charged under the symmetry associated with left-multiplication (e.g. note that s relation between Now boundary conditions areare imposed free parameters on of the the physical form degrees of freedom P does not appear in its covariant derivativetransformation. in ( Note that in question. Iftwo we ends now of demand the that path the solution satisfy the boundary conditions at the Given this reference solution, wedenote now perform the a resulting gauge solution transformation by to ( a flat for the dynamical gaugethis field is indeed permitted by the boundary conditions ofelement interest. of the group a external source to be chosen. However we now have the freedom to pick a However a bulk gauge where of interest to arisechoices at of the desiredfrom solution, the as point explained of inequations view [ of of motion. the When worldline, and this procedure is merely a trick to solve the is simply and we need only determine how The constraints in ( stress that integrating out we multiply ( where the covariant derivatives in question are: JHEP02(2016)033 ). can ) has γ 5.20 (5.27) (5.29) (5.30) (5.25) (5.26) (5.28) (5.24) ) where ). With ). Note ). Thus, 5.23 . 5.2 5.25 5.23 5.17 2 in ( ) , and plug the ? 2 Φ in ( X P . X (∆ 2∆ ) M component. This is ρ 0 ∆ 2 ∼ 0 ρ e component of ( γL L ie 1) factor at the end. 0 4 + 0 L , − , exp( ) takes the form ( . Φ γL p ) is to fluctuate in a manner 1 0 e s 2 − ( ) (∆ = M M  h Φ , 1 X  ) ≡ − ML . 2 γ ∆ ) X L ρ  0 ) 2 X above has no 1 ν 0 ∆ e 1 = 2∆ ) is ρ ? ) Tr( L γL x 2 e 1 − ( P − c sinh 2 matrix representation; see ( 1 ML e L 0 4 + ( to cancel the × log( exp( ML νL ρL f Tr( p Φ  h − , Tr( e  Σ 1 – 29 – ) = X X −  − 2 = 2∆ c ∆ ∆ 1 ) = L ρ ρ 2 1 ν ) to conclude that p ) we can now explicitly compute . This is required to guarantee that ie ie √ 0 shell ρ, X − α to vanish, as is required by the constraint ( = exp = − ( limit. This may seem like a great deal of work to obtain − 1 γL 5.19 ? 5.14 L 2 + 2  P on − Φ to make sure that limit — which means that the interval is very long in units e 2 γ S νL αP r 1   − f = 2∆ ) becomes into ( = h Σ − i f 2 cosh(∆ → ∞ e α  h h ). Taking the trace of both sides we find ) this operation can be implemented explicitly in cases of interest. sinh 5.23 R X , ∆ 5.27 and ρ , ν e = exp 1 component of X 0 = M 2 ∆ L i ρ , use the standard Casimir relation e ρ h ). In this case the gauge parameter e can be decomposed as satisfying ( 5.3 ), we must pick M x ν ( component. Eq. ( , which can in principle be found from integrating the L 1 0 sinh Σ = In the case of SL(2 − i L h f where we have taken thea large ∆ very simple answer.while The minimal, essential greatly reason obscures for the this geometric is description. that the Chern-Simons description, Finally we take the of the cutoff resulting expression for ∆ no with Σ, the parameters satisfy We now pick where all traces are takenthe in help the of this fundamental identity 2 and some algebra we can check that that any This decomposition is helpful asbe we obtained will using pick the (easily checked) identity For illustrative purposes, wegiven perform by the ( computation in the explicit case of the vacuum Using the boundary conditions ( Rather than finding it inthat this allows way, the we note thatgiven the role of the main practical point of the coset construction. h JHEP02(2016)033 (6.1) (5.31) (5.32) is simply ) contains B 2.25 .  ). L π` 4.27 . sin  ), which we quote again below: X L π ∆ ].  axis is preferred. But less manifest L L ) and ( 3.23 5 t − T log 3.46 ∆ B.  vac 0 C ih L Z e 4 Φ ih − ]. 2∆ = ) ] from a background geometry perspective. It is – 30 –  16 1 ¯ ` ` X 15 ) then we find for the full correlation function − U(1) (∆ 5.3 S ) in [ ¯ L L ∼  ` 2.28 tot , then the contribution of the U(1) gauge field S h vac 0 − e iP = ) makes it manifest that the EE 2.18 S ] and the nature of space-time itself [ 45 , 1 ] from a CFT perspective and in [ The main result of this work was, of course, an exact formula for the entanglement One particular feature of WCFTs that was of importance in obtaining these results, and In this work we have shown how to extend these successes both from a standard field Finally, we return to the U(1) portion. This is trivial: an irreducible unitary represen- 14 inclusion of a scaling structure ( entropy of one segment at finite volume in a WCFT, ( classical symmetry ( is the existence ofanomalies a both second preferred for direction. thealgebra Virasoro The shows and that full the U(1) quantum U(1) commutators. algebrabreaking anomaly ( selects the A another generalized fully preferred boost covariant direction writing symmetry. in of This the theory, result this thus provides physical motivation for the Possible connections with quantumin hall [ physics have beena suggested promising in open a direction related to context explore this application further. in coupling the theory to background