BCFT entanglement entropy at large central charge and the black hole interior
PPrepared for submission to JHEP
BCFT entanglement entropy at large central chargeand the black hole interior
James Sully, a Mark Van Raamsdonk, a and David Wakeham a a Department of Physics and Astronomy,University of British Columbia, Vancouver, BC V6T 0C2, Canada
E-mail: [email protected] , [email protected] , [email protected] Abstract:
In this note, we consider entanglement and Renyi entropies for spatial subsys-tems of a boundary conformal field theory (BCFT) or of a CFT in a state constructed usinga Euclidean BCFT path integral. Holographic calculations suggest that these entropiesundergo phase transitions as a function of time or parameters describing the subsystem;these arise from a change in topology of the RT surface. In recent applications to black holephysics, such transitions have been seen to govern whether or not the bulk entanglementwedge of a (B)CFT region includes a portion of the black hole interior and have playeda crucial role in understanding the semiclassical origin of the Page curve for evaporatingblack holes.In this paper, we reproduce these holographic results via direct (B)CFT calculations.Using the replica method, the entropies are related to correlation functions of twist op-erators in a Euclidean BCFT. These correlations functions can be expanded in variouschannels involving intermediate bulk or boundary operators. Under certain sparsenessconditions on the spectrum and OPE coefficients of bulk and boundary operators, we showthat the twist correlators are dominated by the vacuum block in a single channel, withthe relevant channel depending on the position of the twists. These transitions betweenchannels lead to the holographically observed phase transitions in entropies. a r X i v : . [ h e p - t h ] A p r ontents – 1 – Introduction
In this note, we discuss the direct CFT calculation of entanglement and R´enyi entropiesfor an interval (or collection of intervals) in some related 1+1-dimensional systems:1. The vacuum state of a boundary conformal field theory (BCFT) on a half-space.2. The state of a CFT on a circle produced by a Euclidean path integral with a boundaryin the Euclidean past.3. A pair of BCFTs in a thermofield double state.For holographic examples of these systems, calculations making use of the the Ryu-Takayanagi(RT) formula [1] suggest that the entanglement entropy can undergo phase transitions re-lated to a change in topology of the RT surface (for examples, see Figures 1, 4 and 5). Inthis paper, we show that these holographic results can be reproduced with direct BCFTcalculations by assuming large central charge and certain conditions on the CFT spectrumand OPE data. In particular, we show that the assumption of vacuum block dominance for BCFT correlators of twist operators is equivalent to a simple holographic prescription[2, 3] for the duals of BCFTs where the CFT boundary extends into the bulk as a purelygravitational end-of-the-world (ETW) brane with tension related to the boundary entropyof the BCFT (as suggested by [3]). This simple holographic prescription with a purely gravitational ETW brane is notexpected to be valid universally for holographic BCFTs. In many cases, we can have non-vanishing one-point functions for light scalar operators in the BCFT. These translate toback-reacting scalar fields in the dual geometry that we can think of as being sourced bythe ETW brane. In these cases with back-reacting scalars, the entanglement entropy for aninterval in the phase corresponding to a connected RT surface (e.g. Figure 1, top left) is acomplicated function of the interval size and location and is not expected to be reproducedby the vacuum contribution in some channel.On the other hand, the entanglement entropy in the phase corresponding to a discon-nected RT surface is still simple and universal (see equation (3.8)) and reproduced by thevacuum block contribution to the boundary channel. Thus, while vacuum block dominanceis not expected to hold universally for holographic BCFTs, the gravity results indicate thatit does hold in this phase. This suggests a particular sparseness condition (4.39) that shouldhold for the spectrum of any holographic BCFT.
Probing black hole interiors
Our results have various applications to the physics of black hole interiors: • In [5], it was argued that CFT states of type 2 above correspond to black holemicrostates with a specific behind-the-horizon region whose geometry can be deducedby a standard holographic gravity calculation. Holographic calculations using the The philosophy of reproducing gravitational results from vacuum block dominance goes by the amusingname “It from Id” [4]. – 2 –
A A id ΦΦ x x ΦΦ x id Figure 1 . Top: transition in the RT surface from connected to disconnected topology; in blackhole applications, the latter is associated with an entanglement wedge that includes the black holeinterior. Bottom: BCFT interpretation in terms of the two-point function of twist operators usedto compute entanglement entropy via the replica method. Phase transition comes from a switchof dominance between the identity block in a bulk channel to the identity block in a boundarychannel.
Ryu-Takayanagi (RT) formula [1] suggested that in many cases, the black hole interiorcan be probed via the entanglement entropy of sufficiently large subsystems of theCFT. Our direct CFT calculations confirm this expectation, precisely matching theresults of [5]. • In [6], a pair of BCFTs in the thermofield double state was proposed as a microscopicmodel of a black hole in equilibrium with its Hawking radiation, following [7–9]. TheRT surface for a fixed portion of the radiation system exhibits a phase transition as afunction of time as the black hole and radiation system interact. After the transition,the entanglement wedge of this subsystem includes a portion of the black hole interior,suggesting that the black hole has transferred information about its interior to theradiation system through interaction. Our CFT calculations directly confirm thebehavior of entanglement entropy surmised from the holographic calculation.
BCFT methods
The analysis in this paper parallels the computation of R´enyi entropies for multiple in-tervals in a CFT carried out by Hartman [10]. The CFT calculation makes use of thereplica method, where we analytically continue the R´enyi entropies to find the entangle-ment entropy. R´enyi entropies are computed from the CFT path integral on a multi-sheetedsurface, or equivalently, from correlation functions of twist operators in replicated versionof the original BCFT.These correlation functions can be expressed in terms of the basic BCFT data us-ing operator product expansions. Different ways of performing this expansion (different Here, we have a bulk OPE, a bulk-boundary OE, and a boundary OPE. – 3 –channels”) give the same result, but at large c , and under certain conditions on the CFTspectrum, the leading term in a 1 /c expansion of the full result can be reproduced by trun-cating a single channel to include only intermediate states descending from the vacuum.Depending on the locations of the twist operators, this dominant channel can change, lead-ing to a phase transitions in the entanglement entropy. The basic picture is illustrated inFigure 1.A running theme is the use of the doubling trick , and corresponding Ward identities,to simplify the kinematics of the BCFT. This relates the kinematics of BCFT n -pointcorrelation functions on a half-plane to those of 2 n -point correlation functions in a chiralCFT on the full plane. This applies not only to the single interval case, where a two-point function in the BCFT is found to possess the same kinematics as a chiral four-pointfunction, but to the multiple interval case, where the doubling trick can be applied to themonodromy method for computing conformal blocks. Replica gravity calculation
Faulkner [11] simultaneously obtained the results in [10] from the gravity perspective. In-stead of calculating replica partition functions using large central charge CFT methods,the partition functions were calculated using gravitational path integrals in the dual Eu-clidean replica saddle geometries. We will see that Faulkner’s calculation can be carriedover directly to the calculation BCFT entanglement entropies. There, phase transitions inholographic entanglement entropy are mirrored by transitions in the dominant saddle con-tributing to the gravitational path integral. In the phase where the entanglement wedgeof the radiation system includes the black hole interior, we can see explicitly from theFaulkner calculation that the relevant saddle is characterized by “replica wormholes”, assuggested recently in [6, 12, 13].
Outline
In Section 2 below, we provide some basic background on BCFTs and recall various resultsthat we will need for our calculation. In section 3, we recall the properties of gravity dualsfor holographic BCFTs and the holographic calculation of entanglement entropy in thevarious examples mentioned above. In Section 4, we present our main results, the direct(B)CFT calculation of entanglement and Renyi entropies, and an analysis of the conditionson the spectum and OPE coefficients necessary to reproduce the gravity predictions. InSection 5, we generalize our BCFT results to the case of multiple intervals. Section 6discusses the replica calculation of Renyi entropies on the gravity side and the relevance ofreplica wormholes. We finish with a discussion in Section 7.
In this section, we start with a brief review of boundary conformal field theories. Given a CFT, we can define the theory on a manifold with boundary by making achoice of boundary conditions for the fields, and possibly adding boundary degrees of For more detailed reviews of BCFTs, see e.g. [14–19]. – 4 –reedom coupled to the bulk CFT fields. For the theory defined on half of R d or R d − , (e.g. the region x ≥ x ), certain choices of the boundaryphysics give a theory that preserves the subset of the global conformal group mapping thehalf-space to itself, SO( d, ⊂ SO( d + 1 ,
1) for the Euclidean case. These choices define a boundary conformal field theory (BCFT). There are typically many choices of conformallyinvariant boundary condition for a given bulk CFT. We label the choice by an index b .In this paper, we focus on BCFTs defined starting from two-dimensional conformalfield theories. In this case, there is a natural boundary analog of the central charge, knownas the boundary entropy log g b . This may be defined by considering the CFT on a half-space x ≥ b at x = 0. As we review in section 3.1 below, theentanglement entropy of an interval [0 , L ] including the boundary is S = c L(cid:15) + log g b . (2.1)Thus, the boundary entropy gives a boundary contribution to the entanglement entropy.The quantity g b is also equal to the (regulated) partition function for the CFT on a diskwith boundary condition b . Boundary states
For any BCFT, there is a natural family of states | b, τ (cid:105) that we can associate to the parentCFT defined on a unit circle. The wavefunctional (cid:104) φ | b, τ (cid:105) is defined as the Euclidean pathintegral for the CFT on a cylinder of height τ , with boundary condition b at Euclideantime − τ and CFT field configuration φ at τ = 0. We can formally define a boundarystate | b (cid:105) associated with boundary condition b via | b (cid:105) = | b, τ → (cid:105) . (2.2)In terms of | b (cid:105) , we have | b, τ (cid:105) = e − τ H | b (cid:105) , (2.3)since adding δτ to the height of the cylinder corresponds to acting on our state with Eu-clidean time evolution e − δτH . The boundary state itself has infinite energy expectationvalue, but the Euclidean evolution used to define | b, τ (cid:105) suppresses the high-energy compo-nents so that | b, τ (cid:105) is a finite energy state. In general, this state is time-dependent.The overlap of the boundary state | b (cid:105) with the vacuum state is computed via thepath integral on a semi-infinite cylinder. This can be mapped to the disk via a conformaltransformation, so the result is the disk partition function: (cid:104) | b (cid:105) = g b . (2.4) More generally, there can be boundary RG flows between such theories. We can also consider the same theory with different boundary geometries; in this paper, we will onlyconsider geometries that can be mapped to a half-space via a conformal transformation. With this definition, the norm of the states is not equal to 1. This is a version of the global quench considered in the condensed matter literature [19], but we havecompactified the space on which the CFT is defined. – 5 – oundary operators
In addition to the usual CFT bulk operators, a BCFT has a spectrum of local boundaryoperators ˆ O J ( x ), each with a dimension ˆ∆ J . Via the usual radial quantization (takingthe origin to be a point on the boundary), these may be understood to be in one-to-onecorrespondence with the states of the BCFT on an interval with the chosen boundarycondition at each end. The boundary operator dimension is equal to the energy of thecorresponding state on the strip. Symmetries and correlators
