Bootstrapping boundary-localized interactions
Connor Behan, Lorenzo Di Pietro, Edoardo Lauria, Balt C. van Rees
BBootstrapping boundary-localized interactions
Connor Behan , Lorenzo Di Pietro , , Edoardo Lauria and Balt C. van Rees Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe ObservatoryQuarter, Woodstock Road, Oxford, OX2 6GG, UK Dipartimento di Fisica, Università di Trieste, Strada Costiera 11, I-34151 Trieste, Italy INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy CPHT, CNRS, Institut Polytechnique de Paris, France [email protected], [email protected],[email protected], [email protected]
Abstract
We study conformal boundary conditions for the theory of a single real scalar to investigatewhether the known Dirichlet and Neumann conditions are the only possibilities. For this freebulk theory there are strong restrictions on the possible boundary dynamics. In particular,we find that the bulk-to-boundary operator expansion of the bulk field involves at mosta ‘shadow pair’ of boundary fields, irrespective of the conformal boundary condition. Wenumerically analyze the four-point crossing equations for this shadow pair in the case of athree-dimensional boundary (so a four-dimensional scalar field) and find that large rangesof parameter space are excluded. However a ‘kink’ in the numerical bounds obeys all ourconsistency checks and might be an indication of a new conformal boundary condition. a r X i v : . [ h e p - t h ] S e p ontents d/ d systems 26 A.1 bOPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40A.2 Boundary OPE and physical OPE coefficients . . . . . . . . . . . . . . . . . . . . 41
B Three-point function conformal blocks 42
B.1 Blocks in the boundary channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42B.2 Scalar blocks in the bulk channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
C OPE relations and bulk-to-boundary crossing 44
C.1 Derivation of the OPE relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45C.2 Matching with the bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47C.3 Displacement Ward identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
D Crossing equations in a vectorial form 50 Introduction and summary
The classification of conformal boundary conditions for a bulk CFT is a difficult problem. Besideswell-known results for rational boundary conditions in rational CFTs (reviewed in [1]), very littleis known even for relatively simple theories. It is natural to ask whether a systematic approachis feasible – one which does not rely on explicit constructions but leverages instead the modernconformal bootstrap methods [2] (see [3] for a review and [4] for a first application to BCFTwhich relied on results from [5, 6]). A promising methodology is to start from theories that areas simple as possible in the bulk. In this work we pursue precisely such a direction in the casewhere the bulk theory is a single real free scalar field.In any spacetime dimension a free scalar can certainly have Dirichlet or Neumann conformalboundary conditions. The question we try to answer here is whether more general conformalboundary conditions are possible, for example by coupling the bulk scalar to new boundary de-grees of freedom and flowing to the infrared. These putative boundary conditions should modifythe behavior of the scalar near the boundary and produce non-trivial boundary correlators, anal-ogous to those of an interacting one lower-dimensional CFT. We find numerical evidence for atleast one such ‘exotic’ boundary condition in four dimensions, and more generally very strongconstraints on the space of potential conformal boundary conditions. In exploring consistent boundary conditions for a free scalar theory we obtained a very specialset of ‘shadow-related’ crossing symmetry equations, as follows. First of all, the (cid:3) φ = 0 equationof motion implies that the bulk-boundary expansion of the bulk field φ is restricted to contain atmost two operators that we denote as (cid:98) O and (cid:98) O ; their dimensions are ∆ φ and ∆ φ +1 , respectively.At most one of these two operators can vanish, and if so then we are in the Dirichlet or Neumanncase and the two operators are immediately recognizable as the restriction of φ or ∂ ⊥ φ to theboundary. If they are both non-vanishing then the operators can be thought of as a ‘shadowpair’ in the sense of Ferrara, Gatto, Grillo and Parisi [11–14]. Their dimensions match thisobservation since φ + 1 = d − , the dimension of the boundary, but the picture extends totheir three-point functions: for a generic third defect operator (cid:98) O with dimension (cid:98) ∆ and spin l we find the relations (cid:16) l + (cid:98) ∆2 (cid:17) Γ (cid:16) d + l − (cid:98) ∆ − (cid:17) ˆ f (cid:98) O ( l ) = − b /b (cid:16) d + l − (cid:98) ∆ − (cid:17) Γ (cid:16) l + (cid:98) ∆+12 (cid:17) ˆ f (cid:98) O ( l ) , (cid:16) l + (cid:98) ∆2 (cid:17) Γ (cid:16) d + l − (cid:98) ∆2 (cid:17) ˆ f (cid:98) O ( l ) = − b /b Γ (cid:16) d + l − (cid:98) ∆ − (cid:17) Γ (cid:16) l + (cid:98) ∆ − (cid:17) ˆ f (cid:98) O ( l ) , (1.1) The existence of such strong constraints is remarkable. In some cases, for example Maxwell theory in fourdimensions, it is known that the space of conformal boundary conditions is vast, since it includes the space of allCFTs with a U (1) symmetry in three dimensions [7–10]. b b = (cid:115) d − a φ ( d −
2) (1 − d − a φ ) (1.2)and a φ is the one-point function of the operator φ in the presence of the boundary. This relationbetween OPE coefficients agrees with the result obtained from applying a shadow transformationto the relevant three-point functions. In section 2 we derive this equation by demanding theabsence of unphysical singularities in a three-point functions involving two bulk operators φ . Note that the relations as written are still valid when the dimension of (cid:98) O is such that the gammafunctions have poles; this is precisely when the operators are of ‘double-twist’ type and theshadow transformation is singular.The properties of the previous paragraph already lead to a remarkable bootstrap problem. In-deed, up to the special ‘double-twist’ operators there is one spectrum and set of OPE coefficientsthat needs to solve the five different crossing symmetry equations corresponding to the possiblefour-point functions of (cid:98) O and (cid:98) O . This is intriguing in itself, but in the numerical analysis wecan actually impose three more constraints. The first one is related to the Ward identity for thedisplacement operator. The second one is that of locality of the BCFT setup, which translatesto the absence of any vector operators of dimension d in the (cid:98) O × (cid:98) O OPE. Both of these aredescribed in section 2.2.3. The third one is imposed to separate local three-dimensional CFTs,which do not interest us here, from boundary conditions: this requires the scaling dimension ofthe first spin 2 operator to be strictly greater than 3.We have numerically explored the system of crossing equations originating from the (cid:104) (cid:105) , (cid:104) (cid:105) and (cid:104) (cid:105) four-point functions in four bulk dimensions subject to all the above conditions.It might be tempting to conjecture that no non-trivial conformal boundary conditions exist thatmeet such stringent criteria, but surprisingly this is not quite what we find. On the one hand,there does exist a large range of possible values of a φ where the first spin 2 operator musthave a dimension less than about 3.1. Since this value is likely to decrease even further whenincreasing computational power, it is natural to conjecture that it must converge to 3, andthen no conformal boundary conditions would be possible in this range. On the other hand,for a φ near its Neumann value we suddenly find room for interesting physics: as indicated in In [15] the same analysis was carried out for defects with a higher co-dimension in the free scalar theory,leading to similar shadow relations and a proof of triviality in many cases. For non-integer dimensions this setupcan also be used to describe the long-range Ising model, where the relations can be found from the non-localequation of motion [16]. More details can be found in [17, 18] and a first numerical analysis in this context wasdone in [19]. The corresponding four-point functions should be related by the integral transformation that implements theshadow symmetry. However it is not clear to us whether the conformal block decompositions of such shadow-transformed four-point functions are automatically consistent. For example, the integral transformation is sensi-tive to contact terms and it seems unlikely that it can be swapped with the sum over conformal blocks. (cid:98) O and (cid:98) O , we organize them in a way that takes advantage of the exact relations,and we explain the approximations of the resulting ‘superblocks’ that we use in our numericalimplementation. In section 5 we present the numerical results in the case of d = 4 , showing plotsthat involve several different characteristic observables, and in particular we show the kink thatwe mentioned above. We finally discuss possible future directions in section 6. A summary ofthe conventions and various technical results that we used along the way are relegated to theappendices. Consider a free massless scalar field φ in d > dimensions with a planar boundary. We use thecoordinate y ≥ for the direction orthogonal to the boundary, and (cid:126)x for the directions parallelto the boundary. We denote the components of x = ( (cid:126)x, y ) ∈ R d − × R + as x µ , µ = 1 , . . . , d with x d = y , and those of (cid:126)x ∈ R d − as x a , a = 1 , . . . , d − . We are interested in unitary boundaryconditions that preserve the boundary conformal symmetry SO ( d, . In this section we discuss the bulk-to-boundary OPE (bOPE) of the scalar field for a genericconformal boundary condition, and the constraints imposed by crossing symmetry on the bulktwo-point function of the scalar field. 4 .1.1 Bulk-boundary two-point functions
In this section we review the existence of two operators with protected scaling dimension in thebOPE of the free scalar field [4, 22–25]. In a BCFT the bulk-boundary two-point function ofa scalar bulk operator O of scaling dimension ∆ O and a scalar boundary operator (cid:98) O of scalingdimension (cid:98) ∆ (cid:98) O is [5, 6, 4] (cid:104) O ( (cid:126)x, y ) (cid:98) O (0) (cid:105) = b O (cid:98) O y ∆ O − (cid:98) ∆ (cid:98) O ( | (cid:126)x | + y ) (cid:98) ∆ (cid:98) O . (2.1)The bOPE coefficient b O (cid:98) O , which is real for Hermitian operators, is not determined by symmetry.Specializing O to be a free scalar φ of scaling dimension ∆ φ = d − , the equation of motion (cid:3) φ = 0 gives (cid:104) (cid:3) φ ( (cid:126)x, y ) (cid:98) O (0) (cid:105) = (cid:18) d − (cid:98) ∆ (cid:98) O (cid:19) (cid:18) d − − (cid:98) ∆ (cid:98) O (cid:19) b φ (cid:98) O y d − (cid:98) ∆ (cid:98) O +1 ( | (cid:126)x | + y ) (cid:98) ∆ (cid:98) O . (2.2)Therefore the possible scaling dimensions for boundary primaries with b φ ˆ O (cid:54) = 0 are (cid:98) ∆ = d − , (cid:98) ∆ = d . (2.3)Without loss of generality, we can assume there is at most one boundary operator of dimension (cid:98) ∆ with b φ ˆ O (cid:54) = 0 , that we denote as (cid:98) O , and similarly for (cid:98) ∆ , the corresponding operator beingdenoted as (cid:98) O . As observed in [25], the scaling dimensions of these operators add up to d − ,which suggests that the two operators might be thought of as a ‘shadow pair’. In the nextsubsection we will show that also their three-point functions are compatible with such a ‘shadowrelation’. The bOPE of the free scalar is [5, 6, 4] φ ( (cid:126)x, y ) = b C d − [ y, (cid:126) ∇ ] (cid:98) O ( (cid:126)x ) + b y C d [ y, (cid:126) ∇ ] (cid:98) O ( (cid:126)x ) . (2.4)where we defined b i ≡ b φ (cid:98) O i , i = 1 , . The explicit form of the differential operator C (cid:98) ∆ i [ y, (cid:126) ∇ ] isgiven in appendix A. The bOPE can be used to reconstruct bulk correlators starting from theboundary ones. Next, we consider the two-point function (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:105) . (2.5)This correlator is not completely fixed by the symmetry as it depends on the cross-ratio ξ ≡ ( x − x ) y y ≡ | (cid:126)x − (cid:126)x | + ( y − y ) y y . (2.6)5e can compute (2.5) by plugging twice the bOPE (2.4), using the diagonal and unit-normalizedboundary two-punt functions (cid:104) (cid:98) O i ( (cid:126)x ) (cid:98) O j ( (cid:126)x ) (cid:105) = δ ij | (cid:126)x − (cid:126)x | (cid:98) ∆ i , i, j = 1 , (2.7)and resumming the contributions from the descendants. The resulting boundary channel decom-position of (2.5) is (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:105) = 1( y y ) d − (cid:20) b (cid:0) ξ − d/ + ( ξ + 4) − d/ (cid:1) + b d − (cid:0) ξ − d/ − ( ξ + 4) − d/ (cid:1)(cid:21) . (2.8)An alternative way of computing the two-point function (2.5) is to invoke the bulk OPE φ × φ ,namely φ ( x ) φ (0) = ( x ) d/ − + φ (0) + c φφT C T x µ x ν T µν (0) + (cid:88) (cid:96) =4 , ,... c φφ(cid:96) C J (cid:96) x µ . . . x µ (cid:96) J µ ...