geometry, is the existence of two preferred axes. The black holes [ theory and a holographicSuch perspective powerful to results themakes are realm manifest scarce the of when importance Warped it Conformal of comes Field WCFTs to Theory. in non-relativistic possible field applications to theories. physical systems. This vided many insights oncalculation the of nature entanglement of entropy at non-perturbativemiracles possible finite quantum in volume field this (or theory. caseholography, finite that this The temperature) has result exact is furthered has one ourA sparked of understanding deep considerably. brand the understanding Through new ofphases ways the of of meaning thinking quantum and matter about behavior has quantum of changed gravity. entanglement radically entropy the in way different we think about the entropy of This is the desired result and it agrees both with6 ( Discussion It is undeniable that the power of Conformal Field Theories in two dimensions has pro- If we plug in the background value ( tation of a U(1) symmetryand thus is there one-dimensional, is transforming no need byit. to multiplication construct If by an we a auxiliary call phase, quantum-mechanicalits system the integral to U(1) generate along charge the worldline JHEP02(2016)033 ), quite differently ] for CFTs. ], but further work in 3.27 22 ]. This generalization is 47 , 1 46 . Notice also that this term is not ) is at present lacking. Since it con- vac 0 ]. Using the the twist field approach P 6.1 16 ], it might have an important role in the 10 – 31 – ). This is quite interesting, as we typically see volume ¯ ` ` 6= ¯ L L ], this term vanishes. In holography, however, this term is ]. 22 48 ]. The entanglement entropy would still be real even in that case as 10 , it was easy to reduce the calculation to that of a 2 point function given 3.2 ]. Lower Spin Gravity evades these divergences. The reason is that the symmetries 38 It is important to point out that this twist field approach is hard to extend to higher In parallel to the field theory computations discussed above, the same results where One important point is that, as opposed to the case in CFT technology, WCFTs give A deeper interpretation of the first term in ( at this point not available in the WCFT setup and could be subject to future research. is also well knowndifficult, in see for Lifshitz example holography [ where holographic renormalizationdimensions, has where proven we expectsome that form the of holographic minimal calculation surface is as performed in by the calculating Ryu-Takayanagi formula [ stress that this result differs fromSimons the gravity, expected where result divergences in have usualoffs been Warped AdS [ found Einstein-Chern- due toof different WCFT metric allow component for fall differentan couplings Einstein of a Gravity particle holographicand to description implies geometry. of a While one WCFTs, different could UV this attempt behavior assumption responsible is for not the minimal usual divergences. This problem Gravity/WCFT correspondence put forwardin in section [ holographically by the actionAs of a expected, semi-classical there particle isof moving the a in particle geodesic the action warped equation over geometry. obeyed this preferred by path these yields particles the and correct the result. calculation It is important to on a pure stateVirasoro maps and to a the thermal U(1)U(1) density Kac-Moody charged matrix entanglement algebra. with entropy discussed, potentials This for turned example, gives on in a both [ Hilbert forobtained the space from definition a of holographic the perspective. This is a necessary check for the Lower Spin from the usual behaviorand in implications CFT. of It this would behavior. be interesting to understand whata is geometric the meaning origin tointerpretation. U(1) In chemical a potentials WCFT by the providing entanglement a entropy torus of partition a function tilted segment in the cylinder tributes to black holestatistical entropy interpretation in of holography black [ canonical hole interpretation thermodynamics. of these A U(1) short contributionsthis is discussion direction presented on is in [ definitely the needed. micro- is An that important this clue term that we is present actually in this independent current of work Renyi replica index ( UV divergent like the standard secondand term. it It is is proportional only tothe present the space when volume circle of there identification the ( interval isterms a in the misalignment entanglement between entropypresent the for for segment mixed the states. of vacuum interest This (pure) result and state. is, however, universal and CFT. The first termimaginary is, factor however, might more be exotic. upsetting.one The In discussed fact any controlled at that unitary it length examplegenerically is of in non multiplied WCFT, zero [ like by [ the anholographic overall setups predict an imaginary value for The second term is quite reasonable and it agrees with the expected result for a chiral JHEP02(2016)033 ]. 