A two-dimensional BCFT defined on the upper-half plane (UHP) preserves one copy of theVirasoro symmetry algebra, corresponding to transformations δz = (cid:15) ( z ) δ ¯ z = ¯ (cid:15) (¯ z ) ¯ (cid:15) (¯ z ) = (cid:15) (¯ z ∗ ) (2.5)that map the boundary to itself. These correspond to a set of generators˜ L n = L n + ¯ L n . (2.6)In this case, the conformal Ward identity becomes (cid:104) ˜ T ( z ) (cid:89) i O h i ¯ h i ( z i , ¯ z i ) (cid:105) = (cid:88) i (cid:18) h i ( z − z i ) + 1 z − z i ∂∂z i + ¯ h i (¯ z − ¯ z i ) + 1¯ z − ¯ z i ∂∂ ¯ z i (cid:19) (cid:104) (cid:89) i O h i ¯ h i ( z i , ¯ z i ) (cid:105) , where ˜ T ( z ) = (cid:80) n z − n − ˜ L n .The Virasoro symmetry algebra of a BCFT is thus the same as that of a chiral CFTon the whole plane. A consequence is that the kinematics (i.e. the functional form ofcorrelators given the operator dimensions) of the BCFT in the UHP is directly related tothat of a chiral CFT on the whole plane. Correlators (cid:104)O h ¯ h ( z , ¯ z ) · · · O h n ¯ h n ( z n , ¯ z n ) (cid:105) b UHP (2.7)of bulk CFT operators O h k ¯ h k with conformal weights ( h k , ¯ h k ) in the UHP are constrainedto have the same functional form as chiral CFT correlators (cid:104)O h ( z ) · · · O h n ( z n ) O ¯ h (¯ z ) · · · O ¯ h n (¯ z n ) (cid:105) (2.8)of fields O h k and O ¯ h k with chiral weights h k and ¯ h k respectively. More generally, we caninclude boundary operators ˆ O ˆ∆ I ( x I ) in (2.7), where x I is real. In this case, the functional Here, we recall that it is standard to treat z and ¯ z as independent coordinates and consider a com-plexified version of the symmetry algebra for which the infinitesimal transformations are δz = (cid:15) ( z ) and δ ¯ z = ¯ (cid:15) (¯ z ). The non-complexified transformations correspond to taking ¯ (cid:15) (¯ z ) = (cid:15) (¯ z ∗ ) with (cid:15) ( x ) real for real x , or ¯ (cid:15) (¯ z ) = − (cid:15) (¯ z ∗ ) with (cid:15) ( x ) pure imaginary for real x . Of these, the first set preserves the upper halfplane, acting explicitly as δx = ( (cid:15) ( x + iy ) + (cid:15) ( x − iy )) / , δy = − i ( (cid:15) ( x + iy ) − (cid:15) ( x − iy )) / Here, the original theory is defined on the slice where ¯ z = z ∗ , so the operators O ¯ h i (¯ z i ) live on the lowerhalf-plane. – 6 –orm is reproduced by adding chiral operators with h I = ˆ∆ I at z = x I to the chiralcorrelator (2.8). See [20] for a more complete discussion of this constraint, often referredto as the “doubling trick”.We will later make use of this kinematic equivalence to relate conformal blocks for aBCFT on the UHP to chiral conformal blocks on the entire plane. Bulk one-point functions
The doubling trick implies that a primary operator with weights ( h, ¯ h ) is kinematicallyallowed to have a nonvanishing one-point function if h = ¯ h (i.e. for a scalar primary). Inthis case, the one-point function (cid:104)O h,h ( z, ¯ z ) (cid:105) b UHP is constrained to have the same form as achiral two-point function (cid:104)O h ( z ) ¯ O h ( z ∗ ) (cid:105) , so we have (cid:104)O h, ¯ h ( z, ¯ z ) (cid:105) b UHP = A b O | z − z ∗ | h = A b O | y | ∆ O . (2.9)where we take z = x + iy here and below. Once the normalization of the operators is fixedby choosing the normalization of the two-point function in the parent CFT, the coefficient A b O in the one-point function is a physical parameter that depends in general on both theoperator and the boundary condition.Here and everywhere in this paper we will take the expectation value (cid:104)·(cid:105) b UHP to benormalized by the UHP partition function so that (cid:104) (cid:105) b UHP = 1 . (2.10) Bulk-boundary two-point functions
The correlation function (cid:104)O i ( z, ¯ z ) ˆ O I ( x (cid:48) ) (cid:105) b UHP (2.11)of bulk and boundary primary operators is constrained to have the functional form of achiral three-point function (cid:104)O h i ( z ) O I ( x (cid:48) ) O ¯ h i (¯ z ) (cid:105) . (2.12)For a scalar operator O i , this gives (cid:104)O i ( z, ¯ z ) ˆ O I ( x (cid:48) ) (cid:105) b UHP = B biI (2 y ) ∆ i − ∆ I ( y + ( x − x (cid:48) ) ) ∆ I , (2.13)where B biI forms part of the basic data of our BCFT. Taking ˆ O I to be the identity operator,we have from the previous section that B bi = A bi . Boundary operator expansion and OPEs
In the same way that a pair of bulk operators at separated points can be expanded asa series of local operators via the OPE, a bulk operator can be expanded in terms ofboundary operators via a boundary operator expansion (BOE) . For a scalar primary This follows by the same logic of the state-operator mapping and OPE in a CFT. The state producedby a bulk operator can be mapped by an infinite dilation to a local operator at the origin on the boundary.And, as in the OPE, we choose to expand this local operator in terms of a basis of dilation eigenstates. – 7 –perator, symmetries constrain the general form of this expansion to be O i ( z, ¯ z ) = (cid:88) J B bJi (2 y ) ∆ i − ∆ I ˜ C [ y, ∂ x ] ˆ O J ( x )= (cid:88) J B bJi (2 y ) ∆ i − ∆ I ˆ O J ( x ) + desc . , (2.14)where the sum is over boundary primary operators. The differential operator ˜ C determinesthe contribution of descendant operators and depends only on the conformal weights of O i and ˆ O J . The coefficients B bJi are related to the ones appearing in the bulk-boundarytwo-point function by raising the index with the metric g IJ appearing in the boundarytwo-point function (cid:104) ˆ O I ( x I ) ˆ O J ( x I ) (cid:105) = g IJ | x I − x J | I , (2.15)though we will generally assume that we are working with a basis of boundary operatorsfor which g IJ = δ IJ .Below, we will also make use of the ordinary OPE for bulk scalar operators, O i ( z , ¯ z ) O j ( z , ¯ z ) = (cid:88) k ˆ C kij | z − z | ∆ i +∆ j − ∆ k C ∆ i ∆ j ;∆ k [ z , ∂ z ] O k ( z , ¯ z )= (cid:88) k ˆ C kij | z − z | ∆ i +∆ j − ∆ k O k ( z , ¯ z ) + desc . (2.16)Finally, there is also an OPE for boundary fields, but we will not need this in our calcula-tions below. Two-point functions and conformal blocks
We now consider the bulk two-point function. Here, we restrict to scalar primary operatorsof equal dimension ∆ since that is what we will need below. However, in general, bulktwo-point functions in a BCFT can be non-vanishing for any conformal weights ( h , ¯ h )and ( h , ¯ h ). We discuss the general case in detail in Appendix A.By the doubling trick, the BCFT two-point function (cid:104)O ( z , ¯ z ) O ( z , ¯ z ) (cid:105) b UHP (2.17)of scalar operators with dimension ∆ has the same functional form as a four-point functionof chiral operators (cid:104)O ( z ) O ( z ) O (¯ z ) O (¯ z ) (cid:105) , (2.18)where each operator has chiral weight h = ∆ /
2. Making use of (A.1) for the general formof such a correlator, we have that (cid:104)O ( z , ¯ z ) O ( z , ¯ z ) (cid:105) b UHP = (cid:20) η y y (cid:21) ∆ F ( η ) (2.19) To see that this should still be valid in the presence of a boundary, note that in a conformal framewhere the upper-half-plane is mapped to the exterior of a circle surrounding the origin, the presence of theboundary is equivalent to the insertion of an operator at the origin (specifically, the operator associatedwith the state | b, τ (cid:105) described above). – 8 –here F ( η ) is some function of the cross-ratio η = ( z − ¯ z )( z − ¯ z )( z − ¯ z )( z − ¯ z ) . (2.20)The function F can be written more explicitly by making use of either the BOE or thebulk OPE for the operators in (2.19). Using the BOE for each operator in (2.19), the bulktwo-point functions can be expressed as a sum of boundary two-point functions. In thisway, the function F ( η ) in (2.19) may be expressed as F ( η ) = (cid:88) I B bI O B b O I F ( c, ∆ I , ∆ / | η ) (2.21)where the sum is over boundary primary operators and F ( c, ∆ I , ∆ / | η ) gives the contribu-tion of a single boundary primary operator and all of its Virasoro descendants. We showin Appendix A that this function is the usual conformal block appearing in the expansionof a chiral four-point function of operators with equal conformal weight h = ∆ / We can alternatively use the bulk OPE to reduce the BCFT two-point function (2.19)to a sum of one-point functions. This leads to an alternative expression for F ( η ), F ( η ) = (cid:88) i C i O O A bi F ( c, ∆ I , ∆ / | − η ) (2.22)Here, F is the same chiral conformal block as in (2.21), as we show in Appendix A. Theequivalence of the expressions (2.21) and (2.22) is a BCFT version of the usual crossingsymmetry constraints; in this case, we have a relation between bulk OPE coefficients andboundary operator expansion coefficients. In this section, we review the holographic calculation of entanglement entropies for sub-systems of BCFTs with gravitational duals, or for states of holographic CFTs defined viaEuclidean BCFT path integrals. These are the results that we will try to understand viadirect CFT calculations in the next section.
Certain BCFTs have a dual gravitational description. These correspond to holographicCFTs defined on a space M with boundary ∂M , and a boundary condition perhaps obeyingadditional constraints so that the theory remains holographic. The dual geometries areasymptotically AdS with boundary geometry M , but the bulk physics associated with ∂M can be different depending on the choice of boundary condition.For a d -dimensional CFT, we can have an effective “bottom up” description of thegravity dual as a ( d + 1)-dimensional asypmtotically AdS spacetime with an end-of-the-world (ETW) brane extending from ∂M [21–24]. However, in “top down” microscopic In general, the conformal block depends on four external weights; here and below, we will use theshorthand F ( c, h int , h | η ) ≡ F ( c, h int , [ h, h, h, h ] | η ) where the latter is the general expression for the chiralconformal block used in Appendix A. – 9 –xamples (see for instance [25–30]), the dual can be a smooth higher-dimensional geometry.In this case, the ETW brane in the lower-dimensional description represents the smoothdegeneration of an internal dimension.The simplest possible gravitational dual has an ETW brane coupling only to the bulkmetric field. Its action is taken to include a boundary cosmological constant (interpretedas the brane tension) and a Gibbons-Hawking term involving the trace of the extrinsiccurvature. The details of the action and equation of motion, and all the solutions that wewill require in this paper, may be found in [5]. A more general ansatz is an ETW braneaction with coupling to additional bulk fields, e.g. light scalars. We can use the Ryu-Takayanagi (RT) formula [1] to holographically calculate the entan-glement entropy for spatial subsystems. As usual, the entropy (at leading order in the 1 /c expansion) is given as S A = 14 G Area( ˜ A ) , (3.1)where ˜ A is the minimal area extremal surface in the dual geometry homologous to theboundary region A .A new feature of entanglement entropy for holographic BCFTs is that the RT surfacescan end on the ETW brane [3]. Here, we should keep in mind that the ETW braneitself represents a part of the bulk geometry. The homology condition says that the RTsurface X A for a region A on the boundary, together with the region A itself, should bethe boundary of a region Ξ A of the bulk spacetime: ∂ Ξ A = A ∪ X A . But when applyingthis condition, the ETW brane should be considered as part of this bulk spacetime regionΞ A , rather than an additional contribution to the boundary. As a result, we can have adisconnected RT surface for a connected boundary region, as shown in figure 3. As an example, consider the vacuum state of a two-dimensional BCFT on a half-space x >
0. Here, the SO(1 ,
2) symmetry preserved by the BCFT should be reflected in thedual geometry. Generally, this gives a warped product of AdS and an internal space, suchthat the full geometry has an asymptotic region that is locally AdS times some internalspace, with boundary geometry equal to the half-space on which the CFT lives. In general,we can write the metric asd s M = (cid:96) (cid:20) ˆ g ij ( µ )d µ i d µ j + f ( µ ) z (d z − d t ) (cid:21) . (3.2)Microscopic solutions of this type were constructed in [25, 26].We can also give a lower dimensional description (at least in the vicinity of the bound-ary), where we reduce on the internal space so that the internal metric is represented viascalars and vectors. In this case, we can writed s M = (cid:96) (cid:20) d µ + f ( µ ) z (d z − d t ) (cid:21) , (3.3)– 10 – A~ ETW θ Figure 2 . Holographic calculation of entanglement entropy for an interval A containing the bound-ary. The RT surface ˜ A sits at a fixed location on the AdS fibers of the dual geometry. Here A ishomologous to ˜ A since the ETW brane represents a smooth part of full microscopic geometry. where f ( µ ) → cosh ( µ/(cid:96) AdS ) as we approach the asymptotic boundary at µ = −∞ so thatthe metric is asymptotically AdS . In general, the scalar fields in the geometry can befunctions of the coordinate µ .In the simplest effective bulk theory, there is an ETW brane with stress-energy tensor8 πGT ab = − T g ab /(cid:96) AdS [21, 22], and bulk geometry pure AdS, with f ( µ ) = cosh ( µ/(cid:96) AdS ).The brane sits at µ max = arctanh( T ). Here, the coordinate µ is related to the angularcoordinate θ in a polar-coordinate description of Poincar´e-AdS by 1 / cos( θ ) = cosh( µ ), sothe brane goes into the bulk at a constant angle θ = arcsin( T ), as shown in Figure 2. Entanglement entropy for an interval including the boundary
We now consider the entanglement entropy for an interval in the half-space. In the caseof an interval [0 , L ] containing the boundary, we expect the universal form (2.1) for theentanglement entropy. In the holographic calculation with the general metric (3.2), the RTsurface sits at a constant position on the AdS fiber, so the entanglement entropy is S = 14 G (cid:90) z>(cid:15) d d +1 x (cid:112) ˆ g , (3.4)where z is the Fefferman-Graham radial coordinate and d is the dimension of the internalspace. We can regulate this by subtracting off half the area of the entangling surface of aninterval of length 2 L in vacuum AdS, so S = (cid:34) (cid:96) d +1AdS G (cid:90) z>(cid:15) d d µ i (cid:112) ˆ g − (cid:96) d +1AdS G (cid:90) z>(cid:15),x> d d x (cid:112) ˆ g AdS (cid:35) + (cid:96) d +1AdS G (cid:90) z>(cid:15),x> d d x (cid:112) ˆ g AdS , (3.5)where ˆ g AdS is the metric for pure
AdS . In this expression, the term in square brackets hasa finite limit as (cid:15) →
0, independent of L , while the second term gives c log (cid:0) L(cid:15) (cid:1) . Thus, wereproduce (2.1), with the identificationlog g b = lim (cid:15) → (cid:34) (cid:96) d +1AdS G (cid:90) z>(cid:15) d d x (cid:112) ˆ g − (cid:96) d +1AdS G (cid:90) z>(cid:15),x> d d x (cid:112) ˆ g AdS (cid:35) . (3.6)As an example, with a constant tension ETW brane, we havelog g b = (cid:96) AdS G (cid:90) µ max d µ = (cid:96) AdS G arctanh( T ) = c T ) . (3.7)– 11 – Figure 3 . Holographic calculation of entanglement entropy for an interval A away from the bound-ary. The RT surface has two possible topologies, a connected (solid curve) and disconnected (dashedcurves). This is the result of Takayanagi [3] relating the boundary entropy to brane tension.