µ (cid:96) (cid:96) (0) + . . . (2.9)where T µν is the stress tensor, and the operators J µ ...µ (cid:96) (cid:96) with (cid:96) ≥ are the tower of higher-spinconserved currents present in the free scalar CFT. The OPE data involving the stress tensor are[26] c φφT = − d ( d − d − S d , C T = d ( d − S d , S d ≡ Vol ( S d − ) = 2 π d/ Γ (cid:0) d (cid:1) . (2.10)Plugging in (2.5), we write the two-point function as a sum over bulk one-point functions andtheir derivatives. Boundary conformal invariance allows only for scalar bulk one-point functions[5, 6, 4], hence from the φ × φ bulk OPE the only non-trivial contributions are due to (cid:104) φ ( (cid:126)x, y ) (cid:105) = a φ y d − , (cid:104) (cid:105) = a = 1 . (2.11)Resumming the contribution from bulk descendants we obtain the bulk channel decompositionof (2.5): (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:105) = ξ − d ( y y ) d − (cid:104) a φ d − ξ d − ( ξ + 4) − d (cid:105) . (2.12)Equating the two different decompositions (2.8) and (2.12) gives the bulk-to-boundary crossing equation [4]. Since everything else in the equation is fixed, the only dynamical data are the one-point function coefficient a φ on the l.h.s. and the bulk-to-boundary couplings ( b , b ) on ther.h.s. The solution is [4, 24] b = 1 + 2 d − a φ , b = ( d −
2) (1 − d − a φ ) . (2.13)6his result tells us that in any boundary condition for a free scalar the parameter a φ is con-strained by unitarity to lie in an interval − d − = a ( D ) φ ≤ a φ ≤ a ( N ) φ = 12 d − . (2.14)As we indicated above, the boundaries of the interval correspond to the Dirichlet ( b = 0 ) andNeumann ( b = 0 ) boundary condition. These elementary boundary conditions will be discussedin detail in section 3, but in the remainder of this section we will assume that b b (cid:54) = 0 becausewe would like to explore the possibility of more exotic boundary conditions. In this section we consider three-point functions with two insertions of the free scalar φ and ageneric boundary operator (cid:98) O . Note that, by Lorentz invariance, these correlators can be non-vanishing only if the third operator transforms as a symmetric and traceless tensor of SO ( d − .Without loss of generality we can place the boundary operator at infinity and consider (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:98) O ( l ) ( θ, ∞ ) (cid:105) . (2.15)Following the standard procedure [27], in the above expression we contracted the tensor indiceswith a boundary polarization vector θ a as follows (cid:98) O ( l ) ( θ, ∞ ) ≡ θ a . . . θ a l (cid:98) O a ,...,a l ( ∞ ) , θ · θ = 0 . (2.16)We will show that the boundary channel expansion of this correlation function exhibits unphysicalsingularities, which can be removed only if special conditions are met. Therefore these conditionshave to be satisfied in any conformal boundary condition of the free scalar. The bOPE (2.4) allows to completely determine the correlator (2.15) in terms of the three-pointfunctions between the operators (cid:98) O i , i = 1 , , and (cid:98) O ( l ) . Conformal invariance fixes the latterthree-point functions to take the form [26, 27] (cid:104) (cid:98) O i ( (cid:126)x ) (cid:98) O j ( (cid:126)x ) (cid:98) O ( l ) ( θ, ∞ ) (cid:105) = ˆ f ij (cid:98) O ( l ) | (cid:126)x | (cid:98) ∆ i + (cid:98) ∆ j − (cid:98) ∆ P ( l ) (cid:107) (ˆ x , θ ) , (2.17)where (cid:98) ∆ denotes the scaling dimension of the operator (cid:98) O ( l ) which carries SO ( d − spin l . Thedependence on the polarization vector is through the following polynomial P ( l ) (cid:107) (ˆ x , θ ) ≡ ( − ˆ x · θ ) l , ˆ x a ≡ x a | (cid:126)x | . (2.18)7y Bose symmetry ˆ f ij (cid:98) O ( l ) = ( − l ˆ f ji (cid:98) O ( l ) , (2.19)which implies that only even spins l are allowed in (2.17) if i = j .To compute (2.15), we act twice with the bOPE on the boundary three-point functions (2.17).After some algebra to resum the contributions from the descendants, we obtain the followingboundary channel expansion (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:98) O ( l ) ( θ, ∞ ) (cid:105) = P ( l ) (cid:107) (ˆ x , θ ) | (cid:126)x | d − − (cid:98) ∆ × (cid:16) b ˆ f (cid:98) O ( l ) ˆ F (cid:98) ∆ ,l ( w + , w − ) + b b ˆ f (cid:98) O ( l ) ˆ F (cid:98) ∆ ,l ( w + , w − ) + b ˆ f (cid:98) O ( l ) ˆ F (cid:98) ∆ ,l ( w + , w − ) (cid:17) . (2.20)This expression depends on two cross-ratios w ± , which we take as follows: w ± = ( y ± y ) | (cid:126)x | . (2.21)The functions ˆ F ij (cid:98) ∆ ,l ( w + , w − ) are computed in Appendix B.1 and their explicit expressions aregiven in (B.7). In the next section we will study the analyticity properties of the correlator(2.20). Next, we study the same three-point function using the bulk φ × φ OPE. Since the only singularterm in this OPE is given by the identity operator, which does not contribute to the three-pointfunction, we conclude that the three-point function must be free of singularities when the twobulk points coincide. In terms of the cross-ratios w ± , this limit corresponds to w + → ∞ withany fixed w − .As we show in the appendix C, for generic values of their parameters the boundary blocks onthe r.h.s of (2.20) become singular in this limit. These unphysical singularities can be removedif the OPE coefficients are related in the following way ˆ f (cid:98) O ( l ) = κ ( (cid:98) ∆ , l ) ˆ f (cid:98) O ( l ) , κ ( (cid:98) ∆ , l ) ≡ − b Γ (cid:16) l + (cid:98) ∆2 (cid:17) Γ (cid:16) d + l − (cid:98) ∆ − (cid:17) b Γ (cid:16) d + l − (cid:98) ∆ − (cid:17) Γ (cid:16) l + (cid:98) ∆+12 (cid:17) , ˆ f (cid:98) O ( l ) = κ ( (cid:98) ∆ , l ) ˆ f (cid:98) O ( l ) , κ ( (cid:98) ∆ , l ) ≡ − b Γ (cid:16) l + (cid:98) ∆2 (cid:17) Γ (cid:16) d + l − (cid:98) ∆2 (cid:17) b Γ (cid:16) d + l − (cid:98) ∆ − (cid:17) Γ (cid:16) l + (cid:98) ∆ − (cid:17) . (2.22)For certain special values of the parameters ( (cid:98) ∆ , l ) some of the blocks on the r.h.s of (2.20) arethemselves regular. These special values correspond to the poles of the gamma functions in (2.22)and read (see also table 1) 8 (cid:98) ∆ = d + l − . We have that κ ( (cid:98) ∆ , l ) → ∞ while κ ( (cid:98) ∆ , l ) remains finite. This sets ˆ f (cid:98) O ( l ) = ˆ f (cid:98) O ( l ) = 0 , while leaving ˆ f (cid:98) O ( l ) unconstrained. We denote these operators as [ (cid:98) O (cid:98) O ] ,l . • (cid:98) ∆ = d + l +2 n − with n a positive integer. We have κ , ( (cid:98) ∆ , l ) → ∞ and so ˆ f (cid:98) O ( l ) = 0 while ˆ f (cid:98) O ( l ) , ˆ f (cid:98) O ( l ) remain unconstrained. Given that generically they appear in both OPEs, wecould denote these operators both as [ (cid:98) O (cid:98) O ] n,l or [ (cid:98) O (cid:98) O ] n − ,l . For definiteness, we chooseto denote them as [ (cid:98) O (cid:98) O ] n,l . • (cid:98) ∆ = d + l + 2 n − and n ∈ N . We have κ , ( (cid:98) ∆ , l ) = 0 , which sets ˆ f (cid:98) O ( l ) = ˆ f (cid:98) O ( l ) = 0 while leaving ˆ f (cid:98) O ( l ) unconstrained. We denote these operators as [ (cid:98) O (cid:98) O ] n,l . Importantly,all odd-spin operators in (cid:98) O × (cid:98) O are of this type, as can be seen by combining (2.22) withBose symmetry. (cid:98) ∆ − l conditions independent OPE coeff operator d − b b κ ( (cid:98) ∆ , l ) = ∞ ˆ f (cid:98) O ( l ) [ (cid:98) O (cid:98) O ] ,l d + 2 n − , n > b b κ ( (cid:98) ∆ , l ) , b b κ ( (cid:98) ∆ , l ) = ∞ ˆ f (cid:98) O ( l ) , ˆ f (cid:98) O ( l ) [ (cid:98) O (cid:98) O ] n,l d + 2 n − b b κ ( (cid:98) ∆ , l ) , b b κ ( (cid:98) ∆ , l ) = 0 ˆ f (cid:98) O ( l ) [ (cid:98) O (cid:98) O ] n,l Table 1: Table of special multiplets and their selection rules. Recall that b = 1 + 2 d − a φ and b = ( d −
2) (1 − d − a φ ) . The special cases listed above are related to the higher-spin symmetry of the bulk theory, aswe will now explain. We recall that the φ × φ OPE (2.9) contains infinitely many higher-spinconserved currents J (cid:96) , with even spin (cid:96) ≥ and scaling dimensions ∆ (cid:96) = d + (cid:96) − . Theconservation of these currents is generically violated by terms localized on the boundary, leadingto the following Ward identities (cid:104) ∂ µ J µµ ...µ (cid:96) − (cid:96) ( (cid:126)x, y ) . . . (cid:105) = δ ( y ) (cid:104) (cid:98) O µ ...µ (cid:96) − (cid:96) ( (cid:126)x ) . . . (cid:105) . (2.23)In this formula any subset among the (cid:96) − symmetric traceless indices { µ . . . µ (cid:96) − } can betaken to be parallel to the boundary, with the remaining indices being orthogonal, i.e. in the y direction. Therefore, the BCFT generically contains boundary operators D ( l ) (cid:96) and V ( l +1) (cid:96) of spin l and l + 1 , respectively, and protected dimensions (cid:98) ∆ = d + (cid:96) − , where l is an even integer rangingfrom to (cid:96) − . By ‘generically’ we mean that some of these operators might actually be absent9rom the spectrum in special cases. The equations (2.23) can be equivalently rephrased in termsof the bOPE, namely the operators D ( l ) (cid:96) , V ( l +1) (cid:96) have the correct dimensions and spins to appearin the bOPE of the bulk higher-spin current J (cid:96) in a way that is compatible with its conservationin the bulk. Furthermore, spin selection rules and bulk conservation imply that V ( l +1) (cid:96) cannotappear in the bOPE of any J (cid:96) (cid:48) with (cid:96) (cid:48) (cid:54) = (cid:96) , while the only other bulk current besides J (cid:96) that cancontain D ( l ) (cid:96) in its bOPE is J l .The relation to the special cases of (2.22) now stems from the observation that when (cid:96) − l = 2 n ,with n non-negative integer, D ( l ) (cid:96) has the right dimension to be the special operator [ (cid:98) O (cid:98) O ] n,l intable 1. Similarly, when (cid:96) − l = 2 n + 1 with n ∈ N , V ( l +1) (cid:96) has the right dimension to be thespecial operator [ (cid:98) O (cid:98) O ] n,l +1 . We show in general in the section C.2 of appendix C, and for thespecial case (cid:96) = 2 in the next subsection, that in fact whenever the operator D ( l ) (cid:96) is present in thebOPE of J (cid:96) , then it must appear in at least one of either the OPE of (cid:98) O with itself or the OPEof (cid:98) O with itself. Similarly, whenever V ( l +1) (cid:96) is present in the bOPE of J (cid:96) , it must also appear inthe OPE of (cid:98) O with (cid:98) O . The case (cid:96) = 2 deserves special attention because it corresponds to the bulk stress tensor T µν .Then the scalar operator D (0)2 ≡ D is the so-called displacement operator, and we will refer tothe spin 1 operator V (1)2 ≡ V (1) as the flux operator . Their general importance stems from theconservation of momentum P µ along a time coordinate chosen parallel to the boundary. If wesplit x µ = ( τ, (cid:126)z, y ) then ddτ P µ ( τ ) = (cid:90) d d − (cid:126)z (cid:90) ∞ dy ∂ t T tµ ( τ, (cid:126)z, y )= (cid:90) d d − (cid:126)z T yµ ( τ, (cid:126)z, y → , (2.24)where in the second equality we used the conservation to trade the time derivative with a spatialone, and then rewrote the integral of the spatial derivative as a boundary term. Choosing µ = y orthogonal to the boundary we find that the limit y → gives the displacement operator D,which therefore measures the breaking of translations orthogonal to the boundary and mustbe non-zero for any sensible boundary condition. Choosing µ parallel to the boundary, on theother hand, we find the vector operator V (1) and so we conclude that it measures the flux ofenergy into the boundary. Theories with a non-trivial flux operator V (1) (cid:54) = 0 may still have aconserved boundary-translation charge, if there is an additional boundary contribution to thecharge P a tot = P a + (cid:98) P a satisfying ddτ (cid:98) P a ( τ ) = − (cid:90) d d − (cid:126)z V (1) a ( τ, (cid:126)z ) . (2.25)10owever the flux operator must be absent in any local unitary BCFT setup. To see why, notethat the locality condition on the boundary is that (cid:98) P a , if non-trivial, should be expressible asthe integral (cid:98) P a ( τ ) = (cid:90) d d − (cid:126)z (cid:98) t ta ( τ, (cid:126)z ) , (2.26)of a local boundary operator with two indices (cid:98) t ba . The condition (2.25) locally takes the form ∂ b (cid:98) t ba = − V (1) a . (2.27)Moreover, by repeating the argument for the other generators of the conformal group on theboundary, one can easily show that the operator (cid:98) t ba has spin 2, i.e. it is symmetric and traceless.Recalling that V (1) a has scaling dimension d and therefore (cid:98) t ba has scaling dimension d − , wesee that eq. (2.27) with V (1) (cid:54) = 0 is incompatible with the unitarity bound of a spin 2 operatorin d − dimensions. We conclude that indeed in any unitary BCFT locality implies thatV (1) = 0 (2.28)in which case (cid:98) P a is trivial. In practice this means that if we couple a bulk CFT (not necessarilyour free scalar theory) to some local boundary degrees of freedom, perhaps triggering an RGflow to a new conformal boundary condition, then the flux operator must never appear. This isbecause local boundary degrees of freedom should not be able to hold a macroscopic amount ofenergy.It might be instructive to consider some non-local setups that do feature a flux operator. Thefirst is a conformal interface , where there is an entire new CFT living on the half space with y < . In that case the stress tensor for each side ‘sees’ a flux operator, but if the interface setupis local then these two flux operators are in fact the same operator and the interface cannot actas a simultaneous energy sink for both sides. Such a setup can be generalized to the case whereour d − dimensional boundary is at the same time a conformal defect in some d (cid:48) -dimensionalauxiliary space in which it is coupled to an arbitrary d (cid:48) -dimensional bulk CFT, perhaps eventriggering a boundary/defect RG flow to some new conformal configuration. According to thegeneral structure of the operator expansion near a defect of dimension d − in a d (cid:48) -dimensionalCFT, the d (cid:48) -dimensional stress tensor can always provide a vector operator of precisely therequisite dimension (which is d ) and unless the two sides decouple we will observe this as a flux It is curious that the statement of locality, which in the bulk is encoded by the presence of a stress tensor,corresponds to the absence of a specific vector operator (in the bOPE of T µν ) in the BCFT setup. This simply follows from requiring conservation of the d (cid:48) -dimensional stress tensor in the allowed bulk-defectcorrelators with a vector. For a generic defect CFT these two-point functions were classified in [28]. d -dimensional BCFT. Lastly one could also try to create a non-local setup byadding a GFF on the boundary and coupling it, perhaps with other degrees of freedom, to thebulk field. But this scenario is captured by the previous one, because GFFs are just regular localfields in an auxiliary higher-dimensional space (albeit with non-integral d (cid:48) ).The previous discussion applied to any BCFT, but for the free scalar theory there are a fewadditional results that we can derive. To this end we return to the (cid:104) φφ (cid:98) O (cid:105) three-point function ofequation (2.20) and take the third operator to be either the flux operator V (1) or the displacementoperator D. Let us thererefore first consider: (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) V (1) ( θ, ∞ ) (cid:105) . (2.29)The relations (2.22) in this case give that ˆ f V (1) = ˆ f V (1) = 0 and leave ˆ f V (1) undetermined.On the other hand, the bulk channel decomposition of (2.29) receives contribution only from thebulk stress tensor T µν , because – as we explained in the previous subsection for the general case– spin selection rules and bulk conservation imply that that V (1) cannot appear in the bOPE ofany other operator in the OPE (2.9). We find (cid:104) φ ( x ) φ ( x ) V (1) ( θ, ∞ ) (cid:105) = c φφT C T (cid:20) x µ x ν (cid:104) T µν ( x ) V (1) ( θ, ∞ ) (cid:105) + . . . (cid:21) , (2.30)where the ellipses represent contributions of bulk descendants, and the OPE data of the stresstensor are given in (2.10). The two-point function on the r.h.s. is completely fixed by theboundary conformal symmetry up to a single bOPE coefficient b T V (1) [6, 4]. We can furtherrelate b T V (1) to the two-point function coefficient (cid:104) V (1) ( θ , (cid:126)x ) V (1) ( θ , (cid:105) = C V (1) ( θ · I ( (cid:126)x ) · θ ) | (cid:126)x | d , I ab ( (cid:126)x ) ≡ δ ab − x a x b | (cid:126)x | . (2.31)From the exact correlator (2.20) we find (details in appendix C.2) ˆ f V (1) = − S d ( d − b b b T V (1) = − S d ( d − b b C V (1) (2.32)and we conclude that the flux operator appears in the (cid:98) O × (cid:98) O OPE if it also appears in thebOPE of the stress tensor. Next we consider the three-point function involving the displacement operator (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) D ( ∞ ) (cid:105) . (2.33) Note that a similar statement applies for (cid:96) > , namely if the operator V ( (cid:96) − (cid:96) in the bOPE of J (cid:96) vanishesthen the BCFT admits a conserved charge of spin (cid:96) , which is given by a bulk spatial integral of the higher-spinconserved current. These conclusions again have a natural generalization to all the odd-spin protected boundary operators V ( l ) (cid:96) defined in (2.23), see appendix C.2 for more details.
12n this case, the relations (2.22) give that ˆ f D = 0 while leaving ˆ f D and ˆ f D undetermined.Using again the general argument from the previous subsection about spin selection rules andconservation, we have that the only operators in the OPE (2.9) that contribute to the bulkchannel decomposition of (2.33) are φ and the bulk stress tensor. Therefore we have (cid:104) φ ( x ) φ ( x ) D ( ∞ ) (cid:105) = (cid:20) (cid:104) φ ( x ) D ( ∞ ) (cid:105) + . . . (cid:21) + c φφT C T (cid:20) x µ x ν (cid:104) T µν ( x ) D ( ∞ ) (cid:105) + . . . (cid:21) , (2.34)where again the ellipses denote contributions of bulk descendants. Using the Ward identities forthe displacement operator [6], the bulk-boundary two-point functions in the r.h.s. are determinedin terms of the parameter a φ in (2.14), as well as the coefficient C D in the two-point function ofthe displacement operator (cid:104) D ( (cid:126)x ) D (0) (cid:105) = C D | (cid:126)x | d . (2.35)Comparing with the boundary channel correlator (2.20), after some algebra which we relegate toappendix C.3, we find ˆ f D = ( d − (cid:0) a φ d + 2 C D S d (cid:1) d − S d b , ˆ f D = ( d − (cid:0) C D S d − a φ d (cid:1) S d b . (2.36)The unitarity requirement C D ≥ implies that: ˆ f D ≥ ( d − d d − S d a φ b , ˆ f D ≥ − ( d − d S d a φ b . (2.37) φ Another interesting special case of (2.20) arises when the boundary operator is one of the bound-ary modes of φ , i.e. (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:98) O ( ∞ ) (cid:105) , (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:98) O ( ∞ ) (cid:105) . (2.38)On general grounds, due to Bose symmetry, there are four independent boundary OPE coefficientsthat enter these correlators: ˆ f , ˆ f , ˆ f , ˆ f . The latter are further related to each other, bymeans of the three independent constraints provided by regularity of the φ × φ OPE (2.22): ˆ f = − b b Γ (cid:0) d (cid:1) Γ (cid:0) d − (cid:1) ˆ f , ˆ f = 14 b b ( d − d −
4) ˆ f . ˆ f = − b b ( d − d −
4) Γ (cid:0) d (cid:1) Γ (cid:0) d − (cid:1) ˆ f . (2.39)Hence, the bulk three-point function (cid:104) φφφ (cid:105) is completely controlled by a single boundary OPEcoefficient, e.g. ˆ f . The latter can be non-zero only if the boundary condition breaks the Z global symmetry φ → − φ , under which both (cid:98) O i are odd.13 Examples
In this section we explore some examples of conformal boundary conditions for a free scalar. Westart by reviewing the free boundary conditions, i.e. Neumann and Dirichlet, and then constructexamples of interacting boundary conditions using conformal perturbation theory around the freeones. As we will see, these constructions rely on some ad-hoc assumptions on the spectrum ofan additional local 3d sector living on the boundary, which we couple to the bulk, and thereforethey do not prove rigorously the existence of interacting boundary conditions. On the otherhand, they will provide useful benchmarks to compare to our numerical results in section 5.
Suppose the theory is fully described by the free bulk action S = (cid:90) d d − (cid:126)x (cid:90) ∞ dy
12 ( ∂ µ φ ) , (3.1)without any boundary-localized interaction. In order to have a stationary action, besides thebulk equation of motion (cid:3) φ = 0 we need to set to zero the boundary term δS = − (cid:90) d d − (cid:126)x δφ ∂ y φ | y =0 = 0 . (3.2)The two solutions to this condition that preserve boundary conformal invariance areNeumann: ∂ y φ | y =0 = 0 , or (3.3)Dirichlet: φ | y =0 = 0 . We can rephrase these conditions in terms of the bOPE of the scalar field. Namely, in (2.4) wehave b = 0 and φ | y =0 ∝ (cid:98) O in the case of Neumann boundary condition, and b = 0 and ∂ y φ | y =0 ∝ (cid:98) O in the case of Dirichlet boundary condition. In either case, there is only one boundary operatorin the bOPE of φ , and the full set of boundary correlators can be simply characterized as themean-field theory of this operator. This implies that all the correlation functions of these BCFTsare completely disconnected, i.e. they are computed by Wick contractions as products of two-point functions. For this reason we call these boundary conditions ‘free boundary conditions’.We can also consider additional free boundary conditions that are not Neumann and Dirichlet.Such a boundary condition is obtained requiring both (cid:98) O and (cid:98) O to appear in the bOPE (2.4), i.e. Note that using this canonical normalization of the action the operator φ has a different normalizationcompared to the one in equation (2.9), namely φ | (3.1) = (cid:114) Γ ( d − ) π d φ | (2.9) . We will specify which normalization weare using whenever important. b (cid:54) = 0 , and postulating that these operators are two decoupled generalized free fields. Howeverthis implies that there is a spin 1 operator of dimension 4 in the spectrum of the boundarytheory, namely the vector ‘double-trace’ operator in the OPE of (cid:98) O with (cid:98) O , schematically (cid:98) O ∂ a (cid:98) O − (cid:98) O ∂ a (cid:98) O . It is easy to check that this operator also appears in the bOPE of thebulk stress tensor, hence for these boundary conditions we have a non-vanishing flux operatorV (1) (cid:54) = 0 . Therefore, following the discussion in the previous section, these are non-local boundaryconditions. We conclude that the only local free boundary conditions are the familiar Neumannand Dirichlet boundary conditions reviewed above. In order to look for examples of interacting boundary conditions, a natural strategy is to couplethe bulk scalar to a CFT d − living on the boundary. We turn on some relevant interactionbetween the two sectors and then flow to the IR, hoping to reach a non-trivial BCFT fixed point.Concretely, we add to the free bulk action (3.1) a boundary action of the form S ∂ = S CFT d − + (cid:88) I g I (cid:90) y =0 d d − (cid:126)x (cid:98) σ I , (3.4)where (cid:98) σ I are some scalar composites made of φ | y =0 or ∂ y φ | y =0 , depending on whether we startwith Neumann or Dirichlet boundary condition, as well as of local operators of the CFT d − . Inorder to have perturbative control over the resulting RG flow, we will assume that the operators (cid:98) σ I have scaling dimensions (cid:98) ∆ I = d − − (cid:15) I , < (cid:15) I (cid:28) , (3.5)i.e. the deformations are weakly relevant. Then one can systematically expand observables ofthe BCFT at the putative IR fixed point as a series in (cid:15) I .We will further assume that the boundary degrees of freedom are local. Technically, thismeans that in the absence of bulk-boundary couplings, i.e. for g I = 0 , the spectrum of theCFT d − contains a stress tensor (cid:98) t ab , which is a conserved, spin 2 primary operator, of protecteddimension (cid:98) ∆ (cid:98) t = 3 . At the perturbative BCFT fixed point this operator