16 (A.4) (A.1) (A.2) (A.3) ). The 3 . π 1 2 quantized on = 2 1 − ] for any dimension . Coordinates that β 2 31 = T . coordinates is thermal: )   y ,y ) group manifold (AdS . R half 0; see figure , − . ρ y 1 < has a natural Euclidean periodicity − − Tr( e − τ U − x ≡  ) = ,y β πi . ( − half 0 and + 2 ] for the SL(2 Uρ τ > 2 – 32 – 21 = ∼ + ]. , x ,Z x τ  + 7 and note that , dual CFTs. y ,x βH 6 e y that implements on the Hilbert space the coordinate − N e = half  U W ρ describing the system in the τ + = x   ≡ ,y ,y ) coordinates is at finite temperature on the Lorentzian plane, and consider a CFT  y x half half τ, y ρ ρ  t ), i.e. ≡ slices. To start we will compute the entanglement entropy of the half  A.1 x t 0) on the vacuum state. Due to Lorentz invariance, this computation can be x > The density matrix We define Lastly, we have also matched this result from a bulk Chern-Simons description. This The basic observation is thatmatrix this by density matrix the is unitarytransformation related operator ( to the original reduced density Thus the system in the ( However, with respect to theserather coordinates explicit if the we state write of the system looks thermal. This is constant time line ( understood as the entanglementwedge is entropy the of intersection the ofonly right the cover this Rindler region patch wedge. are The right Rindler A Entanglement entropy in CFT In this appendix we review theto Rindler two method dimensional for CFTs. evaluating entanglement This entropy isapplied applied a to summary two of the dimensions; general see results also in [ [ sions related to thisschappelijk project. Onderzoek A.C. (NWO) isconsortium, via supported a a by program Nederlandse Vidi ofCulture Organisatie the grant. and voor NWO Science Weten- that (OCW). This is work funded is by the part Dutch of Ministry the of Education, Delta ITP of higher spin theories and their Acknowledgments We would like to thank Dionysios Anninos, Edgar S´arosiand G´abor Shaghoulian for discus- is the natural languageThe to technology describe needed Lower to Spin performmanifolds Gravity this of in the calculation the techniques is, developed bulk, infull in as the understanding [ of end, advocated this a setup in is generalization [ ofto to crucial extend importance coset as these it ideas provides the and most provide natural arena a fully covariant, democratic and geometric description JHEP02(2016)033 .  1,  y ,y  . (A.9) (A.5) (A.6) (A.7) (A.8) x half ± ρ ) we can max x Diagram for A.1 = f x . Shaded region is Right. 2 ). x 0 and ∆ A.1 ± t → ≡  , = 1 i − + 1 x − − y y . , e e , ) q ) 2 R  β qβ max  ( ( max . Shaded diamond is covered by coordinates x = LT , 2 x Z Z − 3 πc log log c 6 = q = log , x – 33 – = 1 y 1 − in a CFT half , 1 ∆ x − + 1 half EE , + + ≡ = coordinates brings us to an important feature of this S y y . q EE e e L x S  S ± 2 R x = + x to the L is the size of the spatial slice for the Rindler observer in the ± is given by t ≡ y L Diagram for entanglement entropy of the half line in a CFT and π 1 2 Left. . = . As before, due to Lorentz invariance, this is the entanglement of the ). R T As a second example, we compute the entanglement entropy for a segment of length = A.9 x ∆ relate these cutoff procedures:then if the we domain place of the endpoint at and hence method to evaluate entanglement.entropy to For hold, the weEntanglement equality need entropy between to is entanglement be divergent andintroduce rather at a thermal careful the short with distance boundary cutoff,size divergent of whereas of pieces the thermal spatial of entropy interval slices is each and for IR observables. so the divergent due Rindler one the observer. needs infinite Using to the conformal mapping ( malized. With this equality we find that with coordinates. Mapping ment entropy on the half lineThus equals the the thermal Renyi entropy of entropy the is system simply described given by by where the denominator arise from the fact that the original density matrix was not nor- the right Rindler wedge; thisentanglement entropy is of the a region finite covered segmentin by ∆ ( the coordinates in ( Since the von Newman entropy is invariant under unitary transformations, the entangle- Figure 2 JHEP02(2016)033 , 42 A 42 61 (A.10) (A.11) (2006) . Taking 08 L Quantum Phys. Rev. ]. , J. Stat. Mech. d D (2012) 065007 J. Phys. 2 , 2 , JHEP , SPIRE Gen. Rel. Grav. D 86 Fortsch. Phys. , ]. IN , ][ (2015) 111602 SPIRE IN ]. 114 ][ Phys. Rev.  ,  R ]. . The logic follows as above with SPIRE  π 1 IN . 2 ][  SPIRE arXiv:1107.2917 R = ]. Entanglement entropy in Galilean 2 log ]. [ IN 1 ]. [ ∼ − log is related to the IR divergence Warped Conformal Field Theory β c 3   Phys. Rev. Lett. SPIRE hep-th/9403108  ]. 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