Entanglement entropy for an interval away from the boundary
Now consider the holographic calculation of entanglement entropy for an interval [ x , x ]away from the boundary. In general, the CFT result for this entanglement entropy doesnot have a universal form.In the holographic calculation, we can have a phase transition between two differentpossible RT surface topologies: a connected RT surface or a disconnected surface with bothcomponents ending on the ETW brane. The two topologies are shown in Figure 3.Let us consider these phases in more detail. In the disconnected case, the RT surfacecomputing the entanglement entropy of an interval [ x , x ] is the union of the RT surfacesassociated with [0 , x ] and [0 , x ]. Thus, to leading order in large N , we have that S disc[ x ,x ] = S [0 ,x ] + S [0 ,x ] = c (cid:18) x (cid:15) (cid:19) + c (cid:18) x (cid:15) (cid:19) + 2 log g b . (3.8)This result makes use only of the disconnected topology of the RT surface, so is a universalresult for the disconnected phase in any holographic theory.In the connected phase (expected to apply when the interval is sufficiently far fromthe CFT boundary), there is in general no simple universal result for the entanglemententropy. We need to find an RT surface in the dual geometry, and the calculation of thissurface will depend on the details of the metric ˆ g appearing in (3.2).For certain boundary conditions, it may be that the dual gravitational theory is well-described by an ETW brane with only gravitational couplings. In this case, the dualgeometry is locally AdS , and the calculation of entanglement entropy for the interval willbe the same as the holographic calculation of vacuum entanglement entropy for the sameinterval in the CFT without a boundary. Thus, we have S conn[ x ,x ] = c (cid:18) x − x (cid:15) (cid:19) . (3.9)Below, we will try to understand what conditions must be satisfied in the BCFT in orderthat this result is correct. – 12 –n cases where (3.8) and (3.9) give the correct results for the two possible RT-surfacetopologies, the actual entanglement entropy will be computed by taking the minimum ofthese two results. We find that the disconnected surface gives the correct result for theentanglement entropy whenlog (cid:20) (cid:18)(cid:114) x x − (cid:114) x x (cid:19)(cid:21) > g b c , (3.10)so that for a fixed interval size, we have a phase transition as the location of the intervalrelative to the boundary is varied.In the more general case where the bulk geometry is not locally AdS, there is noexplicit result for the entanglement entropy in the connected phase and (3.10) does notapply. However, we expect that the qualitative behavior of the entanglement entropy issimilar, with a transition to the disconnected phase as the interval approaches the boundary.We can view this as a prediction for the behavior of entanglement entropy in holographicBCFTs. One of our main goals below will be to understand the existence of this transitionvia a direct CFT calculation.Before turning to the machinery of CFTs, we review two closely related holographiccalculations. In both cases, the dual geometry involves a black hole, and the transitionin RT surfaces takes us between phases where the entanglement wedge of the CFT regionunder consideration does or does not include a portion of the black hole interior. | b, τ (cid:105) Consider a CFT on S , in the state | b, τ (cid:105) defined via the Euclidean path integral (2.3).We can consider the entanglement entropy for an interval of angular size ∆ θ in this state,at some fixed Lorentzian time. As described in [5, 31], for small enough τ , this is a high-energy pure state of the CFT and the dual geometry is expected to be black hole. Assumingthat the bulk effective gravitational theory for the BCFT involves a purely gravitationalETW brane of tension T , it was shown in [5, 31] that the dual geometry for T > θ , the RT surface lies entirely outsidethe horizon and gives a time-independent entanglement entropy S conn = c (cid:20) τ π(cid:15) sinh (cid:18) π ∆ θ τ (cid:19)(cid:21) , (3.11)where we take the circumference of the CFT circle to be 1.For small enough τ , large-enough interval size ∆ θ , and time t sufficiently close to 0(when the state is prepared) we also have a disconnected phase, where the RT surface isa union of two surfaces at fixed angular position that enter the horizon and terminate onthe ETW brane. Here, we find that S disc = c (cid:20) τ (cid:15)π cosh (cid:18) πt τ (cid:19)(cid:21) + 2 log g b . (3.12)– 13 – (t) H r H r Figure 4 . The dual geometry for | b, τ (cid:105) for sufficiently small τ is a portion of the maximally-extended AdS-Schwarzchild geometry, cut off by a spherically symmetric ETW brane. The pictureson the right show the spatial slice at t = 0 and the connected and disconnected topologies for theRT surface corresponding to a large interval on the boundary circle. This is smaller than the connected result (and thus represents the actual entanglemententropy) when sinh (cid:18) π ∆ θ τ (cid:19) ≥ cosh (cid:18) πt τ (cid:19) e gbc . (3.13)When this condition is satisfied, the entanglement wedge of the interval includes a portionof the black hole interior, and hence the entanglement entropy probes the interior geometry.For late times, the connected phase always dominates. This is consistent with the expecta-tion that the state will thermalize, so that the entanglement entropy for a subsystem givesthe thermal result. In our final example, we take the thermofield double state of two BCFTs, each on a half-space, and consider the entanglement entropy for the subsystem A ( x ) consisting of theunion of the regions [ x , ∞ ) in each CFT, as in Figure 8d. In [6], following [9], it was arguedthat this system provides a model of a two-sided 2D black hole coupled to an auxiliaryradiation system, where the Page time for the black hole is t Page ∼ g b ) /c .While the simple observables in this system are time-independent, the holographiccalculations in [6] revealed that the entanglement entropy for the subsystem A ( x ) increaseswith time, then undergoes a phase transition. After this transition, the entanglemententropy is time-independent and the entanglement wedge of the radiation system includesa substantial portion of the black hole interior. The interpretation is that while no netenergy is exchanged between the black hole and the radiation system, information fromthe black hole escapes into the radiation system until the radiation system contains enoughof it to reconstruct the black hole interior. This is similar to the behaviour of correlators in microscopic models of black hole collapse based onapproximate global quenches of Vaidya type [4]. A phase transition in channel dominance leads to a shiftin the gravitational saddle computing entanglement entropy, which in turn is responsible for maintainingunitarity. We thank Tarek Anous for discussion of this point. – 14 – igure 5 . Dual geometry to the thermofield double state of two BCFTs, showing two possibletopologies for the RT surface for the region corresponding to the union of points in either CFT ata distance larger than x from the boundary. The phase transition is seen most easily via a holographic calculation in the dualthree-dimensional gravity picture. Here, we have a dynamical ETW brane that connectsthe two CFT boundaries, as shown in Figure 5. At early times, the RT surface for A ( x )is connected and does not intersect the ETW brane. The entanglement entropy is S conn = c (cid:18) (cid:15) cosh t (cid:19) . (3.14)At late times, the RT surface is disconnected with components stretching from the bound-ary of each BCFT to the ETW brane. In this case, the entropy is S disc = c (cid:18) (cid:15) sinh x (cid:19) + 2 log g b . (3.15)This is smallest (and thus gives the correct result) for cosh t > e gbc sinh x .In the next section, we will consider the direct CFT calculation of the entanglemententropies for the situations we have just described. As we explain in Section 4.6, while thethree calculations correspond to rather different physical scenarios, the underlying CFTcalculation of the entanglement entropies is are directly related. In this section, we move on to our central task: performing a direct CFT calculation ofentanglement entropy for one or more intervals in the vacuum state of a BCFT on a half-line, for the thermofield double state of two BCFTs, or for the CFT state | b, τ (cid:105) generatedby a Euclidean BCFT path integral. We will argue that with certain assumptions, we candirectly reproduce the holographic results described in the previous section. We begin by briefly recalling the CFT calculation of entanglement entropy (for more details,see [32]). We consider a CFT or BCFT on a spatial geometry M in some state | Ψ (cid:105) , definedby a Euclidean path integral on a geometry H with boundary M . We would like to calculatethe entanglement entropy S A = − tr( ρ A log ρ A ) for a region A ⊂ M .The entanglement entropy can be obtained from a limit of n - R´enyi entropies S ( n ) A : S A = lim n → S ( n ) A , S ( n ) A := 11 − n log Tr[ ρ nA ] . (4.1)– 15 – igure 6 . Left . Three-replica geometry, R , with a local field ϕ . Right . Individual copies s , withboundary conditions for ϕ i implemented by twists Φ , ¯Φ . The matrix elements (cid:104) φ − A | ρ A | φ + A (cid:105) are calculated from the path integral on a space ( ¯ HH ) A formed from gluing two copies of H along the complement of A in M , where we setboundary conditions φ ( x, τ = ± (cid:15) ) = φ ± A on either side of a cut A . The proper normalizationis obtained by dividing by the same path integral without a cut along A .The trace Tr[ ρ nA ] is then obtained by the path integral on a replica geometry R n obtained by gluing n copies of ( ¯ HH ) A across the cut A, with the lower half of the cuton each copy glued to the upper half of the cut on the next copy, as shown in Figure 6.Including the proper normalization in the path integral expression for the density matrixgives Tr[ ρ nA ] = Z n Z n , (4.2)where Z n is the partition function for the CFT on R n .The ratio Z n /Z n can be expressed as a correlation function of twist operators for aCFT/BCFT defined to be the product of n copies of the original theory. A twist operatorΦ n ( z ) inserted at z is defined via the path integral by inserting a branch cut ending at z , across which the fields in the k th copy of the (B)CFT are identified with fields in the( k + 1)-st copy as we move clockwise around the branch point. Similarly, an anti-twistoperator ¯Φ n ( z ) inserts a branch cut ending at z across which fields in the k th copy of theCFT/BCFT are identified with fields in the ( k − unpaired twistoperator, with the branch cut running between the operator insertion and the boundary.For both the CFT and BCFT, deforming the branch cut simply corresponds to changingthe fundamental domain of the replica Riemann surface R n , as in Figure 7. More precisely, the path integral corresponding to the second copy is the one associated with (cid:104) Ψ | ; anycomplex sources in the action should be conjugated. – 16 – igure 7 . Left . Deforming the contour of the fundamental domain of R n for a CFT. Right .Performing the equivalent deformation on R n for a BCFT. Two-point function of the twist operators in a CFT
The correlator (cid:104) ¯Φ n ( x )Φ n ( x ) (cid:105) for the n -copy CFT on the real line with a branch cutrunning between x and x exactly computes the right-hand side of (4.2) for the casewhere A is the interval [ x , x ]. The two point function takes a simple form, since as shownin [17], the twists fields Φ n , ¯Φ n act like scalar primaries with scaling dimension d n := c (cid:18) n − n (cid:19) (4.3)and weights h n = ˆ h n = d n /
2. Thus, we have (cid:104) ¯Φ n ( x )Φ n ( x ) (cid:105) ∼ | x − x | − d n , (4.4)as we will derive again below.To say more about the coefficient, we need to define the twist operators more preciselyby specifying the behavior of the CFT at the branch points. As a specific regularization,we can consider instead the n -copy theory defined on a space obtained by removing a diskof radius (cid:15) centered at each branch point and placing boundary condition labelled by a i atthe i th resulting circular boundary [33]. The resulting path integral geometries for Z n (and Z ) are then smooth. A conformal transformation z (cid:55)→ i log (cid:18) z − x z − x (cid:19) (4.5)maps the original plane to a cylinder defined by the complex plane with identification z ∼ z + 2 π , and the branch cut [ x , x ] mapping to (cid:60) ( z ) = π . It will be convenient for our discussion below to allow different boundary conditions to regulate thedifferent twist operators, but generally we can choose the same one for each. – 17 –or small (cid:15) , the boundaries surrounding the branch points map to z = ± i log[( x − x ) /(cid:15) ] up to corrections of order (cid:15) . Thus, the path integral geoemtry is a cylinder of length τ = 2 log[( x − x ) /(cid:15) ], with boundary condition a , a at the two ends. The replica geometryis defined by gluing n copies of this cylinder along the vertical branch cut, so correspondsto a cylinder with circumference 2 πn . We can write the path integral on this space usingboundary states as Z n = (cid:104) a | e − τ πn H | a (cid:105) (4.6)where H is the Hamiltonian for the CFT on a circle of unit length. For large τ , the operatorinside approaches a projector to the vacuum state e − τ πn H → e − τ πn E | (cid:105)(cid:104) | , (4.7)where E = − πc/ Z n ) (cid:15) → = (cid:18) | x − x | (cid:15) (cid:19) c n (cid:104) a | (cid:105)(cid:104) | a (cid:105) . (4.8)Finally, (cid:104) ¯Φ n ( x )Φ n ( x ) (cid:105) = Z a,(cid:15)n ( Z a,(cid:15) ) n = ( (cid:104) a | (cid:105)(cid:104) | a (cid:105) ) (1 − n ) (cid:18) | x − x | (cid:15) (cid:19) − d n . (4.9)Making use of this in (4.1) and (4.2) gives the standard result for the entanglement entropyof an interval. With our original definition of (cid:15) , we have c/ L/(cid:15) ) + log g a + log g a ,however, it will be convenient to take a = a = a and absorb the last two terms here intothe definition of (cid:15) . One-point function of the twist operator on a half space