gets a small anomalousdimension, which must be non-negative by unitarity, and actually strictly positive if the bulkand the boundary are not decoupled. We refer to this spin 2 operator at the interacting fixedpoint as ‘pseudo stress tensor’. In the next subsection we show how to compute the leading ordercontribution to this anomalous dimension for a rather generic interaction of the form (3.4), using This includes the case of ‘no boundary’, or more precisely the ‘trivial interface’, where the theory on thefull R d is re-interpreted as a BCFT. In that case a φ = 0 so according to (2.13) this corresponds to b = 1 and b = ( d − . a φ and C D , defined in eqns. (2.11) and(2.35), respectively.Typically, when computing (B)CFT observables in perturbation theory, one first computesthe corrections as a function of the coupling constants, and then plugs the value of the couplingconstants at the fixed point, obtained by solving for the zeroes of the beta functions. However, byrestricting to the case with a single bulk-boundary coupling, we can also avoid the computationof the beta function and simply assume that a perturbative fixed point exists. This is sufficientbecause we can consider ratios of the leading order corrections to the observables mentionedabove, in such a way that the coupling cancels out from the ratios. It would be interesting,but much more laborious, to actually compute the beta functions in terms of the data of theCFT d − . This would actually be necessary if one wants to verify the existence of the fixed point,or consider higher order corrections/multiple bulk-boundary couplings. The beta function neededin this setup starts at cubic order in the coupling, and the coefficient of the cubic term is givenby a regularized integral of the four-point function of the deformation, see e.g. [29, 17] and also[30] for the case of 1d CFTs. We now consider a slightly more specific bulk-to-boundary interaction, with a single coupling, ofthe form S ∂ = S CFT d − + g (cid:90) y =0 d d − (cid:126)x (cid:98) Ω (cid:98) χ . (3.6)In the expression above, (cid:98) χ denotes an operator in the CFT d − and (cid:98) Ω is any local boundaryoperator built out of φ | y =0 or ∂ y φ | y =0 , depending on whether we are perturbing a Neumann orDirichlet free boundary condition. The assumption (3.5) in this case takes the form (cid:98) ∆ (cid:98) Ω + (cid:98) ∆ (cid:98) χ = d − − (cid:15), < (cid:15) (cid:28) . (3.7)In the presence of the interaction (3.6) the conservation and the tracelessness of the stress tensor (cid:98) t ab of the CFT d − is violated as follows ∂ a (cid:98) t ab = g (cid:98) Ω ∂ b (cid:98) χ , (cid:98) t aa = g (cid:98) ∆ (cid:98) χ (cid:98) χ . (3.8) The computation of the beta functions for bulk-boundary couplings in terms of the data of the CFT d − was performed in [31] for some examples of perturbations around Dirichlet and Neumann. Some perturbativeconstructions of interacting boundary conditions for free theories can also be found in [32, 33, 25]. g ∝ (cid:15) , we have two seemingly problematic features in theabove equations, namely the divergence is not expressed in terms of a primary operator of theundeformed theory, and the operator does not have only a spin 2 component because the traceis non-zero. Both these issues are solved by defining the ‘corrected’ operator (cid:98) τ ab = (cid:98) t ab − g (cid:98) ∆ (cid:98) χ ( d − δ ab (cid:98) Ω (cid:98) χ . (3.9)Taking the divergence we then obtain ∂ a (cid:98) τ ab = gd − (cid:16) ( d − − (cid:98) ∆ (cid:98) χ ) (cid:98) Ω ∂ b (cid:98) χ − (cid:98) ∆ (cid:98) χ (cid:98) χ∂ b (cid:98) Ω (cid:17) . (3.10)The new operator (cid:98) τ ab is a symmetric traceless tensor, and its divergence (3.10) is a primary spin1 operator of the undeformed theory, making the recombination of the multiplets manifest. Notethat (3.10) is a manifestation in perturbation theory of the locality condition that we discussedin 2.2.3. If the boundary degrees of freedom were non-local they would not have the operator (cid:98) t ab and then the right hand side of (3.10) would be a primary operator of spin 1 and protecteddimension d (it is easily checked that indeed this operator would appear in the bulk-to-boundaryOPE of the bulk stress tensor).We can exploit the recombination to compute the leading order anomalous dimension of (cid:98) τ ab at the interacting fixed point. Let us consider computing the two-point function (cid:104) ∂ a (cid:98) τ ab ( (cid:126)x ) ∂ c (cid:98) τ cd (0) (cid:105) . (3.11)On the one hand, we can take derivatives of the two-point function of (cid:98) τ ab , which is fixed byboundary conformal invariance to be [26] (cid:104) (cid:98) τ ab ( (cid:126)x ) (cid:98) τ cd (0) (cid:105) = C (cid:98) τ ( g ) I ab,cd ( (cid:126)x ) | (cid:126)x | (cid:98) ∆ (cid:98) τ ( g ) ,I ab,cd ( (cid:126)x ) ≡
12 [ I ac ( (cid:126)x ) I bd ( (cid:126)x ) + I ad ( (cid:126)x ) I bc ( (cid:126)x )] − d − δ ab δ cd . (3.12)The definition of I ab was given in (2.18), and we introduced C (cid:98) τ ( g ) = C (0) (cid:98) τ + O ( g ) , (cid:98) ∆ (cid:98) τ ( g ) = d − (cid:98) γ (cid:98) τ ( g ) = d − (cid:98) γ (1) (cid:98) τ g + O ( g ) . This is the correct scaling with (cid:15) if the three-point function of the operator (cid:98) Ω vanishes for the free boundaryconditions, as in the examples we will consider below. One can also consider cases in which the three-pointfunction of (cid:98) Ω is non-vanishing, e.g. (cid:98) Ω = φ | y =0 for a perturbation of Neumann, in which case g ∝ (cid:15) at the fixedpoint. In any case the precise scaling does not affect any result in this section. C (0) (cid:98) τ is the ‘central charge’ of the CFT d − that the bulk scalar couples to, i.e.the coefficient appearing the two-point function of the stress tensor t ab before we turn on theinteraction. On the other hand we can compute (3.11) at the leading order in g by directly usingthe r.h.s. of (3.10). By comparing the two results, we find (cid:98) γ (1) (cid:98) τ = 2 (cid:98) ∆ (cid:98) χ ( d − − (cid:98) ∆ (cid:98) χ )( d + 1)( d − C (0) (cid:98) Ω C (0) (cid:98) χ C (0) (cid:98) τ , (3.13)where C (0) (cid:98) O denotes the coefficient of the two-point function of the boundary operator (cid:98) O computedat g = 0 . With the canonical normalization (3.1) of the bulk action we have C (0) φ = Γ (cid:0) d − (cid:1) π d , C (0) ∂ y φ = Γ (cid:0) d (cid:1) π d . (3.14)We note that the leading order anomalous dimension is essentially controlled by the centralcharge C (0) (cid:98) τ of the CFT d − . We now further specialize to the case in which the free boundary condition is Dirichlet, and theoperator (cid:98) Ω is ∂ y φ | y =0 , namely we take a deformation of the form S ( D ) ∂ = S CFT d − + g (cid:90) y =0 d d − (cid:126)x ∂ y φ (cid:98) χ . (3.15)The interaction term leads to the the following modified Dirichlet boundary condition φ | y =0 = − g (cid:98) χ . (3.16)In this case the condition (3.5) gives (cid:98) ∆ (cid:98) χ = d − − (cid:15) , with < (cid:15) (cid:28) . As we discussed above, weassume the existence of a perturbative fixed point with g ∝ (cid:15) . Plugging in eq. (3.13) we obtain (cid:98) γ (1) (cid:98) τ = Γ (cid:0) d + 1 (cid:1) π d ( d + 1) C (0) (cid:98) χ C (0) (cid:98) τ . (3.17)Let us now consider the leading order correction to the one-point function coefficient a φ ( g ) a φ ( g ) = − − d + δa φ ( g ) = − − d + δa (1) φ g + O ( g ) . (3.18) This can be obtained by varying the action (3.15), supplemented by the boundary term (cid:82) φ∂ y φ , with respectto ∂ y φ . δa (1) φ must be non-negative as a consequence of the unitarity bound (2.14). Tocompute its value, note that the modified Dirichlet boundary condition (3.16) determines thebOPE coefficient b to be b = − g (cid:115) π d Γ (cid:0) d − (cid:1) C (0) (cid:98) χ (cid:0) O ( g ) (cid:1) . (3.19)Plugging this result in the crossing relations (2.13), we find δa (1) φ = 2 − d π d C (0) (cid:98) χ Γ (cid:0) d − (cid:1) . (3.20)Having obtained two observables we can form a ratio that does not depend on the value of thecoupling at the putative fixed point, namely (cid:98) γ (cid:98) τ ( g ) δa φ ( g ) = (cid:98) γ (1) (cid:98) τ δa (1) φ + O ( g ) = 2 d − Γ (cid:0) d − (cid:1) Γ (cid:0) d + 1 (cid:1) π d ( d + 1) 1 C (0) (cid:98) τ + O ( g ) . (3.21)This quantity depends on the central charge C (0) (cid:98) τ of the CFT d − that the bulk scalar couples to.Next, we consider the leading order correction to the coefficient C D in the two-point functionof the displacement operator C D = C ( D ) D + δC D ( g ) = C ( D ) D + δC (1) D g + O ( g ) , (3.22)where C ( D ) D denotes the value at the free Dirichlet boundary condition. The displacement operatorin this theory is [5, 6] D = (cid:18)
12 ( ∂ y φ ) −
12 ( ∂ a φ ) + 14 d − d − ∂ a φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) y =0 . (3.23)Note that this formula makes sense even for an interacting boundary condition if we interpretthe composite operators on the right hand side as products of φ | y =0 and ∂ y φ | y =0 , made finiteby subtracting all the singular terms in the OPE. This is because D ( (cid:126)x ) = lim y → T yy ( (cid:126)x, y ) andthe bulk operator T yy is always equal to (3.23). (The scaling dimension of D is guarantueedto come out correctly because φ | y =0 and ∂ y φ | y =0 have protected dimensions.) This observationallows us to easily compute the two-point function of D in conformal perturbation theory for themodified Dirichlet condition (3.16). We find that the contributions from φ | y =0 are O ( g ) whereas This formula is simply obtained by appropriately normalizing the operators involved, namely (cid:98) O needs tohave unit-normalized two-point function and the bulk scalar field needs to have unit-normalized contribution ofthe identity in the bulk OPE, see also footnote 8. ∂ y φ | y =0 is corrected already at O ( g ) by the interaction term (3.15)and is given by: (cid:104) ∂ y φ (0 , (cid:126)x ) ∂ y φ (0 , (cid:126)x (cid:48) ) (cid:105) = Γ (cid:0) d (cid:1) π d | (cid:126)x − (cid:126)x (cid:48) | d + g (cid:32) Γ (cid:0) d (cid:1) π d (cid:33) (cid:90) d d − (cid:126)u | (cid:126)x − (cid:126)u | d (cid:90) d d − (cid:126)u (cid:48) | (cid:126)x (cid:48) − (cid:126)u (cid:48) | d C (0) (cid:98) χ | (cid:126)u − (cid:126)u (cid:48) | d − + O ( g )= Γ (cid:0) d (cid:1) π d | (cid:126)x − (cid:126)x (cid:48) | d (cid:32) − g π d C (0) (cid:98) χ Γ (cid:0) d − (cid:1) (cid:33) + O ( g ) . (3.24)Note that the integrals have a power-law UV divergence for (cid:126)u ∼ (cid:126)x and (cid:126)u (cid:48) ∼ (cid:126)x (cid:48) that we subtracted.As a check, the result (3.24) implies b = (cid:112) d − (cid:32) − g π d C (0) (cid:98) χ Γ (cid:0) d − (cid:1) (cid:33) + O ( g ) , (3.25)which is in perfect agreement with the correction (3.20) that we computed for a φ and the crossingrelations (2.13). Using (3.24) to compute the two-point function of ( ∂ y φ ) (cid:12)(cid:12) y =0 and therefore ofD, we obtain C ( D ) D = Γ (cid:0) d (cid:1) π d , δC (1) D = − ( d − (cid:0) d (cid:1) C (0) (cid:98) χ π d . (3.26)We can then form another ratio of observables that is independent of the coupling at the putativeperturbative fixed point δC D ( g ) δa φ ( g ) = δC (1) D δa (1) φ + O ( g ) = − d − Γ (cid:0) d (cid:1) π d + O ( g ) . (3.27)Note that this ratio does not depend on any data of the CFT d − and therefore it is a universalresult for deformations of the form (3.15) of the Dirichlet boundary condition. As a final example, we consider deformations of the Neumann free boundary condition by thefollowing interaction S ( N ) ∂ = S CFT d − + g (cid:90) y =0 d d − (cid:126)x φ (cid:98) χ . (3.28)The interaction gives rise to the following modified Neumann boundary condition ∂ y φ | y =0 = g (cid:98) χ . (3.29)20he condition (3.5) now gives (cid:98) ∆ (cid:98) χ = d − (cid:15) , with < (cid:15) (cid:28) , and again we will assume the existenceof a perturbative fixed point with g ∝ (cid:15) . Plugging in eq. (3.13) we obtain (cid:98) γ (1) (cid:98) τ = Γ (cid:0) d − (cid:1) π d dd + 1 C (0) (cid:98) χ C (0) (cid:98) τ . (3.30)To compute the variation of the parameter a φ we use the same strategy as in the previousexample, namely it follows from the modified Neumann condition that b = g (cid:115) π d Γ (cid:0) d − (cid:1) C (0) (cid:98) χ (cid:0) O ( g ) (cid:1) , (3.31)and using the crossing relations (2.13) this gives δa (1) φ = − − d π d C (0) (cid:98) χ Γ (cid:0) d (cid:1) . (3.32)Note that this has an opposite sign compared to eq. (3.20), in agreement with the unitaritybounds (2.14). The coupling-independent ratio then is (cid:98) γ (cid:98) τ ( g ) δa φ ( g ) = (cid:98) γ (1) (cid:98) τ δa (1) φ + O ( g ) = − d − Γ (cid:0) d − (cid:1) Γ (cid:0) d + 1 (cid:1) π d ( d + 1) 1 C (0) (cid:98) τ + O ( g ) . (3.33)Like in the previous example we now compute the correction to C D , again using the definition(3.23) as the starting point. The main difference is that in this case the leading-order correctioncomes from the second and third terms in eq. (3.23), namely those involving φ | y =0 , while thefirst term involving ∂ y φ | y =0 only starts contributing at subleading order O ( g ) . We will thenonly need the two-point function of φ | y =0 up to O ( g ) corrections, that is (cid:104) φ (0 , (cid:126)x ) φ (0 , (cid:126)x (cid:48) ) (cid:105) = Γ (cid:0) d − (cid:1) π d | (cid:126)x − (cid:126)x (cid:48) | d − + g (cid:32) Γ (cid:0) d − (cid:1) π d (cid:33) (cid:90) d d − (cid:126)u | (cid:126)x − (cid:126)u | d − (cid:90) d d − (cid:126)u (cid:48) | (cid:126)x (cid:48) − (cid:126)u (cid:48) | d − C (0) (cid:98) χ | (cid:126)u − (cid:126)u (cid:48) | d + O ( g )= Γ (cid:0) d − (cid:1) π d | (cid:126)x − (cid:126)x (cid:48) | d (cid:32) − g π d C (0) (cid:98) χ Γ (cid:0) d (cid:1) (cid:33) + O ( g ) . (3.34)The integrals have a power-law UV divergence for (cid:126)u ∼ (cid:126)u (cid:48) that we subtracted. As a check, from(3.24) we obtain b = √ (cid:32) − g π d C (0) (cid:98) χ (cid:0) d (cid:1) (cid:33) + O ( g ) , (3.35)21hich, upon substitution in the crossing equations (2.13), gives a correction to a φ in agreementwith (3.32). Using (3.34) we obtain C ( N ) D = Γ (cid:0) d (cid:1) π d , δC (1) D = − Γ (cid:0) d (cid:1) C (0) (cid:98) χ π d . (3.36)Comparing with (3.26) we see that the value at the free boundary condition is the same forNeumann and Dirichlet, while the leading correction differs by a factor of d − . Taking the ratiowith δa φ we get δC D ( g ) δa φ ( g ) = δC (1) D δa (1) φ + O ( g ) = − d − Γ (cid:0) d (cid:1) π d + O ( g ) , (3.37)which notably is the same as the one obtained for the deformation of Dirichlet in eq. (3.27).Like in that example, this ratio is universal for deformations of the form (3.28) of the Neumannboundary condition, because it does not depend on data of the CFT d − . In this section we present the crossing equation for the mixed system of four-point functions ofthe boundary modes of φ , namely (cid:104) (cid:98) O i ( (cid:126)x ) (cid:98) O j ( (cid:126)x ) (cid:98) O m ( (cid:126)x ) (cid:98) O n ( (cid:126)x ) (cid:105) . (4.1)The crossing equations for a generic mixed system of scalars, labelled by indices i, j, m, n , werederived in [34] and read (cid:88) (cid:98) O ( l ) [ ˆ f ij (cid:98) O ( l ) ˆ f mn (cid:98) O ( l ) F ij,mn ∓ , (cid:98) ∆ ,l ( u, v ) ± ˆ f mj (cid:98) O ( l ) ˆ f in (cid:98) O ( l ) F mj,in ∓ , (cid:98) ∆ ,s ( u, v )] = 0 , (4.2)where u = x x x x and v = x x x x . The functions F ij,mn ± , (cid:98) ∆ ,l are the following combinations of theordinary s -channel conformal blocks g (cid:98) ∆ ij , (cid:98) ∆ mn (cid:98) ∆ ,l F ij,mn ± , (cid:98) ∆ ,l ( u, v ) ≡ v ( (cid:98) ∆ m + (cid:98) ∆ j ) g (cid:98) ∆ ij , (cid:98) ∆ mn (cid:98) ∆ ,l ( u, v ) ± u ( (cid:98) ∆ m + (cid:98) ∆ j ) g (cid:98) ∆ ij , (cid:98) ∆ mn (cid:98) ∆ ,l ( v, u ) . (4.3)Note that not all equations in this system (4.2) are independent, since F ij,mn ± , (cid:98) ∆ ,l ( u, v ) = F mn,ij ± , (cid:98) ∆ ,l ( u, v ) , F ij,ij ± , (cid:98) ∆ ,l ( u, v ) = F ji,ji ± , (cid:98) ∆ ,l ( u, v ) , F ij,kk ± , (cid:98) ∆ ,l ( u, v ) = F ji,kk ± , (cid:98) ∆ ,l ( u, v ) . (4.4) Recall that [35, 36] g ∆ , ∆ ∆ ,(cid:96) ( u/v, /v ) = ( − (cid:96) v ∆342 g − ∆ , ∆ ∆ ,(cid:96) ( u, v ) = ( − (cid:96) v − ∆122 g ∆ , − ∆ ∆ ,(cid:96) ( u, v ) , ˆ f ij (cid:98) O ( l ) = ( − l ˆ f ji (cid:98) O ( l ) . (4.5)If we specialize all these ingredients to our problem where i, j, m, n ∈ { , } then we find 7independent crossing equations: (cid:88) (cid:98) O ( l ) ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) F , − , (cid:98) ∆ ,l ( u, v ) , (cid:88) (cid:98) O ( l ) ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) F , − , (cid:98) ∆ ,l ( u, v ) , (cid:88) (cid:98) O ( l ) ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) F , − , (cid:98) ∆ ,l ( u, v ) , (cid:88) (cid:98) O ( l ) ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) F , − , (cid:98) ∆ ,l ( u, v ) , (cid:88) (cid:98) O ( l ) ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) F , − , (cid:98) ∆ ,l ( u, v ) , (cid:88) (cid:98) O ( l ) ( − l ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) F , ∓ , (cid:98) ∆ ,l ( u, v ) ± ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) F , ∓ , (cid:98) ∆ ,l ( u, v ) . (4.6)In our case the operators must also obey the OPE relations (2.22). Imposing those, the systemof equations can be rewritten as follows (details can be found in appendix D) (cid:126)V + (cid:88) l = even ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) (cid:126)V + , (cid:98) ∆ ,l + (cid:88) l = odd ,... (cid:98) ∆= d + l − nn =0 , ,... ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) (cid:126)V − , (cid:98) ∆ ,l + (cid:88) (cid:96) ∈ N (cid:88) even l<(cid:96) (cid:98) ∆= d + (cid:96) − (cid:16) ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) (cid:17) (cid:126)V , (cid:98) ∆ ,l (cid:32) ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) (cid:33) . (4.7)The quantities (cid:126)V ± , (cid:98) ∆ ,l , (cid:126)V , (cid:98) ∆ ,l are 7-component vectors defined in (D.5), (D.6) and (cid:126)V , (cid:98) ∆ ,l are vectorsof × matrices defined in (D.7). Beside the identity, the first line accounts for ‘unprotected’primaries, i.e. operators of generic scaling dimension away from the poles of the gamma functionsin (2.22). The last two lines take into account the tower of ‘protected’ operators listed in table1, which can have both odd and even spin. The indices of the OPE coefficients are contractedaccording to the conventions of Appendix A. 23 .2 Implementing the exact relations We will numerically ‘bootstrap’ a set of crossing equation in the sense of [2]. For most problems,the fastest program available for this task is the semidefinite program solver
SDPB [37]. We haveused the recent version [38] which supports the ‘hotstarting’ algorithm suggested in [39]. Inorder to use
SDPB efficiently, the ingredients of the crossing equations must be approximated asrational functions of the scaling dimension such that all poles of odd order lie at or below theunitarity bound. This is always possible when the basis functions are conformal blocks and werefer the reader to [34] for details. The complication in this work is that the vector (cid:126)V + , (cid:98) ∆ ,l in (4.7)has several occurences of (cid:98) ∆ which are not in conformal blocks.As shown in Appendix D, we must consider linear combinations in which the coefficients arevarious products of κ ( (cid:98) ∆ , l ) and κ ( (cid:98) ∆ , l ) – the functions from (2.22). While these types of blockswere first introduced for studying the long-range Ising model, in the numerical analysis of [19]only the last two components of (D.6) were used. In addition, κ ( (cid:98) ∆ , l ) κ ( (cid:98) ∆ , l ) was treated as avector with > discrete evaluations, thereby eschewing some of the benefits of semidefiniteprogramming. In this work, we do not need to limit ourselves to those crossing equations thatinvolve only the product κ ( (cid:98) ∆ , l ) κ ( (cid:98) ∆ , l ) where b and b cancel out. Thanks to (2.13), the otherproducts only introduce one new parameter and it has a clear physical meaning in the BCFTcontext. To remedy the second problem, we need to discuss the approximation theory of gammafunctions.To start, it is easily verified that κ ( (cid:98) ∆ , l ) = − b b κ ( (cid:98) ∆ , l ) κ ( (cid:98) ∆ , l ) = − b b ( (cid:98) ∆ + l − d + l − (cid:98) ∆ − κ ( (cid:98) ∆ , l ) (4.8)with κ ( (cid:98) ∆ , l ) = Γ (cid:16) l + (cid:98) ∆2 (cid:17) Γ (cid:16) d + l − (cid:98) ∆ − (cid:17) Γ (cid:16) d + l − (cid:98) ∆ − (cid:17) Γ (cid:16) l + (cid:98) ∆+12 (cid:17) . (4.9)As such, a rational approximation for κ ( (cid:98) ∆ , l ) will cover the cases of κ ( (cid:98) ∆ , l ) , κ ( (cid:98) ∆ , l ) and With infinitely many poles above the unitarity bound it is clear that any rational approximation for κ ( (cid:98) ∆ , l ) is going to have significant errors in the semi-infinite range of allowed values for (cid:98) ∆ . Extremely large valuesof (cid:98) ∆ should however be unimportant for the numerical results, and for a finite window of values a rationalapproximation is perfectly feasible. We have attempted to account for this in practice by using the simplercrossing equations (4.6) to cover the ‘tail’ of the more constraining crossing equations. Each time we approximate κ ( (cid:98) ∆ , l ) with (cid:98) ∆ ∈ [∆ , ∞ ) , we allow these extra conformal blocks, multiplying independent f ij (cid:98) O coefficients, tohave an exchanged scaling dimension in [∆ + 20 , ∞ ) . ( (cid:98) ∆ , l ) κ ( (cid:98) ∆ , l ) . The most expensive step for our purposes will be the Weierstrass formula Γ( z ) = e − γz z ∞ (cid:89) k =1 (cid:16) zk (cid:17) − e z/k (4.10)which introduces a new series of poles for each gamma function. Unlike [19], which suggestedusing (4.10) on the full function, we will only apply it to the − (cid:98) ∆ part of κ ( (cid:98) ∆ , l ) .The + (cid:98) ∆ part, since it is regular, should be approximated with one of the many expressionsfor the Wallis ratio. This is a quantity which has attracted interest for hundreds of years due tothe application of calculating π . In particular, we note the asymptotic formula Γ( z + 1)Γ (cid:0) z + (cid:1) (cid:46) (cid:114) z + 14 + 132 z + 8 , z → ∞ . (4.11)It was found in [40] that (4.11) is the n = 1 case in a sequence of approximants that have theschematic form ( z n + . . . ) n . We cannot use these higher radicals due to the requirement that κ ( (cid:98) ∆ , l ) be a rational function but it is still possible to make (4.11) arbitrarily accurate. Onesimply applies the functional equation n times to arrive at Γ( z + 1)Γ (cid:0) z + (cid:1) (cid:46) (cid:0) z + (cid:1) n ( z + 1) n (cid:114) z + n + 14 + 132 z + 32 n + 8 , z → ∞ . (4.12)We have not found it necessary to choose a large value of n . For example, even when z = ,Weierstrass does not become better than (4.11) until k = 36 .We will now quote expressions for the two factors of κ ( (cid:98) ∆ , l ) . Using (4.12) with n = 1 , Γ (cid:16) (cid:98) ∆+ l (cid:17) Γ (cid:16) (cid:98) ∆+ l +12 (cid:17) = (cid:98) ∆ + l + 1)( (cid:98) ∆ + l )( (cid:98) ∆ + l + 2) Γ (cid:16) (cid:98) ∆+ l +42 (cid:17) Γ (cid:16) (cid:98) ∆+ l +32 (cid:17) ≈ (cid:34) (cid:98) ∆ + l + 1)( (cid:98) ∆ + l )( (cid:98) ∆ + l + 2) (cid:35) (cid:98) ∆ + 8(2 l + 5) (cid:98) ∆ + 8 l + 40 l + 518(2 (cid:98) ∆ + 2 l + 5) . (4.13)The singular part requires a cutoff which we call k max . Γ (cid:16) d + l − (cid:98) ∆ − (cid:17) Γ (cid:16) d + l − (cid:98) ∆ − (cid:17) ≈ e γ k max (cid:89) k =1 e k (cid:34) k max (cid:89) k =0 (cid:98) ∆ − d − l − k + 1 (cid:98) ∆ − d − l − k + 2 (cid:35) ≈ k max k max (cid:89) k =0 ( (cid:98) ∆ − d − l − k + 1) ( (cid:98) ∆ − d − l − k + 2) . (4.14)While it is optional to resum the exponent in the second step, the logarithmic behaviour of theharmonic series makes it convenient. 