In a BCFT, the twist operators also have a non-vanishing one-point function, related tothe R´enyi entropies for an interval [0 , x ] in the vacuum state of the BCFT on a half-space x ≥
0. We can calculate this using the regularization defined above.We will consider a BCFT on the UHP with boundary condition b at (cid:61) ( z ) = 0, withthe twist operator at z = x + iy regulated by boundary condition a . The conformaltransformation z (cid:55)→ i log (cid:18) z − z ∗ z − z (cid:19) (4.10)maps the upper half-plane to a cylinder defined by the complex plane with identification z ∼ z + 2 π , where the boundary along the real axis maps to the interval [0 , π ] on the realaxis. The circle of radius (cid:15) regulating the twist operator maps (in the limit of small (cid:15) ) toa second end of the cylinder at (cid:61) ( z ) = log(2 y /(cid:15) ).Thus, the one-point function is Z n / ( Z ) n , where Z n is the partition function on acylinder of circumference 2 πn and height τ = log(2 y /(cid:15) ). Using the second equation in We recall that the boundary state was defined using a circle of length 1. Scaling the cylinder to havethis circumference, the length becomes τ / πn . – 18 –4.9), we have that (cid:104) Φ n ( z , ¯ z n ) (cid:105) = Z n ( Z ) n = ( (cid:104) a | (cid:105)(cid:104) | b (cid:105) ) (1 − n ) (cid:12)(cid:12)(cid:12)(cid:12) y (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) − d n , (4.11)Interpreting the (cid:60) ( z ) direction as Euclidean time, this gives tr( ρ n ) for an interval [0 , y ]in the vacuum state of a BCFT on a half-space. From (4.1), the entanglement entropyassociated with this density matrix is S = c (cid:18) y (cid:15) (cid:19) + log( g a ) + log( g b ) . (4.12)The term log( g a ) in the regulator can be absorbed by a redefinition of (cid:15) to give the result(2.1). On the other hand, boundary entropy term log( g b ) is physical. It is equal to thedifference between the BCFT entanglement entropy and the half the entanglement entropyin the parent CFT for an interval of length 2 y , with twist regularization fixed. We are now ready for our main calculation. We consider the correlator on the UHP of atwist operator at z and an anti-twist operator at z . As discussed in Section 2, we canexpress the two-point function here either as a sum of bulk one-point functions (the bulkchannel ), or as a sum of boundary two-point functions (the boundary channel ), using thebulk OPE or the BOE respectively. We now consider these expressions explicitly. Boundary channel for the two-point function
The boundary channel for the BCFT two-point function is obtained by first expanding eachoperator using the BOE, so that the bulk two-point function becomes a sum of boundarytwo-point functions. The contribution of two-point functions involving all the operators ina multiplet of the Virasoro symmetry sums to a conformal block. Using the general result(2.19) with (2.21), we find (cid:104) Φ n ( z , ¯ z ) ¯Φ n ( z , ¯ z ) (cid:105) b UHP = (cid:20) η y y (cid:21) d n (cid:88) I B b Φ I B b ¯Φ I F ( c, ∆ I , d n | η ) (4.13)where η = ( z − ¯ z )( z − ¯ z )( z − ¯ z )( z − ¯ z ) . (4.14)and where I indexes untwisted boundary operators in the n -fold product theory. As wereview in Appendix B, the BOE coefficients here can be expressed in terms of correlatorsof boundary operators in the original BCFT. Writing the cross ratio explicitly in termsof real coordinates, we have η = 4 y y ( x − x ) + ( y + y ) = 1 − ( x − x ) + ( y − y ) ( x − x ) + ( y + y ) , (4.15) Note that this is the same redefinition as the previous subsection. To avoid cluttering our notation further, we will generally leave n implicit in our BOE coefficients B . – 19 –o we see that η is a real number in [0 , η → z and z aremuch closer to each other than the boundary and η → z and z aremuch closer to the boundary than to each other.Consider the contribution from the term where only the boundary identity operator iskept in each BOE (2.14). This is equal to the disconnected term in the two-point functionthat factorizes into the product of one-point functions, and hence (cid:104) Φ n ( z , ¯ z ) ¯Φ n ( z , ¯ z ) (cid:105) b UHP , = B b Φ B b ¯Φ [4 y y ] d n = g − n ) b (cid:15) d n [4 y y ] d n , (4.16)where we have read off B b Φ from (4.11). In general, this contribution should dominate thecorrelator in the limit η →
0, where the two operators approach the boundary.
Bulk channel for the two-point function
We can obtain an alternative expression for the two-point function using the bulk OPEto express the product Φ n ( z , ¯ z ) ¯Φ n ( z , ¯ z ) as a sum of bulk operators. This reduces thetwo-point function to a sum of one-point functions.Using the general result (2.19) with (2.22) for this bulk-channel expression for thetwo-point function, we obtain (cid:104) Φ n ( z , ¯ z ) ¯Φ n ( z , ¯ z ) (cid:105) b UHP = (cid:20) η y y (cid:21) d n (cid:88) i C i Φ n ¯Φ n A bi F ( c, h i , d n | − η )= (cid:20) − η | z − z | (cid:21) d n (cid:88) i C i Φ n ¯Φ n A bi F ( c, h i , d n | − η ) , (4.17)where i indexes untwisted operators in the n -fold product CFT. Again, it will be usefulbelow to note the contribution where we keep only the bulk identity operator term in theOPE (2.16): (cid:104) Φ n ( z , ¯ z ) ¯Φ n ( z , ¯ z ) (cid:105) b UHP , = C Φ n ¯Φ n A b | z − z | d n = (cid:15) d n | z − z | d n , (4.18)where we have used A b = 1 and C Φ n ¯Φ n = (cid:15) d n from (4.9). This contribution shoulddominate the correlator in the limit η →
1, where the two operators approach each otheraway from the boundary.
We now use our results to calculate the Renyi entropy for an interval A = [ y , y ] for thevacuum state of a BCFT on a half space y >
0. This is related to the two-point functionof twist operators on the upper half-plane as e (1 − n ) S ( n ) A = (cid:104) Φ n ( z , ¯ z ) ¯Φ n ( z , ¯ z ) (cid:105) b UHP , (4.19)where we take z = (0 , y ) and z = (0 , y ).– 20 – ulk and boundary limits First, consider the R´enyi entropy in the limits η → η →
1, where the twist operatortwo-point function is given by (4.16) and (4.18) respectively. In this case, we find that S ( n ) A = c n + 1 n log (cid:18) y (cid:15) (cid:19) + c n + 1 n log (cid:18) y (cid:15) (cid:19) + 2 log g b η → c n + 1 n log (cid:18) | y − y | (cid:15) (cid:19) η → . (4.20)Taking the n → S A = c (cid:18) y (cid:15) (cid:19) + c (cid:18) y (cid:15) (cid:19) + 2 log g b η → c (cid:18) | y − y | (cid:15) (cid:19) η → . (4.21)We see that these precisely match the holographic results (3.8) and (3.9).The result (3.8) is expected to be valid for any holographic CFT in some finite intervalaround η = 0 where the RT surface is disconnected, while the result (3.9) is expected tobe valid in a finite interval around η = 1 in the case where the holographic theory canbe modelled by a purely gravitational ETW brane. Thus, the results (4.21) have a muchlarger range of validity than we would naively expect from the CFT point of view. Wewould now like to understand from the CFT perspective how this larger range of validityfor the vacuum results can arise. Entropies at large c We begin with the general expressions (4.13) and (4.17) for the twist operator two-pointfunction. General closed-form expressions for the conformal blocks are not known, but inthe semiclassical limit c → ∞ , the chiral conformal blocks exponentiate [34]: F ( c, h int , h | η ) c →∞ = exp (cid:20) − c f (cid:18) h int c , hc , η (cid:19)(cid:21) . (4.22)The exponent f is called the semiclassical block . In our case of identical external weights,recursion relations for the block allow one to commute the limits c → ∞ and h int /c, h/c → light internal operators h int = O ( c )are just the vacuum (semiclassical) block: f (cid:18) hc , η (cid:19) ≡ f (cid:18) , hc , η (cid:19) . (4.23)We can apply these results to our two-point function of twist operators, for which all ofthe external dimensions are d n /
2, and the central charge of the replicated CFT is nc . There is a beautiful but non-rigorous argument for exponentiation from Liouville theory, using theexplicit structure constants [35, 36] and the path integral. We refer the interested reader to the clearaccount in [37]. In general, this depends on the set of external weights, but our notation takes into account that all ofthe external weights are identical. – 21 –e find that the c → ∞ limit of the expressions (4.13) and (4.17) for the twist operatortwo-point function in the boundary and bulk channels become (cid:104) Φ n ( z , ¯ z ) ¯Φ n ( z , ¯ z ) (cid:105) b UHP = (cid:18) η y y (cid:19) d n ˆ D L e − nc f ˆ0 ( dn nc ,η ) + (cid:88) J H B b Φ J B b ¯Φ J e − nc f (cid:18) ˆ∆ Jnc , dn nc ,η (cid:19) (4.24)= (cid:18) η y y (cid:19) d n D L e − nc f ( dn nc , − η ) + (cid:88) j H C j Φ n ¯Φ n A bj e − nc f (cid:16) hjnc , dn nc , − η (cid:17) , (4.25)where J H , j H range over heavy internal operators, and ˆ D L and D L are degeneracy factorsmultiplying the vacuum channel:ˆ D L = (cid:88) J L B b Φ J B b ¯Φ J , D L = (cid:88) j L C j Φ n ¯Φ n A bj . (4.26)As c → ∞ , the sums (4.24) and (4.25) should be dominated by the exponential withsmallest exponent, if the coefficients of the exponential in the sum are not too large. Moreprecisely, let us now make two assumptions:1. The contribution of all heavy internal operators, in a neighbourhood around η = 0 or η = 1 in the respective channel, is exponentially suppressed in c . We will take heavyto mean any operators whose dimension scales as O ( c ) or greater.2. The degeneracy factors ˆ D L , D L are given by the vacuum contribution times somemultiplicative correction that does not change the leading exponential in c behaviour.If the neighborhoods described in the first assumption meet at some point η n ∗ , so thatthey cover the entire interval η ∈ [0 , c behaviour of thecorrelator is given by the larger of the vacuum block contribution in the boundary channelor the vacuum block contribution in the bulk channel for the entire interval η ∈ [0 , vacuum block dominance .Under our first assumption of vacuum block dominance, the R´enyi entropy for aninterval [ y , y ] is given by S ( n ) A = c n + 1 n log( y + y ) + c nn − f (cid:18) (cid:18) − n (cid:19) , y y ( y + y ) (cid:19) + 11 − n log ˆ D L η < η n ∗ c n + 1 n log( y + y ) + c nn − f (cid:18) (cid:18) − n (cid:19) , ( y − y ) ( y + y ) (cid:19) + 11 − n log D L , η > η n ∗ (4.27) where η n ∗ is the value of η at which the lower expression becomes larger than the upperone. In the limit n →
1, the behavior of the semiclassical vacuum block follows from theresult that for small α = ( n − /
12 [10], f ( α, η ) = 12 α log η + O ( α ) , (4.28)as we derive in Section 5.2. – 22 –nder our second assumption of vacuum block dominance, we have that (at order c )lim n → − n log ˆ D L = lim n → − n log( B b Φ B b ¯Φ )= − c (cid:15) + 2 log g b lim n → − n log D L = lim n → − n log( C Φ n ¯Φ n A b )= − c (cid:15) up to contributions O ( c ). Note that, by keeping the boundary entropy term, we areassuming that it, too, is O ( c ).Using these results and the results for the semiclassical blocks gives S A = lim n → S ( n ) A = c (cid:18) y (cid:15) (cid:19) + c (cid:18) y (cid:15) (cid:19) + 2 log g b η < η ∗ c (cid:18) | y − y | (cid:15) (cid:19) η > η ∗ (4.29)where η ∗ is the value of η where the two expressions coincide. These are exactly the results(4.21) we obtained keeping only the contributions from boundary and bulk identity opera-tors. Thus, we see that the assumption of vacuum block dominance provides the extendedrange of validity for the formulas in (4.21), so that the results match our gravitationalcalculation with a purely gravitational ETW brane. Our expression in (4.29) now matches precisely the gravitational calculation, (3.8) and(3.9) for all η , at leading order in c . Following the previous work for bulk CFTs [10, 39],let us now explore what constraints our vacuum block dominance assumptions place onthe spectrum and OPE data of the BCFT. We begin with the disconnected phase in the boundary channel that dominates in a neigh-bourhood of η = 0. Our first assumption held that the contribution of heavy boundaryoperators was exponentially suppressed in c and does not contribute at leading order. Wewill examine this claim in a cascading series of steps, from heaviest to lightest operators.First, looking at operators of dimension O ( c α ) for α >