25he expressions (4.13) and (4.14) have been implemented as a patch for the helper program PyCFTBoot [41]. For the poles exhibited here, which are two units apart, we have taken k max = 20 .The poles coming from conformal blocks [34] are only one unit apart so we take k max = 40 forthose. The standard way to account for poles in (cid:98) ∆ is to absorb them into the OPE coefficientsof (4.7) so that crossing symmetry becomes a statement about polynomials. The most desirabletype of problem for SDPB is one in which these polynomials can be expressed in terms of anorthonormal basis [37]. Recently, [42] gave an example of a problem which cannot be optimizedin this way. In our case, this privileged basis of polynomials is again unavailable due to the 20double poles of (4.14) that are above the unitarity bound. For this reason, we have opted tostill use the simple crossing equations (4.6) for spins above a certain cutoff l . For most of thebounds in the next section, this is l = 4 while some of them have been redone with l = 6 .Seeing almost no difference, we conclude that the exact relations for l = 0 and l = 2 are doingmost of the work.The other limitation of our approach is that the square root in (4.12) can only be eliminatedwhen the κ i ( (cid:98) ∆ , l ) appear quadratically. This forces us to drop (cid:104) (cid:98) O (cid:98) O (cid:98) O (cid:98) O (cid:105) which is linear in κ ( (cid:98) ∆ , l ) and (cid:104) (cid:98) O (cid:98) O (cid:98) O (cid:98) O (cid:105) which is linear in κ ( (cid:98) ∆ , l ) . According to standard lore, the boundsshould be unaffected as these two correlators exchange the same operator families as the otherthree. d/ d systems Let us collect the constraints used for the numerical bootstrap analysis. First, we take thecrossing equations for (cid:104) (cid:98) O (cid:98) O (cid:98) O (cid:98) O (cid:105) , for (cid:104) (cid:98) O (cid:98) O (cid:98) O (cid:98) O (cid:105) and (cid:104) (cid:98) O (cid:98) O (cid:98) O (cid:98) O (cid:105) given in (4.6). Theseapply to any (possibly non-local) CFT containing the operators (cid:98) O and (cid:98) O . Second, we havethe exact OPE relations (4.13) and (4.14) which are necessary for a solution of these crossingequations to be an admissible boundary condition for a massless free scalar in d dimensions.These conditions reduce the crossing equations to equation (4.7) at the cost of introducing a newparameter, a φ , that our bounds will depend on. Notice that this in particular implies that theodd-spin operators can only have the scaling dimensions of the generalized free theory. Third,the boundary spectrum cannot have a stress tensor, so it is natural to demand that the firstspin 2 operator has a dimension (cid:98) ∆ (cid:98) τ strictly larger than d − . Fourth, we should demand thatthe flux operator V (1) of dimension d is absent to avoid interfaces and other possible sources ofnon-locality on the boundary, see the discussion in 2.2.3. Fifth, we have the Ward identities forthe displacement operator (2.36) which restrict ˆ f D and ˆ f D to a curve parametrized by C D .We will set d = 4 throughout in order to work with a correlator system that involves d conformal blocks. As discussed in [43, 44], similar problems with d blocks often require more26xperimentation with the gaps being imposed. These works are concerned with maximizing thegap in the scalar sector, and indeed, we can provide a nice preview of our results by doing thesame. Figure 1 bounds the dimension of the lightest exchanged scalar, which we call (cid:98) ∆ (cid:98) ε , asa function of the pseudo stress tensor dimension (cid:98) ∆ (cid:98) τ . To obtain this plot we scanned over allthe allowed values of a φ . The so obtained blue region is clearly smaller than the pink region,obtained without imposing the OPE relations, or the single correlator region delineated by theupper black line. Three further comments are worthwhile. Figure 1: A plot showing the upper bound on the dimension of (cid:98) ε , the first scalar, other thanthe identity, seen by any of the OPEs in our correlator system. The unshaded region is the onethat follows from a single correlator (cid:68) (cid:98) O (cid:98) O (cid:98) O (cid:98) O (cid:69) . The pink region, which is more restrictive,uses the multi-correlator system but the only inputs it uses from the exact relations are theodd-spin operator dimensions given in table 1. The blue region, more restrictive again, followsfrom a genuine use of the exact relations. Since these depend on a φ , we have extremized (cid:98) ∆ (cid:98) ε over a third axis which is not shown. First, we observed that much of the constraining power came from our fourth constraint,i.e. the exclusion of the dimension d vector V (1) = [ (cid:98) O (cid:98) O ] , from the spectrum. In fact, if wewere to reinstate just this vector then the blue region would expand to almost the same sizeas the pink region. We emphasize that the OPE relations are essential to meaningfully imposethis constraint: they prevent the appearance of vector operators of dimensions very close to d (1) . Furthermore, because of the fake primaryeffect [45, 46] the block for V (1) can be mimicked in our numerical analysis by a spin 2 operatorof dimension 3, and therefore the constraint that (cid:98) ∆ (cid:98) τ > (strictly) is also essential to ensurethat it is really absent. This latter argument relies on the observation that, for a spin 2 operatorwhose dimension (cid:98) ∆ → , the corresponding combination of blocks that enters in the crossingequation (4.7) is: κ ( (cid:98) ∆ , κ ( (cid:98) ∆ , g , (cid:98) ∆ , ( u, v ) − g − , (cid:98) ∆ , ( u, v ) = 45( (cid:98) ∆ − g − , , ( u, v ) + . . . (5.1)by virtue of the OPE relations discussed in section 4, whose notation we follow here. Thereforewe can recover a vector operator if we assume that its overall coefficient ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) ∝ ( (cid:98) ∆ − .Second, for the Dirichlet or Neumann boundary conditions either (cid:98) O or (cid:98) O vanishes so insome sense they are not within the reach of our numerical analysis. On the other hand we canfind a more general solution including both (cid:98) O and (cid:98) O as independent GFFs, with any value of a φ . This solution does contain the vector V (1) , and because of the fake primary effect we justdiscussed it corresponds to the point with (cid:98) ∆ (cid:98) ε = 2 and (cid:98) ∆ (cid:98) τ = 3 , which is well within the allowedregion.Third, one may check that these bounds are saturated by the following two extremal solutions.The point ( (cid:98) ∆ (cid:98) τ , (cid:98) ∆ (cid:98) ε ) = (4 , represents the ‘single GFF’ solution where (cid:98) O is a GFF and (cid:98) ε = (cid:98) O = (cid:98) O . This satisfies our crossing equations because it consists entirely of protected operators in(4.7). Also, the aforementioned vector is indeed absent from the spectrum of primaries because [ (cid:98) O (cid:98) O ] , is a descendant of (cid:98) O in this theory. Since the bound in figure 1 can only decreaseas a function of (cid:98) ∆ (cid:98) τ , we can be confident that it will stop changing once it hits (cid:98) ∆ (cid:98) ε = 2 . Inthe blue plot, this turns out to happen well before (cid:98) ∆ (cid:98) τ = 4 . We can also understand the point ( (cid:98) ∆ (cid:98) τ , (cid:98) ∆ (cid:98) ε ) ≈ (3 , . : here the four-point function of (cid:98) O can be the extremal solution for a localthree-dimensional CFT with (cid:98) ∆ = 1 , which according to [47] has (cid:98) ∆ (cid:98) ε ≈ . , and then (cid:98) O can bea disconnected GFF. This setup satisfies all of the constraints we have imposed (we are of coursenot imposing (cid:98) ∆ (cid:98) τ > here) except for the absence of V (1) , which again manifests itself as a spin2 operator of dimension 3.We find it plausible that, with infinite computational power, the drop from (cid:98) ∆ (cid:98) ε ≈ . becomesinfinitely sharp leading to a value of (cid:98) ∆ (cid:98) ε = 2 almost everywhere. The remainder of this section isabout what lies below (cid:98) ∆ (cid:98) ε = 2 . The bulk has a global reflection symmetry φ → − φ under which (cid:98) O and (cid:98) O are odd but (cid:98) ε is even. Since (cid:98) ∆ (cid:98) ε < always, and (cid:98) ∆ (cid:98) ε ≤ seems likely, any non-trivial boundary condition must be (strongly) unstable even forRG flows that preserve the Z symmetry. The cases of Dirichlet and Neumann are not included in this discussionbecause as we explained in these cases one should remove many operators from the spectrum. The Neumanncondition is also (strongly) unstable due to the operator φ , while in the Dirichlet case the leading Z evendeformation is ( ∂ y φ ) so this boundary condition is stable. .1 A universal bound Figure 2: Bounds on the dimension of the leading spin 2 operator (cid:98) τ over the range − < a φ < with our best estimate for the allowed region shaded in blue. The curves have n max = 5 , , , inthe notation of [48, 49]. As for the number of derivative components being kept in each crossingequation, these correspond to , , , respectively. The dotted line shows the maximumpossible value for (cid:98) ∆ (cid:98) τ from leading order conformal perturbation theory under the assumptionthat the Ising model is the 3d CFT with the lowest central charge. The next parameter to introduce is a φ , which through (2.13) determines b and b , in order tomore fully exploit the exact relations. When scanning over a φ , it is instructive to first determineits value for the two extremal solutions at ( (cid:98) ∆ (cid:98) τ , (cid:98) ∆ (cid:98) ε ) = (4 , and at ( (cid:98) ∆ (cid:98) τ , (cid:98) ∆ (cid:98) ε ) ≈ (3 , . discussedabove. In the first we found a spin 2 boundary operator with (cid:98) ∆ (cid:98) τ = 4 corresponding to anunprotected block in the first line of (4.7). Since the overall coefficient of this combination is ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) , (2.22) tells us that we must be in a situation where κ (4 , does not blow up.As shown in the middle row of table 1, this can only happen for an (cid:98) O double-trace if b = 0 .Consequently, this solution sits at a φ = . In the other extremal solution, the spin 2 operatorwith (cid:98) ∆ (cid:98) τ = 3 needs to be absent from (cid:98) O × (cid:98) O since (cid:98) O is a GFF. The only way to make thiscompatible with (2.22) is to have a φ = − . Proceeding to intermediate values of a φ , it is useful29o maximize (cid:98) ∆ (cid:98) τ since this can be interpreted as a measure of how non-local a CFT is. Figure 2presents this result. The four different lines correspond to four different search spaces, giving asense of how close we are to having an optimal bound. Every other plot in this section uses thenumber of components corresponding to the second most restrictive region in figure 2.Figure 2 enables a comparison with the results of conformal perturbation theory, particularlyaround a φ = − where the bound is very strong and we can make a meaningful comparisonwith the deformation of the Dirichlet boundary conditions by a putative 3d CFT with a scalaroperator with (cid:98) ∆ (cid:98) χ = 1 − (cid:15) , that we studied in section 3.2.2. Recall that, according to equation(3.17), the anomalous dimension of the spin 2 operator depends on the unperturbed centralcharge C (0) (cid:98) τ . Even though we do not know a theory with an operator that can play the role of (cid:98) χ ,it is clear that the central charge of a unitary 3d CFT cannot be arbitrarily small. In fact, webelieve that it is not unreasonable to assume that the 3d Ising CFT with C (0) (cid:98) τ ≈ . C free (cid:98) τ (5.2)is the theory with the lowest possible central charge. An early indication for this conjecturewas the local minimum corresponding to the Ising CFT found in [50, 51], and recently in [52] arigorous lower bound was found that, with sufficient numerical precision, is likely to lie betweenabout . C free (cid:98) τ and . C free (cid:98) τ . For us this implies that C (0) (cid:98) τ > . C free (cid:98) τ ⇒ (3.21) (cid:98) γ (cid:98) τ < . δa φ (5.3)as a bound on the anomalous dimension of the first spin 2 operator. In figure 2 it follows thatevery such example must lie below the dotted line. The possibilities obeying (5.3) are all withinthe allowed region for now, but we will see that many of them are ruled out when we add moreconstraints. We now come to the most striking feature of figure 2 which is the jump near the right handside. Since the convergence appears to be rapid in this vicinity, we can be confident that thecoordinate at which the curve flattens again is not tending towards a φ = . In other words,there is a kink at ( a φ , (cid:98) ∆ (cid:98) τ ) ≈ (0 . , . which obeys the exact relations and cannot be oneof the free boundary conditions. If it is truly a new boundary condition then it must obey afurther constraint that we have not yet imposed: the Ward identity (2.36) for the displacementoperator. This is one of the reasons we would like to investigate the spectrum at the kink inmore detail. We are using conventions such that C free (cid:98) τ = π . a φ , (cid:98) ∆ (cid:98) τ , C D ) = (0 . , . , . l (cid:98) ∆ ˆ f O ˆ f O ˆ f O . − . . − . . − . . − . . − . . . See (5.8) .
466 0 . . − . . . . . − . . − . . − . . . . . . . − . . − . a φ , (cid:98) ∆ (cid:98) τ , C D ) = (0 . , . , . l (cid:98) ∆ ˆ f O ˆ f O ˆ f O . − . . − . . − . . − . . − . . . See (5.10)-(5.11) .
859 0 . . − . . . . . − . . − . . − . . . . . . . − . . − . Table 2: The low-lying spectrum at two points in ( a φ , (cid:98) ∆ (cid:98) τ , C D ) space. The point associatedwith the left table is still visible after projecting down to just ( a φ , (cid:98) ∆ (cid:98) τ ) – it is the kink in figure2. Due to our maximization choice, we see every possible operator with odd spin. Protectedoperators (the ones with integer scaling dimensions) of even spin have vanishing mixed OPEcoefficients and they start above the leading twist. Unprotected operators have their OPEcoefficients related by (2.22) and thus we have shown calculated values in red. The extremal spectrum
The extremal functional method [20, 21] allows for the extractionof an approximate spectrum and OPE coefficients for any point on the boundary of an allowedregion. We have done so at two points: the first is the kink in figure 2 and the second involvesa tuning of the displacement central charge C D using the procedure that will be explained inthe next subsection. The CFT data for operators with (cid:98) ∆ < . are listed in table 2. The blacknumbers were obtained from the output of the script in [53] and the red numbers were computedusing the OPE relations. Our OPE coefficients are defined such that g (cid:98) ∆ ,l ( z, z ) ∼ ( − l (cid:20) z (1 + √ − z ) (cid:21) (cid:98) ∆ + . . . (5.4)for the standard cross-ratios z and ¯ z approaching zero along the diagonal.Notice that the most stable result is obtained after maximizing an OPE coefficient near theboundary of the plot to make the functional as close to extremal as possible. We have chosen tooptimize the coefficient of the V (3)4 operator, which is a spin 3 operator of dimension 6. Our reasonfor doing so is to avoid another interference from the fake primary effect: much like V (1)2 can be31imicked by a spin 2 operator of dimension almost 3, the V (3)4 operator can be mimicked by aspin 4 operator of dimension almost 5. However this scenario is unnatural, not only because ofthe existence of an operator very close to the unitarity bound but also because the absence of V (3)4 would imply that the higher-spin charge corresponding to the bulk spin 4 current is preserved bythe boundary. We do not expect such an ‘integrable’ boundary, and our optimization minimizesthe chances of an unwanted spin 4 operator taking the place of V (3)4 .After going through this maximization, a reassuring feature we observe is that there is noeven spin l operator with (cid:98) ∆ = d + l − , as anticipated in the range of the sum in (4.7). Such ablock, if present, would have to be treated with (cid:126)V ,d + l − ,l because the exact relations degenerateat this point. But indeed, bulk spin (cid:96) currents only have boundary modes up to l = (cid:96) − in thebOPE and there is no reason for l = (cid:96) to be present as a protected operator. The displacement Ward identity
With the spectrum in hand we can investigate whetherthe Ward identity (2.36) for the displacement operator is satisfied. However this is again a rathersubtle business, this time because there might be other scalar operators of dimension 4. Thecorrect procedure is as follows.Consider all the scalar operators of dimension 4 in the putative extremal solution at the kink.One of these operators is the displacement operator, and its coefficients ˆ f D and ˆ f D must obey(2.36), i.e. they are constrained to lie on a curve parameterized by C D . Every other operator isthen not a displacement operator, meaning that it must be absent from the bulk-to-boundaryOPE of the stress tensor: b T D (cid:48) = 0 for any ‘non-displacement’ D (cid:48) . Repeating the arguments inappendix C.3 this leads to the condition that: ˆ f D (cid:48) = b φ D (cid:48) b , ˆ f D (cid:48) = − b φ D (cid:48) b . (5.5)with b φ D (cid:48) arbitrary. So from a physical perspective (2.36) and (5.5) are the equations to bechecked.On the numerical side of things we do not get these coefficients so cleanly; instead we aregiven the elements of the matrix corresponding to (cid:88) O = D , D (cid:48) (cid:32) ˆ λ O ˆ λ O ˆ λ O ˆ λ O ˆ λ O ˆ λ O (cid:33) , (5.6)where ˆ λ ij O denotes the OPE coefficient with unit-normalized two-point function of O , and it isup to us to cook up a series of OPE coefficients ˆ λ ij O for operators D and D (cid:48) in order to fit thisdata. (The switch from ˆ f ij O to ˆ λ ij O is deliberate: numerically we obtain OPE coefficients forunit-normalized operators, so the coefficients in (2.36) should really be scaled by √ C D .)32or the spectrum on the left hand side of table 2 we numerically obtain a matrix of rank 1whose factorization yields (cid:32) ˆ λ D ˆ λ D (cid:33) = (cid:32) . . (cid:33) . (5.7)Since we have only one operator this must be the displacement, so we can check compatibilitywith the Ward identity (2.36). Remarkably we find that it is well obeyed with C D = 0 . – astrong indication that the extremal solution is actually physical! We also obtain that (cid:32) ˆ f D ˆ f D (cid:33) = (cid:32) . . (cid:33) (5.8)for the OPE coefficients.For the spectrum on the right hand side of table 2 we find that (cid:88) O = D , D (cid:48) (cid:32) ˆ λ O ˆ λ O ˆ λ O ˆ λ O ˆ λ O ˆ λ O (cid:33) = (cid:32) . . . . (cid:33) . (5.9)This is a matrix of rank two and we need more than one operator. It is natural to try to see ifwe can fit it with one displacement and one non-displacement operator. Notice that the matrixhas three independent entries but for the two operators we only have the two parameters C D and b φ D (cid:48) . Two simple approaches can be taken at this point. In the first approach, we demand thata non-displacement is exactly present and extract b φ D (cid:48) : (cid:32) ˆ λ D (cid:48) ˆ λ D (cid:48) (cid:33) = b φ D (cid:48) b − b φ D (cid:48) b ⇒ b φ D (cid:48) = 1 . , (cid:32) ˆ λ D ˆ λ D (cid:33) = ± (cid:32) . . (cid:33) . (5.10)To fit the rest of the matrix we need the given displacement OPE coefficients, which lead to C D = 0 . from ˆ λ D and C D = 0 . from ˆ λ D . Alternatively, in the second approach,we demand that one of the outer products is an exact displacement and extract C D and OPEcoefficients for the non-displacement: (cid:32) ˆ λ D ˆ λ D (cid:33) = S C D +16 a φ S b √ C D S C D − a φ S b √ C D ⇒ C D = 0 . , (cid:32) ˆ λ D (cid:48) ˆ λ D (cid:48) (cid:33) = ± (cid:32) . − . (cid:33) . (5.11)This leads to b φ D (cid:48) = 1 . from ˆ λ D (cid:48) and b φ D (cid:48) = 1 . from ˆ λ D (cid:48) . Although the smallmismatches in both approaches might be due to numerical errors, it seems reasonable to concludethat this solution is not as physical as the solution on the left hand side of table 2.33 .2 Local boundary conditions As with numerical bounds on the gap, the exact relations also lead to significant improvementsfor bounds on OPE coefficients. Consider again the (unit-normalized) displacement operatorwhich appears with the coefficients ˆ λ D and ˆ λ D . To constrain them, we set (cid:32) ˆ λ D ˆ λ D (cid:33) (cid:55)→ ˆ λ D (cid:32) cos θ sin θ (cid:33) (5.12)as in [54], then apply standard methods for bounding the magnitude of an OPE-space vector[55]. Figure 3 shows the results of this exercise for different values of a φ .A first thing to note is once more the importance of the exact OPE relations in (2.22). Withoutthem the allowed region would certainly be the union of all the regions in figure 3. However for a φ → we observe an unbounded growth in the vertical direction (note the different verticalscales), and therefore ˆ λ D is really only bounded by virtue of the OPE relations.The dotted line represents the combinations of OPE coefficients that obey the Ward identity(2.36), parameterized by the displacement central charge C D . As discussed above, the non-trivialfact about the kink in figure 2 was that it happened to sit on this line. The intersection of thedotted line with the allowed region also translates into a lower and upper bound for C D for eachvalue of a φ . This is shown as the pink region in figure 4. This bound is certainly valid butrather crude: it does not take into account the restriction (5.5) on additional scalar operatorsof dimension 4 that are not the displacement. To do better we can assume a fixed displacementoperator with a certain C D in the crossing equations by replacing (cid:126)V (cid:55)→ (cid:126)V + 1 S C D (cid:16) S C D +16 a φ b S C D − a φ b (cid:17) (cid:126)V , , S C D +16 a φ b S C D − a φ b (5.13)and removing the scalar of dimension 4 from the special operators in the crossing equation (4.7).We then bisect in C D to find the allowed region, and this leads to the much improved blue regionin figure 4.One may wonder if the blue region allows for other scalar operators of dimension 4 that arenot the displacement operator. The answer is that it does, because such operators lie in theallowed continuum of operators. Furthermore, ˆ f D (cid:48) ˆ f D (cid:48) = κ (4 , κ (4 ,
0) = − b b , (5.14)which implies that the limit of a scalar operator as ˆ∆ → in the continuum is actually exactly aD (cid:48) operator whose OPE coefficients automatically obey (5.5). So fixing C D not only allows one34 a) a φ = − . (b) a φ = − . (c) a φ = − . (d) a φ = 0 . (e) a φ = 0 . (f) a φ = 0 . Figure 3: Six allowed regions for the OPE-space vector of the unit-normalized displacement.The dotted line shows the physical locus for ˆ λ D and ˆ λ D , i.e. (2.36) divided by √ C D . Whenthis line becomes vertical (defining a unique ˆ λ D in order for ˆ λ D to be finite), it saturates ourbound. This does not quite happen in the opposite limit of the line becoming horizontal. Notethat in the GFF example there are two candidates for the displacement. Both [ (cid:98) O (cid:98) O ] , and [ (cid:98) O (cid:98) O ] , are compatible with these bounds if we treat them as different operators that satisfy ˆ λ D ˆ λ D = 0 . to single out a displacement operator for which the Ward identities (2.36) are obeyed, it alsoensures that (5.5) holds for every other scalar of dimension 4.We can compare figure 4 to the conformal perturbation theory results (3.27) and (3.37).The lines corresponding to potential perturbative fixed points saturate the lower bound on C D once the proper constraints on dimension 4 scalars are imposed. These emanate from the points ( a φ , C D ) = (cid:0) ± , π (cid:1) at which the upper and lower bounds are forced to meet by the Wardidentity. The other point that can be explained analytically is the origin which is associated withno boundary at all. This point has to be allowed by the pink region since it corresponds to addingzero in (5.13). Once we classify dimension 4 scalars into displacements and non-displacements,it appears that there are no longer any nearby solutions that would allow us to see this point inthe blue region. To see that they cannot arise from a GFF construction, consider the explicit35 igure 4: An asymmetric plot showing the minimum and maximum C D as a function of a φ .In the blue region, all dimension 4 scalars not singled out by (5.13) are constrained to satisfy b T D (cid:48) = 0 . No such constraint is made in the pink region which leads to weaker bounds. Thedotted lines give the predictions of conformal perturbation theory which are model-independentat leading order. A slight kink in the upper right corner looks well positioned to be identifiedwith the kink in figure 2. displacement operator (3.23). We may rewrite it asD = (cid:115) C D ( d − b + b (cid:16) ( d − b [ (cid:98) O (cid:98) O ] , + b [ (cid:98) O (cid:98) O ] , (cid:17) (5.15)by using the bulk two-point function (2.8) to relate the double-traces of φ and ∂ y φ to thoseinvolving (cid:98) O and (cid:98) O . The rules of GFF then allow us to go from (5.15) to ˆ f D = b ( d − √ d − (cid:115) C D ( d − b + b , ˆ f D = b (cid:115) C D ( d − b + b . (5.16)For a generic a φ , there is no C D which can make both of these coefficients satisfy the Wardidentity. 