1, we find that agreement withthe gravity calculation seems to place rather weak constraints on the BCFT. In particular,the convergence of the boundary OPE can be used in an exactly analogous manner tothe convergence of the bulk OPE [40] to show that the contribution of all operators ofdimension ˆ∆ > O ( c ) is exponentially suppressed in the central charge.We then need only worry about operators up to dimension O ( c ). Define ρ b,n ( δ ) d δ to be the number of untwisted n -fold product boundary operators with dimensions ˆ∆ ∈ c [ δ, δ + d δ ], and define a measure of the average twist-operator BOE coefficients by | B n ( δ ) | = (cid:80) ˆ∆ I ∈ c [ δ,δ +d δ ] ¯ B b Φ I ¯ B b ¯Φ I (cid:80) ˆ∆ I ∈ c [ δ,δ +d δ ] . (4.30)– 23 –here we have introduced ¯ B b ¯Φ I = (cid:15) − d n g n − b B b ¯Φ I to remove a universal prefactor that appearsin all the BOE coefficients (see Appendix B). We can use the known small η expansion ofthe semiclassical block [10], f ( h int , h ext , η ) = 6(2 h ext − h int ) log η − h int η + O ( η ) , (4.31)to write the bracketed expression in (4.24) as e − nc (cid:16) − n (cid:17) log η/(cid:15) +2(1 − n ) log g b (cid:90) O (1)0 d δ ρ b,n ( δ ) | B n ( δ ) | e cδ log η + cδη + O ( η ) . (4.32)In this expression, the heavy operators will not contribute to the order c entanglement en-tropy if the integral over of any region bounded away from zero is exponentially suppressedin c as compared to the integral near zero. This constrains the product of the densityof operators appearing in the twist OPE and their OPE coefficients so as not to grow soquickly as to overcome the suppression from the block. For η (cid:28)
1, this requireslog (cid:0) ρ b,n ( δ ) | B n ( δ ) | (cid:1) < cδ log( η − ) for δ (cid:38) . (4.33)In particular, requiring the CFT calculation to agree with the gravity result in an interval0 < η < ˆ η (cid:28) ρ b,n ( δ ) | B n ( δ ) | grows more slowly than exp( cδ log(1 / ˆ η ).Extending to a larger range with ˆ η not necessarily much less than 1 gives a strongerconstraint, but the exact form requires more detailed knowledge of the semiclassical block.Let us then focus on the lower limit of this integral and consider only operators ofdimension less than O ( c α ) for α <
1, where we can approximate the semiclassical blockby the vacuum block for all operators, up to O ( c α − ) corrections. The gravity calculationpredicts that the leading exponential in c behavior of the result matches the vacuum channelcontribution, so we require that (cid:88) I L ¯ B b Φ n I ¯ B b ¯Φ n I (4.34)is subexponential in c . In Appendix B, we recall that the coefficients ¯ B b Φ n I can be ex-pressed in terms of n -point correlations functions of light boundary operators in the originalBCFT, so this constraint can be translated into a constraint on the spectrum and n -pointfunctions of the original BCFT. We consider the case n = 2 in more detail below. We can largely repeat the above analysis in the bulk channel. Again, we find only weakconstraints on operators of dimension O ( c α ) for α >
1. The convergence of the bulk OPEcan be used now precisely as in [40] to show that the contribution of all operators of It is also interesting to consider the constraints on the CFT assuming that we have a conventionalgravitational theory with a usual semiclassical expansion. In this case, the corrections to the entropyare expected to be of order c (as opposed to some larger power of c or log c ). In this case, we wouldobtain stronger constraints on the BCFT. However, for this paper, we focus on the constraints arising fromdemanding that the order c terms in the entropies match with the classical gravity calculation. – 24 –imension ∆ > O ( c ) is exponentially suppressed in the central charge. We then need onlyworry about operators up to dimension O ( c ).Define ρ n ( δ ) d δ to be the number of bulk untwisted n -fold product operators withdimensions ∆ ∈ c [ δ, δ + d δ ] and AC b,n ( δ ) = (cid:80) ∆ i ∈ c [ δ,δ +d δ ] ¯ C i Φ¯Φ A bi (cid:80) ∆ i ∈ c [ δ,δ +d δ ] , (4.35)where ¯ C i Φ¯Φ = (cid:15) − d n C i Φ¯Φ . Then in this channel, we have e − nc (cid:16) − n (cid:17) log(1 − η ) /(cid:15) (cid:90) O (1)0 d δ ρ n ( δ ) AC b,n ( δ ) e cδ log(1 − η )+ cδ (1 − η )+ O ((1 − η ) ) . (4.36)For heavy operators to not contribute to the order c entanglement entropy when 1 − η (cid:28) ρ n ( δ ) AC b,n ( δ )) < cδ log(1 − η ) − for δ (cid:38) . (4.37)This is analogous to the boundary channel condition, but with η → − η .For operators with dimension O ( c α ) for α <
1, assumption 2 must hold in order tomatch with from gravity with a purely gravitational ETW brane. This requires that forthe light operators, the sum (cid:88) i L ¯ C i Φ¯Φ A bi (4.38)should be sub-exponential in c . We have now spelled out explicitly a set of conditions on a BCFT that will ensure that thedirect BCFT calculation of entanglement entropy matches with the gravity results in theholographic model with a purely gravitational ETW brane. However, we recall that thedisconnected phase result (3.8) is universally valid for any holographic BCFT. Assumingthat entanglement entropy has such a disconnected phase for some interval η ∈ [0 , η ∗ ],as it does for the simple model, suggests that vacuum block dominance should hold forany holographic BCFT in an interval η ∈ [0 , η n ], where the upper end of the interval maydepend on the Renyi index n .From the results in the previous subsection, this implies a constraintlog (cid:0) ρ b,n ( δ ) | B n ( δ ) | (cid:1) < cβ ∗ δ + O ( c a ) , a < , (4.39)where the quantities in the left side were defined in (4.30) and the preceeding paragraph.Here our knowledge of the semiclassical block was not sufficient to fix the O (1) coefficient β ∗ in this bound. In addition, we have a constraint (4.34) on the light operators. We takethese bounds to be novel constraints on which BCFTs can possibly have a gravitationaldual.Although we found an analogous boundlog ( ρ n ( δ ) AC b,n ( δ )) (cid:46) cγ ∗ δ + O ( c a ) , a < A bi . While the disconnected phase is universal and depends only on the boundaryentropy, the connected phase depends on the gravitational background (e.g. whether wehave backreacting scalars in the solution dual to the BCFT vacuum). The vacuum solutionfor the bulk CFT is unique, but in contrast, there is no unique gravitational solutionconsistent with the symmetries of the BCFT.A useful diagnostic for the non-universal behaviour of entropy and the bulk back-ground is when light operators have large, O ( c ), expectation values that backreact on thegravitational solution: (cid:104)O i ( x, y ) (cid:105) b UHP = A bi (2 y ) ∆ , A bi ∼ c . (4.41)Consistency with the large- c factorization in the bulk then implies there is a large family of“multi-trace” operators of the schematic form O m with expectation values (cid:104)O m (cid:105) b UHP ∼ c m .When calculating the twist correlation function, this tower of operators must be resummedinto a new semiclassical block, just as with the gravitational Virasoro descendants. For aBCFT, the form of the semiclassical block is theory-dependent and hence non-universal.Thus, in the bulk channel vacuum-block dominance is not required by the theory. Wemust choose to restrict to those boundary states without semiclassical expectation valueswhere non-universal contributions can be ignored. The constraint (4.39) involves both the spectrum of boundary operators in the n -copytheory and the BOE coefficients for twist operators in this theory. As we review in appendixB, both of these can be related to the spectrum and OPE data for boundary operators inthe single-copy BCFT; we can make use of these relations to convert the constraint (4.39)to a direct statement about the single-copy BCFT.In particular, consider the case of n = 2, where the branched geometry (includinga regulator boundary for the twist operator as above) is conformal to the annulus. TheVirasoro primaries appearing in the the n = 2 twist BOE, analogously to the bulk CFTcase in [39, 41, 42], contain products of base primaries of the form O I = O i ⊗ O i , (4.42)For these operators, as we show in appendix B, the BOE coefficients are¯ B b Φ I ¯ B b ¯Φ I = 16 − i , (4.43)identical to the bulk case in [39, 41, 42] up to the non-standard normalization of the twistoperators induced by the boundary. Taking into account only these primaries, we have aconstraint from (4.34) that (cid:88) i L − i , (4.44) The same limitation holds for previous bulk CFT calculations. When light bulk operators have largeexpectation values that backreact on the geometry, the entanglement entropy of a region is no longeruniversal and is not determined by vacuum block dominance. – 26 –s sub-exponential in c , where the sum is over light boundary primary operators in theoriginal BCFT. This will be true if the number of light boundary primaries in the basetheory is also sub-exponential in c .Note that the BOE also contains primaries composed of products of descendants inthe base theory, such as O i ⊗ L − O i − h i +1 / h i L − O i ⊗ L − O i + L − O i ⊗ O i . (4.45)These are primaries with respect to the orbifold Virasoro generators: L m ⊗ ⊗ L m , (4.46)but are generated by even powers of the antisymmetric linear combinations L m ⊗ − ⊗ L m . (4.47)To estimate the number of such primaries, we use the Hardy-Ramanujan Formula [43],which gives an asymptotic estimate for the number of descendants (partitions p ( k )) at agiven level k : p ( k ) ∼ e π √ k/ . (4.48)If the density of light primaries for a single BCFT is sub-exponential, as above, including thecontribution of the extra primaries in the 2-copy BCFT not of the form (4.42) generates nonew contributions exponential in c . Thus, we do not get a substantially stronger constraintfrom their inclusion. We have seen that under the assumption of vacuum block dominance, the direct BCFTcalculation of entanglement entropy for an interval in the vacuum state of the theory on ahalf-line matches exactly with the gravity calculation using an effective bulk theory witha purely gravitational ETW brane.In this section, we show that essentially the same BCFT calculation allows us toreproduce the gravitational results (3.11) and (3.12) for the entanglement entropy of aninterval in a Euclidean time-evolved boundary state | b, τ (cid:105) , and the results (3.14) and(3.15) for the entanglement entropy of the auxiliary radiation system in BCFT model of atwo-sided black hole coupled to a radiation bath.From the CFT point of view, the three examples — half-space ( § § § , ¯Φat the boundaries of these blue regions, and calculate entanglement entropy from theircorrelator.In our first example, the entanglement entropy is calculated via the two-point functionof twist operators on the half-plane { z = x + iy : (cid:61) ( z ) = y > } . The result for a– 27 – ba d Figure 8 . Relation between Euclidean path integrals for the different BCFT setups in which weare calculating entanglement entropy. In (a), we have the half-space x >
0. In (b), we have thepath integral preparing the boundary state | b, τ (cid:105) . We decompactify from a circle to a line to obtainthe global quench geometry (c), which is equivalent (under a global conformal transformation) to(a). In (d), we have the thermofield double state of (a). holographic BCFT is (cid:104) ¯Φ n ( z )Φ n ( z ) (cid:105) b UHP = min (cid:40) (cid:104) | b (cid:105) n − (cid:12)(cid:12)(cid:12)(cid:12) y y (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) − d n , (cid:12)(cid:12)(cid:12) z (cid:15) (cid:12)(cid:12)(cid:12) − d n (cid:41) , (4.49)for twist scaling dimension d n . The calculation in § w = f ( z ) = 1 z − i/ − i. (4.50)This maps the boundary of the half-plane to the circle in Figure 8d. Since twists areprimary CFT operators (by definition), the correlator in the thermofield double geometryis related to the half-plane correlator by (cid:104) ¯Φ n ( w )Φ n ( w ) (cid:105) b TFD = | w (cid:48) ( z ) w (cid:48) ( z ) | − d n (cid:104) ¯Φ n ( z )Φ n ( z ) (cid:105) b UHP . (4.51)For a symmetric interval with w , = ± x + it , (4.49) and (4.51) agree with the holographiccalculation in the n → | b, τ (cid:105) in § τ is a trivial compaci-fication of the covering space in which we “unwind” the angular coordinate. At leadingorder in large c , the entanglement entropy for an interval of fixed length on the circle isthen independent of the circle’s size. The limit of an infinite circle is the line. This gives the global quench geometry, asin Figure 8c. For small τ , this is dual to a planar black hole. Unlike the cylinder geom-etry, the global quench is conformally equivalent to the half-plane, under the coordinatetransformation κ = 2 τ π log z . (4.52)The correlator of twists is then (cid:104) ¯Φ n ( κ )Φ n ( κ ) (cid:105) b GQ = | κ (cid:48) ( z ) κ (cid:48) ( z ) | − d n (cid:104) ¯Φ n ( z )Φ n ( z ) (cid:105) b UHP . (4.53) This is the phenomenon of large- N volume independence. See, e.g., [44]. – 28 –nce again, combining this with (4.49) gives the same result as the HRT formula [5]. The generalization of our BCFT results from a single interval to multiple intervals closelyparallels the generalization of the CFT result from two intervals to multiple intervals [10].We start with the holographic calculation, and then discuss how to obtain the results fromthe monodromy method in the BCFT.
Consider a collection of k disjoint intervals A = (cid:116) i A i , A i = [ x i − , x i ], in the vacuumstate of a BCFT on the half-space x ≥
0, with an associated minimal surface X A . A giventopology for X A geodesically (and without intersection) pairs each endpoint x i to either(a) another endpoint x j , or (b) the brane. Morally, we can view the latter as pairing x i toan image point x ∗ i placed on a mirror image of the bulk theory across the brane.Thus, the possible topologies top( X A ) can equally be described by symmetric geodesicpairings of 2 k intervals, of which there are (cid:0) kk (cid:1) . Assuming that the gravity dual theoryis described via a purely gravitational ETW brane, so that the local geometry is pure AdS,the two types of geodesics have (regulated) lengths (cid:96) ij G N = c (cid:16) x ij (cid:15) (cid:17) , (cid:96) mm ∗ G N = c (cid:18) x m (cid:15) (cid:19) + log g b . Hence, the holographic result is S A = min top( X A ) G N (cid:88) ( ij ) (cid:96) ij + (cid:88) ( mm ∗ ) (cid:96) mm ∗ = min top( X A ) (cid:88) ( ij ) c (cid:16) x ij (cid:15) (cid:17) + (cid:88) ( mm ∗ ) c (cid:18) x k (cid:15) (cid:19) + log g b , (5.1)where ( ij ) denotes paired endpoints in the half-space and ( mm ∗ ) image-paired endpoints.As a concrete example, take the interval A = [ x , x ]. The explicit expression forholographic entanglement entropy is then S A = min (cid:26) c (cid:18) x − x (cid:15) (cid:19) , c (cid:18) x x (cid:15) (cid:19) + 2 log g b (cid:27) = min (cid:110) S conn A , S disc A (cid:111) , recovering our results from Section 3.2.1. The calculation is similar in other vacuum ge-ometries. We can also include a boundary-centred interval [0 , x ], which forces at least oneimage-paired geodesic. Arbitrary non-intersecting geodesic pairings of n intervals are counted by Catalan numbers (cid:0) k (cid:1) / ( k +1).Each yields k +1 symmetric pairings on 2 k intervals, giving our result. We thank Chris Waddell for discussionof this point. When A = [0 , x ] (cid:116) A (cid:116) · · · (cid:116) A k − , X A has (cid:0) k +1 k (cid:1) possible topologies. This can established by similarcombinatorics to the non-boundary case. – 29 – .2 BCFT calculation for multiple intervals To calculate the entanglement entropy of A = (cid:116) A i in the BCFT on a half-space, we cansimply calculate a correlator of k twist and anti-twist operators on the Euclidean UHP andanalytically continue. We will therefore focus on the UHP calculation. As above, we canuse kinematic doubling to write the correlator as (cid:42) k (cid:89) i =1 Φ n ( z i − , ¯ z i − ) ¯Φ n ( z i , ¯ z i ) (cid:43) b UHP = (cid:42) k (cid:89) i =1 Φ n ( z i − ) ¯Φ n (¯ z i − ) ¯Φ n ( z i )Φ n (¯ z i ) (cid:43) . (5.2)As in the single interval case, we have some choice about the order in which we performbulk OPE or BOE expansions of the twist correlator. We can regard this sequence of choicesas a fusion channel E , analogous to the s- and t-channels in the single interval case. A givenfusion channel has a natural expansion in terms of a set of cross-ratios, (cid:126)η , and higher-pointconformal blocks: (cid:42) k (cid:89) i =1 Φ n ( z i − ) ¯Φ n (¯ z i − ) ¯Φ n ( z i )Φ n (¯ z i ) (cid:43) b UHP = N ( (cid:126)η ) (cid:88) (cid:126)h,(cid:126) ∆ C E ,(cid:126)h,(cid:126) ∆ e − nc f ( (cid:126)h,d n / nc,(cid:126)η ) , (5.3)where we have taken the semiclassical limit, and C E ,(cid:126)h,(cid:126) ∆ is a product of OPE and BOEcoefficients depending on the internal weights (cid:126)h, (cid:126) ∆. Here N ( (cid:126)η ) is just a standard prefactor.Having related the UHP correlator to a chiral correlator, the higher-point blocks can beobtained from the standard monodromy method. We briefly summarize this method here,following [10]. (We discuss the method in slightly more detail in Appendix C.) Readersfamiliar with the monodromy method may freely jump ahead to (5.10).The monodromy method begins with a powerful trick: instead of the desired 2 k -pointfunction, consider a (2 k + 1)-point function, where we have added an additional operator, χ (1 , ( z ), which is taken to be a null descendant of a primary operator θ ( z ). The nulloperator must decouple and the correlator must vanish. The vanishing of the correlatoris expressed as the differential equation (writing χ ( z ) as a differential operator acting on θ ( z )) Θ (cid:48)(cid:48) ( z ) + T ( z )Θ( z ) = 0 , (5.4)where Θ( z ) is the correlatorΘ( z ) = (cid:42) θ ( z ) (cid:89) i Φ n ( z i − ) ¯Φ n (¯ z i − ) ¯Φ n ( z i )Φ n (¯ z i ) (cid:43) , (5.5)and T ( z ) is T ( z ) = (cid:88) i (cid:26) h n /c ( z − z i ) + 6 h n /c ( z − ¯ z i ) + ∂ z i z − z i + ∂ ¯ z i z − ¯ z i (cid:27) . (5.6) Strictly speaking, this operator is only guaranteed to exist in Liouville theory. However, as the block isa kinematic object, we expect the form not to depend on whether this operator exists in our theory or not. – 30 –or a given channel E , in an appropriate limit of the cross ratios η → η E , we expect thisto be dominated by the exchange of the lightest possible operator, generally the identityand its descendants. We thus make the ansatz that the correlator is given byΘ( z ) ≈ ψ ( z | z i , ¯ z i ) e − nc f E , (5.7)to leading order in c . Here f E is the semiclassical vacuum block for the original 2 k -pointfunction and ψ ( z | z i , ¯ z i ) is thought of as a ‘wavefunction’ for the inserted operator. In thiscase, we can rewrite T ( z ) T ( z ) = (cid:88) i (cid:26) h n /c ( z − z i ) + 6 h n /c ( z − ¯ z i ) − c i z − z i − ¯ c i z − ¯ z i (cid:27) , (5.8)where the c i are accessory parameters : c i = ∂f E ∂z i , ¯ c i = ∂f E ∂ ¯ z i = c i . (5.9)If we know the accessory parameters, we can integrate (5.9) to find the block f E . Todetermine these parameters, the monodromy method then uses the fact that a solution ofthe differential equation must have monodromies around any set of points that is consistentwith the corresponding operator being exchanged in the block. This constraint can be usedto fix the accessory parameters. In general, this cannot be done analytically. However, itis possible to find the parameters explicitly for twist operators in the n → n = 1, one finds f E = (cid:88) ( ij ) α log | z ij | + (cid:88) ( mm ∗ ) α log z mm ∗ + O ( α ) . (5.10)Here, the channel E pairs some twists to anti-twists on the same half-plane, and some twiststo their images on the opposite half-plane. We have denoted the pairs by ( ij ) and ( mm ∗ )respectively, and note that (as expected from the CFT case [10]) channels biject with thetopologies of Section 5.1, so we can view E ∈ top( X A ). We illustrate the correspondencebetween channels, trivial cycles, and the bulk RT surfaces in Figure 9.To calculate the entanglement entropy, we also need to compute C E . This is easilydone, since the OPE coefficients for vacuum exchange are always unity, while the BOEalways gives the one-point function of twists (4.11). If there are M image pairs ( mm ∗ ), wehave C E = [ g − nb ] M = g − αMb . (5.11)We can recover the factors of (cid:15) from the one-point functions (4.8) and (4.11). From (5.7),the entanglement entropy in the limit η → η E is then S A = lim α → (cid:18) c α f E − α log C E (cid:19) = (cid:88) ( ij ) c (cid:18) | z ij | (cid:15) (cid:19) + (cid:88) ( mm ∗ ) c (cid:16) z mm ∗ (cid:15) (cid:17) + log g b . (5.12)– 31 – igure 9 . Left.
Vacuum exchange in two different channels for k = 3 twists on the UHP. Trivialcycles cut through identities. Middle.
The monodromy cycles to be trivialized in the doubledpicture of the BCFT.
Right.
The corresponding RT topologies in the bulk with an ETW brane.
The corrections to (C.4) are in α and not in z ij . It follows that in finite regions around η E , expression (5.12) is the full entanglement entropy to leading order in c .If we make the assumption of vacuum block dominance as in Section 4.3, we canupgrade (5.12) to precisely reproduce (5.1): S A = min top( X A ) (cid:88) ( ij ) c (cid:18) | z ij | (cid:15) (cid:19) + (cid:88) ( mm ∗ ) c (cid:16) z mm ∗ (cid:15) (cid:17) + log g b . (5.13)This follows because vacuum dominance in a channel E implies the vacuum contributionis larger in other channels. Thus, we have a derivation of the full RT formula in a BCFTdual to AdS with an ETW brane, to the same level of generality as the CFT case [10]. In this paper, we have seen how phase transitions in holographic BCFT entanglemententropies, originally understood using the HRT formula, can be understood directly inthe BCFT via the exchange of dominance between bulk and boundary channels in thetwo-point function of twist operators. The twist operator correlation function calculatesthe R´enyi entropies as the partition function for the BCFT on a replica manifold. Ourcalculation indicates that the R´enyi entropies themselves, or the replica partition functions,also have a phase transition. It is interesting to understand the gravitational origin ofthese transitions. Here, we use the fact that the CFT partition function on the replicamanifold should be equal to the partition function of the gravity theory in which thespacetime geoemtries are constrained to be asymptotically AdS, with boundary geometryequal to the replica manifold. Note that the parameter values at which this happens depends on the replica index. – 32 –n [6], inspired by [12, 13], it was argued that the transitions in the gravitational pathintegral arise because we can have various topologies for the ETW brane whose boundary isthe disconnected set of n CFT boundaries, where n is the replica index. In the saddle-pointapproximation (which gives the leading contribution to the partition function in the 1 /c expansion), the logarithm of the CFT partition function is equal to the gravitational actionfor the bulk configuration with least action. The transition occurs when the gravitationconfiguration with least action changes as the parameters specifying the interval are varied.In this section, we will confirm these expectations in an explicit example where thegeometries can be understood in detail. We consider the R´enyi entropy for a single interval[ x , x ] in the vacuum state of a BCFT on a half-line x >
0. We assume that the BCFTand the chosen boundary condition correspond to a dual gravitational theory that has aneffective three-dimensional description as gravity with a negative cosmological constant andan gravitational ETW brane with zero tension. In this simple case, the dual geometries thatwe need can be obtained very simply from the geometries that contribute to the calculationof the R´enyi entropies for a pair of intervals [ − x , − x ] ∪ [ x , x ] in the vacuum state ofour parent CFT on a real line. These geometries have a Z symmetry corresponding tothe transformation x → − x in the CFT that exchanges the intervals.It follows from symmetry that the bulk surface fixed by this transformation has zeroextrinsic curvature. Thus, if we consider a new geometry defined as half of the previousgeometry, with a zero-tension ETW brane replacing the Z symmetric surface, the new ge-ometry will satisfy the gravitational equations in the bulk and at the ETW brane. The newgeometry is thus a valid saddle-point geometry for the BCFT R´enyi entropy calculation.It is plausible that all of the geometries we need can be obtained in this way.Happily, the geometries for the two-interval calculation have already been discussedin detail by Faulkner in [11]. There, it was understood that the phase transition in R´enyientropies indeed arises from a transition in the lowest-action gravitational solution. Wecan check that under this transition, the hypersurface fixed by the Z symmetry changestopology, becoming connected when the intervals are close to each other. We illustrate thisfor the case of the second Renyi entropy in Figure 10. Thus, the ETW brane in the gravityversion of the BCFT calculation takes on a replica wormhole geometry in the gravitationalsaddle that computes the entanglement entropy of an interval close to the boundary.There is an alternative intuitive picture for the transition in the ETW-brane topologythat leads to the phase transition in R´enyi entropies. Consider again the two-intervalcalculation, but now reinterpret the x coordinate as a Euclidean time coordinate τ . In thiscase, the CFT path-integral on the τ < n CFTs. This is illustrated in Figure11. The Z symmetric surface in the dual gravitational geometry is the τ = 0 surface thatgives the initial data for the time-symmetric Lorentzian geometry dual to this state.In the limit where our interval moves toward τ = −∞ , the path integral will give thevacuum state of n CFTs, so the dual geometry is n copies of AdS and the Z symmetric To see this, recall that the extrinsic curvature can be defined as the Lie derivative of the hypersurfacemetric with respect to a normal vector n to the hypersurface. This switches sign under n → − n , but theextrinsic curvature is invariant under this operation because of the symmetry. – 33 – X XX X XXX
Figure 10 . CFT and bulk depictions of the path integral for computing the second R´enyi entropyof two intervals. We indicate a Z identification, whose fixed-point locus is drawn as a red line,that relates this to a BCFT whose boundary is the red line. (a) When the twist operators are nearthe Z -line (in red), the corresponding bulk solution will fill in the cycle indicated by the greendashed line. The Z -line and the bulk contractible cycle intersect each other. (b) When the twistoperators are far from the Z -line, the bulk contractible cycle (green) is now homologous to the Z -line on each sheet. (c) The corresponding bulk solution when the twist operators are near the Z -line. The bulk Z -surface connects the two Z -lines on each sheet. (d) The bulk solution whenthe twist operators are far from the Z -line. There is now a disconnected bulk Z -surface for each Z -line on the boundary. τ=0τ Figure 11 . Entangled state of n CFTs produced by inserting twist and anti-twist operators intothe Euclidean path integral for the vacuum state. As the branch cut is moved toward τ = 0, theentanglement between the CFTs becomes large, and the τ = 0 spatial slice of the dual geometrybecomes a connected multi-boundary wormhole. In the original picture, this surface is the ETWbrane geometry in the gravitational calculation of R´enyi entropy for an interval close to the BCFTboundary. surface has n disconnected components. On the other hand, as the interval moves closerto τ = 0, the path integral produces a state with more and more entanglement between– 34 –he n CFTs.
At some point, we have a phase transition similar to the Hawking-Pagetransition, where the τ = 0 spatial slice in the dual geometry becomes connected andthe Lorentzian geometry dual to our state is a multi-boundary wormhole. In our BCFTapplication, this τ = 0 spatial slice becomes the ETW brane geometry, so we see thatthe Euclidean wormholes in the replica calculation can be directly related to the usualappearance of wormholes in the gravity dual of highly-entangled states of holographicsystems. Starting with the vacuum state of a 1+1-dimensional CFT, the geometry of a putative bulkdual is fixed by symmetry to be AdS , up to internal dimensions. We can also fix AdS usingthe RT formula: it is the unique bulk geometry whose minimal surfaces correctly reproducethe universal result for the entanglement entropy of a single interval. The RT formula makes non-universal predictions for two or more intervals, so we can go in the other direction anddetermine the class of holographic CFTs which reproduce these non-universal gravitationalresults. As shown in [10], vacuum block dominance guarantees that the twist-antitwistcorrelators used to calculate entanglement entropy agree with the holographic value forany number of intervals. Vacuum dominance places explicit constraints on the spectrumand OPE coefficients of a holographic CFT.The logic for a CFT with boundary is similar. Symmetry, or the universal resultfor the entanglement entropy of a boundary-centred interval, restricts us to a class ofSO(1 , cut off by an ETWbrane with purely gravitational couplings [3], though we emphasize this is not the mostgeneral bulk dual consistent with ground-state symmetry.In this paper, we have taken the next step of transforming non-universal gravitationalpredictions from these geometries with a purely gravitational ETW brane into a constrainton holographic BCFTs. To match the holographic predictions for a non-centred interval, orindeed any number of intervals, vacuum dominance in both the BCFT bulk and boundarychannels is necessary and sufficient. From a kinematic perspective, this follows from thedoubling trick and the remaining copy of the Virasoro algebra. But the implications forthe BCFT spectrum and OPE coefficients are more subtle. We have made some precisestatements above, but expect there is more juice to be squeezed from this particular lemon.For instance, it might be possible to finesse the spectral constraints along the lines of[45], though the CFT machinery required is potentially quite different. It would also beinteresting to investigate the additional constraints that arise from assuming not only thatthe BCFT calculations reproduce the leading O ( c ) entropies, but also that the subleading For example, in the n = 2 case, we get a path integral similar to the one that produces the thermofielddouble state, and moving the interval closer to τ = 0 corresponds to increasing the temperature. In the limit where the interval hits τ = 0, we have a state where the left half of CFT k is connected tothe right half of CFT k +1 , so the entanglement of the system formed from the left and right half of CFT k is infinite. – 35 –orrections to the entropies are order c as we expect from a conventional gravitationaltheory with a semiclassical expansion.We have argued that the expression (3.8) for the small η entanglement entropy isuniversal in holographic CFTs (assuming that the RT surface is disconnected in someinterval [0 , η d ] as in the simple model), so the constraints associated with vacuum blockdominance for an interval around η = 0 should be expected to hold much more generally, forany holographic BCFT with a disconnected RT surface phase at small η . It seems plausiblethat any holographic BCFT should have such a phase, though it would be interesting tofind a direct argument.Our results have several interesting consequences and applications. First, they putthe AdS/BCFT proposal of [3] on firmer microscopic footing, exhibiting explicit conditionson a BCFT under which a locally AdS geometry with a purely gravitational ETW branecaptures the microscopic ground-state entanglement entropies. The gravity calculationsallow the RT surface to end on an ETW brane, so our results also confirm this aspect ofTakayanagi’s proposal. Our work also has direct applications to the physics of black holes. For black holesdual to CFT states prepared by a Euclidean path integral on the cylinder with conformallyinvariant boundary conditions, the phase transition in entanglement entropy for a non-centred interval leads to a period of Lorentzian time where boundary observers with accessto suitably large boundary regions can see behind the horizon [5].Similarly, for the thermofield-double state of a BCFT on a half-line, the phase tran-sition in entanglement entropy corresponds to a transition in bulk entanglement wedge toinclude part of the black hole interior [6]. Treating this as a model of a black hole in equilib-rium with its Hawking radiation, after this transition part of the interior is reconstructablefrom the radiation. The brane is therefore playing a similar role to the “quantum extremalislands” which restore unitarity of the Page curve [7–9]. We have argued above that thephase transition in entanglement entropy can be directly related to topological changes inbulk replica wormholes, and it is of obvious interest to explore this connection further.
Acknowledgments
We would like to thank Tarek Anous, Thomas Hartman, Eliot Hijano, Alex May, DominikNeuenfeld, and Chris Waddell for useful discussions. DW is supported by an InternationalDoctoral Fellowship from the University of British Columbia. MVR is supported by theSimons Foundation via the It From Qubit Collaboration and a Simons Investigator Award.This work is supported in part by the Natural Sciences and Engineering Research Councilof Canada. From the lower-dimensional perspective, this is a modification of the usual homology condition, thoughno modification is required if we take the higher-dimensional perspective that the ETW brane represents asmooth part of the full bulk geometry. – 36 –
BCFT two-point functions from Virasoro conformal blocks.
In this appendix, we briefly review the structure of four-point functions of chiral operatorsand their expansion in terms of Virasoro conformal blocks, and then argue that the sameobjects form the building blocks of two-point functions in boundary conformal field theories.
Chiral four-point functions and conformal blocks
In a 2D CFT, for operators φ i with chiral dimensions h i , the global conformal symmetryimplies that the four-point function takes the form (cid:104) φ ( z ) φ ( z ) φ ( z ) φ ( z ) (cid:105) = (cid:18) z z (cid:19) h − h (cid:18) z z (cid:19) h − h η h + h ( z ) h + h ( z ) h + h F ( η ) (A.1)where z ij = z i − z j and F is some function of the single cross-ratio η = z z / ( z z ).We can define F as F ( η ) = lim z ∞ →∞ ( − h + h + h + h z h ∞ (cid:104) φ (0) φ ( η ) φ (1) φ ( z ∞ ) (cid:105) . (A.2)We can express F in terms of the OPE data for the CFT and a standard set of functionsby expanding the products φ ( z ) φ ( z ) and φ ( z ) φ ( z ) using (2.16). In this case, thefour point function reduces to a sum of two-point functions of intermediate operators, F ( η ) = (cid:88) i C i C i F ( c, h ; [ h , h , h , h ] | η ) , (A.3)The conformal blocks F ( c, h ; [ h , h , h , h ] | η ) are specific functions which depend only onthe central charge, the dimensions h i of the external operators, and the “internal” dimen-sion h . These give the contribution to the four-point function from a primary operator ofweight h and all of its Virasoro descendants. The block has a simple behavior in the limit η →
0, where we have F ( c, h ; [ h , h , h , h ] | η → ∼ η h − h − h (A.4) BCFT two-point function
We now consider the two-point function of bulk operators in a BCFT defined on the upper-half-plane (cid:104)O ( z , ¯ z ) O ( z , ¯ z ) (cid:105) b UHP . (A.5)As we discussed in section 2, this has the kinematics of a chiral four-point function. Wewill show this somewhat more carefully here, and see that we can expand the two-pointfunction in either a bulk channel or a boundary channel in terms of the chiral conformalblocks defined above. – 37 – ne-point functions for scalar Virasoro descendants To begin, it will be useful to compute one-point functions for scalar global primaries thatare themselves Virasoro descendants. Consider, in particular, a Virasoro primary operator O h, ¯ h ( z, ¯ z ) and state | h, ¯ h (cid:105) for the CFT on S associated to it by the state-operator corre-spondence. We denote an operator O α,βh, ¯ h (0) which creates the Virasoro descendants of thisstate V hα ¯ V ¯ hβ | h, ¯ h (cid:105) . Here, V hα and V ¯ hβ are polynomials in L − n and ¯ L − n respectively chosenso that these states give an orthonormal basis of the Verma module: (cid:104)O α (cid:48) ,β (cid:48) h, ¯ h ( ∞ ) O α,βh, ¯ h (0) (cid:105) = (cid:104) h, ¯ h | V h, † α (cid:48) ¯ V ¯ h, † β (cid:48) V hα ¯ V ¯ hβ | h, ¯ h (cid:105) = δ αα (cid:48) δ ββ (cid:48) . (A.6)We can re-express the same operators in terms of local operators O α i ,β i h, ¯ h ( z i , ¯ z i ) at arbitrarypoints z , z by the use of a global conformal transformation that maps ( ∞ ,
0) to ( z , z ).We then have (cid:104)O α ,β h, ¯ h ( z , ¯ z ) O α ,β h, ¯ h ( z , ¯ z ) (cid:105) = δ α α δ β β . (A.7)Note that the form of each local descendant operator depends explicitly on both points z , z , and not just implicitly on one point through the local primary. These operators areonly orthogonal precisely at the points z , z (and form an orthogonal basis of operators inthe ‘North-South’ quantization between these two points).Next, we require a somewhat more refined version of the doubling trick. We have seenthat a correlator (cid:104)O h ¯ h ( z , ¯ z ) · · · O h n ¯ h n ( z n , ¯ z n ) (cid:105) b UHP (A.8)of bulk CFT operators O h k ¯ h k with conformal weights ( h k , ¯ h k ) is constrained to have thesame functional form as chiral CFT correlators (cid:104)O h ( z ) · · · O h n ( z n ) O ¯ h (¯ z ) · · · O ¯ h n (¯ z n ) (cid:105) (A.9)Similarly, a correlator of descendants (cid:104)O α ,β h ¯ h ( z , ¯ z ) · · · O α n ,β n h n ¯ h n ( z n , ¯ z n ) (cid:105) b UHP (A.10)takes the same functional form as (cid:104)O α h ( z ) · · · O α n h n ( z n ) O β ¯ h (¯ z ) · · · O β n ¯ h n (¯ z n ) (cid:105) . (A.11)Then, taking O h ( z ) to be a primary operator in some CFT such that (cid:104)O h,h ( z, ¯ z ) (cid:105) b UHP = A bh (cid:104)O h (¯ z ) O h ( z ) (cid:105) = A bh | z − ¯ z | h (A.12) Recall that the conjugation operation used to define the dual operator at infinity is an inversion inradial quantization, and so the operator is rescaled by the conformal transformation. We keep the rescalingimplicit. Note that it’s not necessary for such a CFT to exist, since we are only making statements aboutkinematics. – 38 –e have that (cid:104)O α,βh,h ( z, ¯ z ) (cid:105) b UHP = A bh (cid:104)O αh ( z ) O βh (¯ z ) (cid:105) = δ αβ A bh (A.13)where here the descendant indices are labeling the orthogonal basis of states for the pairof points z, ¯ z . Bulk channel expression for the two-point function
We can now derive a bulk-channel expression for the two-point function (A.5). First wewill use the bulk state-operator map (bulk OPE) to insert a complete set of bulk states (inthis ‘North-South’ quantization between z and ¯ z ) (cid:104)O ( z , ¯ z ) O ( z , ¯ z ) (cid:105) b UHP = (cid:88) i,α,β (cid:104)O ( z , ¯ z ) O ( z , ¯ z ) O α,βi (¯ z , z ) (cid:105)(cid:104)O α,βi ( z , ¯ z ) (cid:105) b UHP . (A.14)Using the form of the boundary one-point function (A.13), we can rewrite this as (cid:104)O ( z , ¯ z ) O ( z , ¯ z ) (cid:105) b UHP = (cid:88) { i | h i =¯ h i } ,α,β ˆ C i A bi (cid:104)O h ( z ) O h ( z ) O αh i (¯ z ) (cid:105)(cid:104)O αh i ( z ) O ¯ h (¯ z ) O ¯ h (¯ z ) (cid:105) , (A.15)where we have pulled out the dynamical information in the OPE coefficients and expec-tation values. The three-point functions, as written, are now purely kinematic, i.e. theyrepresent the functional dependence of such a three-point function where the overall coef-ficient is taken to be one. Each sum over Virasoro descendants now can be seen to givea standard chiral Virasoro conformal block F ( c, h ; [ h , h , ¯ h , ¯ h ] | z ), so that the two-pointfunction can be expanded in this bulk channel as (cid:104)O ( z , ¯ z ) O ( z , ¯ z ) (cid:105) b UHP = (cid:18) z ∗ z ∗ (cid:19) h − h (cid:18) z ∗ z ∗ (cid:19) ¯ h − ¯ h z h + h ( z ) h + h ( z ∗ ∗ ) ¯ h +¯ h × (cid:88) i ˆ C i A bi F ( c, h i ; [ h , h , ¯ h , ¯ h ] | z ) , (A.16)and where we have written the conformal block in terms of the cross-ratio z = z z ∗ ∗ z ∗ z ∗ . (A.17) Boundary channel expression for the two-point function
We can similarly expand the two-point function in the boundary channel . Here we insert acomplete set of states corresponding to the expansion of the bulk operators in terms of theboundary operator expansion. The boundary state-operator mapping gives a complete setof states in terms of boundary operators which appear in representations of the surviving– 39 –iagonal Virasoro symmetry. We thus insert a complete set of orthonormal states of theform (cid:104)O ( z , ¯ z ) O ( z , ¯ z ) (cid:105) b UHP = (cid:88) I,α (cid:104)O ( z , ¯ z ) ˜ V ˆ h I α | ˆ h I (cid:105) b UHP (cid:104) ˆ h I | ˜ V ˆ h I † α O ( z , ¯ z ) (cid:105) b UHP . (A.18)Using the doubling trick to account for the representation of bulk operators under theboundary Virasoro operators, we can rewrite this as (cid:104)O ( z , ¯ z ) O ( z , ¯ z ) (cid:105) b UHP = (cid:88) I,α B b h B b h (cid:104)O h ( z ) O ¯ h ( z ∗ ) V ˆ h I α | ˆ h I (cid:105)(cid:104) ˆ h I | V ˆ h I † α O h ( z ) O ¯ h ( z ∗ ) (cid:105) , (A.19)where we have pulled out the dynamical information in the coefficients. The remainingthree-point functions, as written, are now purely kinematic. Again we recognize thatthis sum over Virasoro descendants is the standard bulk chiral Virasoro conformal block F ( c, h ; [ h , ¯ h , h , ¯ h ] | η ), giving (cid:104)O ( z , ¯ z ) O ( z , ¯ z ) (cid:105) b UHP = (cid:18) z ∗ z (cid:19) h − ¯ h (cid:18) z z ∗ (cid:19) ¯ h − h η h +¯ h ( z ∗ ) h +¯ h ( z ∗ ) h +¯ h × (cid:88) I B b I B b I F ( c, h ; [ h , ¯ h , h , ¯ h ] | η ) (A.20)where we have used the cross-ratio η = 1 − z . (A.21) B Boundary operator expansion for twist operators
In this section, we relate the boundary operator expansion of the twist operator Φ n inan n -copy BCFT to n -point functions of boundary operators in the original BCFT. Ourdiscussion here is directly parallel to the discussion in Section 4 of [39] on contributions tothe OPE coefficients of CFT twist operators.Via radial quantization, a twist operator inserted at z into an n -copy BCFT can beunderstood to give rise to some entangled state of this n -copy BCFT on an interval. Bythe state-operator correspondence, the same state can be obtained by the insertion of someoperator at the origin. A basis of boundary operators for the n -copy BCFT may be writtenas O I ⊗ · · · ⊗ O I n , where O I are a basis of boundary operators in the original BCFT. Thus,we can write thatΦ n ( x + iy ) = (cid:88) { I k } | y | d n − (cid:80) k ∆ Ik B Φ n I ··· I n O I ⊗ · · · ⊗ O I n ( x ) . (B.1)When the operators O I are primary, the coefficient B Φ n I ··· I n can be defined according to(2.13) via the bulk-boundary two-point function as B Φ n I ··· I n = 2 d n − (cid:80) k ∆ Ik (cid:104) Φ n ( z = i ) O I ⊗ · · · ⊗ O I n (0) (cid:105) = 2 d n − (cid:80) k ∆ Ik (cid:104) Φ n ( i ) (cid:105) (cid:104) Φ n ( i ) O I ⊗ · · · ⊗ O I n (0) (cid:105)(cid:104) Φ n ( i ) (cid:105) – 40 – 2 − (cid:80) k ∆ Ik (cid:15) d n g − nb (cid:104) Φ n ( i ) O I ⊗ · · · ⊗ O I n (0) (cid:105)(cid:104) Φ n ( i ) (cid:105) . To compute the ratio of correlators in the last line, consider the conformal transformation w ( z ) = i (cid:18) ( z + i ) n + ( z − i ) n ( z + i ) n − ( z − i ) n (cid:19) . (B.2)This takes the UHP to the n -sheeted UHP associated with the insertion of our twistoperator. The points x k ≡ cot (cid:18) π k − n (cid:19) k = 1 , . . . , n (B.3)map to the origin on the various sheets. By this conformal transformation, we have that (cid:104) Φ n ( i ) O I ⊗ · · · ⊗ O I n (0) (cid:105)(cid:104) Φ n ( i ) (cid:105) = (cid:89) k (cid:18) d w d z ( x k ) (cid:19) − ∆ Ik (cid:42)(cid:89) k O I k ( x k ) (cid:43) (B.4)For the points x k where w ( z ) = 0, we have thatd w d z ( x k ) = nx k + 1 = n sin (cid:18) π k − n (cid:19) (B.5)Combining everything, we have that B Φ n I ··· I n = 2 − (cid:80) k ∆ Ik (cid:15) d n g − nb (cid:89) k (cid:20) ] n sin (cid:18) π k − n (cid:19)(cid:21) − ∆ Ik (cid:42)(cid:89) k O I k ( x k ) (cid:43) . (B.6)It is useful to note that the explicit dependence on c appears as a universal prefactor, B Φ n I ··· I n = (cid:15) d n g − nb ¯ B Φ n I ··· I n . (B.7)For n = 2, we see that the correlator vanishes unless I = I , and we have that¯ B Φ II = 116 ∆ I . (B.8)For n = 3, we have that ¯ B Φ IJK = C IJK (∆ I +∆ J +∆ K ) , (B.9)where we have used the standard result for a CFT three-point function. C Monodromy method
Here we continue the discussion of accessory parameters from (5.9) in the main text to givea more complete description of the calculation of the semiclassical blocks.There are only 2 k − , R ) invari-ance imposes three (real) constraints. Explicitly, these constraints are (cid:88) i (cid:60) ( c i ) = (cid:88) i (cid:60) (cid:18) c i z i − h n c (cid:19) = (cid:88) i (cid:60) (cid:18) c i z i − h n z i c (cid:19) = 0 , – 41 –he real part of the usual SL(2 , C ) constraints.If we know the accessory parameters, we can integrate to find the block f E . To deter-mine these parameters, we transport a pair of solutions (cid:126) Θ( z ) = [Θ + ( z ) , Θ − ( z )] T around apoint z c where an OPE or BOE is to be performed. The null decoupling equation (5.4)applied to the three-point function implies that the 2 × M performingthe transport, (cid:126) Θ( z ) (cid:55)→ M (cid:126) Θ( z ), gives tr M = − π Λ c ) , Λ c = (cid:114) − h n c . (C.1)The number of independent monodromies to tune equals the number of internal primaries,2 k − so we have the right number of monodromy constraints to fix our accessoryparameters c i .In general, we cannot analytically solve for the accessory parameters. Luckily, however,it is possible to find them explicitly for twist operators in the n → α = ( n − /
12. Entanglement entropy is obtained from R´enyi entropies in the limit α →
0, and since h n = c ( n + 1) α/ n = cα → z i , ¯ z i . As a result, the equation (5.4) decouples into a sum ofindependent monodromy equations, depending on which cycles the channel E trivializes.To illustrate, suppose E involves a pairing between twists Φ n ( z i ) and ¯Φ n ( z j ). We mustchoose the accessory parameters to make the monodromy around z i , z j trivial. Since thisdecouples from the other problems as α →
0, we can simply focus on the contribution T ij ( z ) = 6 h n c (cid:20) z − z i ) + 1( z − z j ) (cid:21) − c i z − z i − c j z − z j + c.c.= 6 α (cid:20) z − z i ) + 1( z − z j ) − z j ( z − z i ) (cid:21) − c i z − z i + c i z i z j ( z − z j ) + c.c. , (C.2)where “c.c” stands for complex conjugate terms, and in (C.2), we used the constraint (cid:60) ( c i z i + c j z j ) = 6 α . To obtain a trivial monodromy around z i , z j (and the image cycleenclosing ¯ z i , ¯ z j ), it is sufficient for T ij ( z ) to be regular at infinity. This is equivalent to thesum of residues at simple poles vanishing, and hence c i + ¯ c i = 12 α | z ij | + O ( α ) , (C.3)where O ( α ) corrections arise because the equations only strictly decouple for α = 0. Thecalculation is analogous for a twist paired with its image, but the contribution T mm ∗ ( z )involves only two insertions at z m and ¯ z m . To see this, we suppose the leading term in Θ( z ) ∼ ( z − z c ) κ . Plugging this into (5.4), we find that κ ( κ −
1) = − h n /c , with two solutions κ ± . These pick up factors e πiκ ± after traversing a loop z = z c + (cid:15)e iθ ,leading to (C.1). See [10] for details. An exchange channel E is a cubic tree with 2 k leaves and 2 k − E = 4 k −
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