36 igure 5: The maximum possible (cid:98) ∆ (cid:98) τ for several points in the most interesting region of figure4. Planes are inserted below points with the same value of a φ for visibility. The red point hasits spectrum shown in the left columns in table 2. After producing universal bounds in the ( a φ , (cid:98) ∆ (cid:98) τ ) and ( a φ , C D ) planes, it is natural to tryscanning in all three parameters. This means choosing a grid of points in the allowed blue regionof figure 4 and maximing the spin 2 gap at each one. The best feature of this plot is that everypoint with (cid:98) ∆ (cid:98) τ > is guaranteed to obey all the constraints given above: exact OPE relations,no stress tensor, no flux operator, and the Ward identity for the displacement operator. For a φ ≥ . , which is the vicinity of the kink, the results are shown in figure 5.Before we discuss this figure, let us comment first on the analysis for more general a φ and forwhich the data is not shown. This analysis indicated that the maximum spin 2 gap, so the pointson the boundary of figure 2, correspond to the largest possible values of C D , so the points nearthe upper boundary of figure 4. On the other hand, near the lower boundary of figure 4 the spin 2gap remains very close to . Since the perturbative line in figure 4 is near this lower boundary, itindicates that the corresponding line in figure 2 must actually be quite a bit flatter than the slopedetermined by the Ising model central charge (5.3). In short, the (non-rigorous!) extrapolationof the one-loop analysis to small but finite values of δa φ indicates that the Dirichlet boundarycondition can only be driven to a weakly coupled fixed point if the 3d CFT that triggers the RGflow has a large central charge.Let us return to figure 5. Our best candidate for a non-trivial boundary condition, the left of37able 2, may be found by hugging the upper edge of figure 4 and looking for where (cid:98) ∆ (cid:98) τ jumps. Assuggested by the extremal spectrum, this happens at ( a φ , (cid:98) ∆ (cid:98) τ , C D ) = (0 . , . , . andcorresponds to the red point in the figure. We see that the two-dimensional kinks in previousfigures have become a three-dimensional feature: a cliff appears to develop around this point.The spectrum on the right of table 2 corresponds to ( a φ , (cid:98) ∆ (cid:98) τ , C D ) = (0 . , . , . which appears to be a more generic point in this three parameter space. We originally chosethis point by hugging the lower edge of pink region in figure 4, i.e. by bisecting in (cid:98) ∆ (cid:98) τ withoutthe constraint (5.5) for non-displacement scalars of dimension 4. This produces a jump at ( a φ , C D ) = (0 . , . in that plot. However, re-interpreting the extra dimension 4 scalarsfound in that solution as discussed above shifted C D from . to . . (As discussed above,this is under the assumptions that the Ward identities hold for this point.)Notice also that (cid:98) ∆ (cid:98) τ → rather smoothly as C D approaches its lowest possible value. Ac-cording to the dashed line in figure 4 this is where we could find potential weakly coupled fixedpoints from the Neumann end. As for the Dirichlet end discussed above, one might take this asan indication that the anomalous dimension of the three-dimensional stress tensor cannot be toobig. We set out to investigate whether a free real scalar field could have conformal boundary conditionsother than Dirichlet or Neumann. The bulk equation of motion restricted the two- and three-point functions of φ so strongly that we found that all non-trivial boundary conditions mustsupport a shadow pair of boundary operators of dimensions ∆ φ and ∆ φ + 1 . The numericalanalysis in four bulk dimensions (so three boundary dimensions) of correlation functions of thisshadow pair yielded interesting results. On the one hand, for a large range of values of a φ (theone-point function of the bulk φ operator) there must be a spin 2 operator relatively close tothe unitarity bound, providing some evidence for the absence of interesting boundary conditions.On the other hand, for a φ near its upper bound of / this maximal value shoots up and weobserved an interesting kink in the data at about a φ = 0 . with a spin 2 operator of dimension . and C D approximately equal to . . More numerical data is provided in table 2. Thiscould be a new conformal boundary condition for the free scalar field.It the future it would be interesting to see whether the shadow relations can be exploredanalytically rather than numerically. Indeed, one could ask whether the shadow transform ˜ O ( x ) = (cid:90) d d y x − y ) d − ∆) O ( y ) (6.1)can be applied to four-point functions and conformal blocks? As we have seen, the shadow38ransformation is singular for three-point functions when the scaling dimension of the thirdoperator is of double-twist type, so it is not entirely obvious that shadow transforming one ormore operators in a consistent four-point function automatically leads to another consistent four-point function. Our expectation is instead that contact terms will become important becausethey get magnified to non-trivial functions by the shadow transformation.It would of course also be interesting to understand the possible new conformal boundarycondition that corresponds to the kink in our numerical plots. One approach would be to tryto ‘move’ the kink by changing the problem. For example, we could try different spacetimedimensions d or generalize the problem to N > free scalar fields. These would of course beinteresting studies in their own right, but if we can dial a parameter like d or N to a value wherethe kink merges with a free boundary condition then that would provide strong indications for apossible perturbative approach to the problem. Another approach would be to put the free bulktheory in AdS: then we can add a mass term to the bulk field which would continuously changethe scaling dimensions but which is not expected to spoil the conformality of the boundary anda conformal bootstrap analysis should always be possible [56].An obvious direction for the future is to try to extend the analysis of this paper to otherexamples of free theories in the bulk, such as the free scalar in other spacetime dimensions, thefree fermion in any d or the free vector in d = 4 . In the latter case it would be extremelyinteresting to see if there is any signature of the continuous family of boundary conditions [10]in the bootstrap approach, perhaps along the lines of the previous bootstrap study of conformalmanifolds in [57].More generally, the ‘landscape’ of boundary conditions for a given CFT d is a wide openproblem. It therefore remains an interesting target for further explorations. The subject iseven richer because, as this paper exemplifies, we need to modify the usual crossing symmetryequations in surprising ways when defects, boundaries, or interfaces are taken into account. Acknowledgements
We thank the organizers of the ‘Bootstrap 2019’ and the ‘Boundaries and Defects in QuantumField Theory’ workshops at Perimeter Institute where part of this work was carried out. Wefurther thank P. Liendo and X. Zhao for collaboration on related topics and S. Cremonesi, M. DelZotto, D. Dorigoni, D. Gaiotto, C. Herzog, M. Lemos, M. Meineri, P. Niro and E. Trevisanifor interesting discussions. Research at Perimeter Institute is supported by the Governmentof Canada through Industry Canada and by the Province of Ontario through the Ministry ofResearch & Innovation. EL and BvR are supported by the Simons Foundation grant O ( N ) invariant conformal boundary conditions for multiple free scalars were studied recently in [33, 25]. A Conventions
A.1 bOPE
Consider a scalar bulk operator O , not necessarily free. The bOPE of O is completely determinedby SO ( d, symmetry, up to a certain collection of CFT data [5, 6] O ( (cid:126)x, y ) = (cid:88) (cid:98) O (cid:88) n b O (cid:98) O y ∆ O − (cid:98) ∆ (cid:98) O (cid:16) − y (cid:126) ∇ (cid:17) n n ! (cid:16) (cid:98) ∆ (cid:98) O + − d (cid:17) n (cid:98) O ( (cid:126)x ) . (A.1)One can check that the expression above reproduces the bulk-to-boundary correlators (2.1), onceapplied to the boundary two-point functions (cid:104) (cid:98) O ( (cid:126)x ) (cid:98) O (cid:48) (0) (cid:105) = (cid:98) C (cid:98) O (cid:98) O (cid:48) | (cid:126)x | (cid:98) ∆ (cid:98) O , (cid:98) C (cid:98) O (cid:98) O (cid:48) = δ (cid:98) O (cid:48) (cid:98) O (cid:98) C (cid:98) O (cid:98) O , (A.2)and using that b O (cid:98) O (cid:98) C (cid:98) O (cid:98) O = b O (cid:98) O . We will take unit-normalized boundary two-point functions,except for the protected operators that can appear in the bOPE of the bulk conserved currents J (cid:96) . Such operators, collectively denoted by (cid:98) J ( l ) (cid:96) (with l = 0 , . . . (cid:96) − ) have their normalizationfixed by the Ward identities (2.23), and therefore the coefficients in their two-point functions arephysical (cid:104) D ( (cid:126)x ) D (0) (cid:105) = C D | (cid:126)x | d , (cid:104) (cid:98) J ( l ) (cid:96) ( (cid:126)x, z ) (cid:98) J ( l ) (cid:96) (0 , z ) (cid:105) = C (cid:98) J ( l ) (cid:96) ( z · I (ˆ x ) · z ) l ( (cid:126)x ) | d + (cid:96) − . (A.3)When the bulk operator O is a free scalar φ , as we explained in 2.1.1, the scaling dimensionsof its boundary modes (cid:98) O i are completely determined by the bulk equation of motion. The40xpression (A.1) becomes (compare to (2.4)) φ ( (cid:126)x, y ) = (cid:88) i =1 , (cid:88) n b i y ∆ φ − (cid:98) ∆ i (cid:16) − y (cid:126) ∇ (cid:17) n n ! (cid:16) (cid:98) ∆ i + − d (cid:17) n (cid:98) O i ( (cid:126)x ) . (A.4)Conventionally, we choose unit normalization for the boundary modes of φ (cid:104) (cid:98) O i (0) (cid:98) O j ( ∞ ) (cid:105) = δ ij , (A.5)such that b iφ = b φi ≡ b i . A.2 Boundary OPE and physical OPE coefficients
We denote generic boundary operators as (cid:98) O k , where the label k indicates collectively all possibleindices of the local operator. The OPE between two boundary operators (cid:98) O i is (up to boundarydescendants) (cid:98) O i ( (cid:126)x ) (cid:98) O j (0) ∼ (cid:88) k ˆ f ijk | (cid:126)x | (cid:98) ∆ i + (cid:98) ∆ j − (cid:98) ∆ k (cid:98) O k (0) + . . . (A.6)The boundary two-point functions are normalized as in (A.2). We use the Zamolodchikov metric (cid:98) C (cid:98) O (cid:98) O (cid:48) to raise and lower indices of ˆ f ijk s. Concretely (sum over repeated indices) (cid:104) (cid:98) O i ( (cid:126)x ) (cid:98) O j ( (cid:126)x ) (cid:98) O m ( ∞ ) (cid:105) = ˆ f ijk (cid:98) C km | x | (cid:98) ∆ i + (cid:98) ∆ j − (cid:98) ∆ m ≡ ˆ f ijm | x | (cid:98) ∆ i + (cid:98) ∆ j − (cid:98) ∆ m . (A.7)With these conventions, we have that the displacement operator, whose normalization is takenas in (A.3), enters the generic boundary OPE (A.6) as (cid:98) O i ( x ) (cid:98) O j (0) ⊃ ˆ f ij D | x | (cid:98) ∆ i + (cid:98) ∆ j − d D (0) + . . . , (A.8)and a generic boundary four-point function as (cid:104) (cid:98) O i (0) (cid:98) O j ( x ) (cid:98) O k (1) (cid:98) O m ( ∞ ) (cid:105) ⊃ ˆ f ij D ˆ f km D (cid:104) D (0) D ( ∞ ) (cid:105) (1 + . . . )= ˆ f ij D ˆ f km D g (cid:98) ∆ ij , (cid:98) ∆ kl D ( u, v )= ˆ f ij D ˆ f km D C D g (cid:98) ∆ ij , (cid:98) ∆ kl D ( u, v ) . (A.9)In the equation above we introduced the conformal blocks, which are normalized as (5.4). Al-ternatively we can think of D to be unit-normalized, such that the physical boundary OPEcoefficient is (cid:104) (cid:98) O i (0) (cid:98) O j ( x ) (cid:98) O k (1) (cid:98) O m ( ∞ ) (cid:105) ⊃ ˆ λ ij D ˆ λ km D g (cid:98) ∆ ij , (cid:98) ∆ kl D ( u, v ) , ˆ λ ij D = ˆ f ij D √ C D (A.10)Similar remarks apply for other protected operators that can appear in the bOPE of the bulkconserved currents J (cid:96) . 41 Three-point function conformal blocks
In this appendix we derive the conformal block decomposition of the free field φ three-pointfunction with a generic boundary operator (cid:98) O ( l ) (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:98) O ( l ) ( θ, ∞ ) (cid:105) . (B.1)We will obtain closed-form expressions for all the conformal blocks exchanged in the boundarychannel of this three-point function. We also compute some bulk channel blocks, while leaving amore complete study for the future. B.1 Blocks in the boundary channel
We start from the blocks in the boundary channel. As we explained in the main text (seesection 2.2), the expansion of the correlator (B.1) in a basis of boundary conformal blocks canbe obtained by acting twice with the bOPE on the generic three-point functions (cid:104) (cid:98) O i ( (cid:126)x ) (cid:98) O j ( (cid:126)x ) (cid:98) O ( l ) ( θ, ∞ ) (cid:105) = ˆ f ij (cid:98) O ( l ) | (cid:126)x | (cid:98) ∆ i + (cid:98) ∆ j − (cid:98) ∆ P ( l ) (cid:107) (ˆ x , θ ) , (B.2)and then resumming the contributions from boundary descendants. The polynomials P ( l ) (cid:107) aredefined in (2.18). Applying the bOPE (2.4) and using the identity ∇ n(cid:126)x (cid:18) ( − (cid:126)x · θ ) l | (cid:126)x | t (cid:19) = 4 n ( t ) n (cid:18) t − l − d − (cid:19) n ( − (cid:126)x · θ ) l | (cid:126)x | t +2 n , (B.3)we can rewrite (B.1) as (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:98) O ( l ) ( θ, ∞ ) (cid:105) = (cid:88) i,j =1 , b i b j ˆ f ij (cid:98) O ( l ) P ( l ) (cid:107) (ˆ x , θ ) × (cid:88) m,n ( − m + n m ! n ! y (cid:98) ∆ i − ∆ φ +2 n y (cid:98) ∆ j − ∆ φ +2 m | (cid:126)x | − κ ij +2 m +2 n − l × ( − κ ij ) m ( − κ ij + m ) n (cid:16) − κ ij − ˆ h − l (cid:17) m (cid:16) − κ ij + m − ˆ h − l (cid:17) n (cid:16) (cid:98) ∆ i − ˆ h (cid:17) n (cid:16) (cid:98) ∆ j − ˆ h (cid:17) m . (B.4)In the above formula we introduced κ ij ≡ −
12 ( (cid:98) ∆ i + (cid:98) ∆ j − (cid:98) ∆ + l ) , ˆ h ≡ d − . (B.5)42he infinite sum in the r.h.s. of (B.4) can be explicitly performed, and the result takes the form (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:98) O ( l ) ( θ, ∞ ) (cid:105) = P ( l ) (cid:107) (ˆ x , θ ) | (cid:126)x | d − − (cid:98) ∆ × (cid:16) b ˆ f (cid:98) O ( l ) ˆ F (cid:98) ∆ ,l ( w + , w − ) + b b ˆ f (cid:98) O ( l ) ˆ F (cid:98) ∆ ,l ( w + , w − ) + b ˆ f (cid:98) O ( l ) ˆ F (cid:98) ∆ ,l ( w + , w − ) (cid:17) . (B.6)The quantities ˆ F ij (cid:98) ∆ ,l are hypergeometric functions of the cross-ratios w ± (defined in (2.21)) ˆ F (cid:98) ∆ ,l ( w + , w − ) = 12 (cid:34) F (cid:32) − (cid:98) ∆ − l , d − l − (cid:98) ∆2 ; 12 ; − w − (cid:33) + ( w + ↔ w − ) (cid:35) , ˆ F (cid:98) ∆ ,l ( w + , w − ) = 12 (cid:34)(cid:0) ( − l − (cid:1) w − / F (cid:32) − (cid:98) ∆ − l , d + l − (cid:98) ∆ −
12 ; 32 ; − w − (cid:33) + (cid:0) ( − l + 1 (cid:1) w +1 / F (cid:32) − l − (cid:98) ∆2 , d + l − (cid:98) ∆ −
12 ; 32 ; − w + (cid:33)(cid:35) , ˆ F (cid:98) ∆ ,l ( w + , w − ) = 12( (cid:98) ∆ + l − d − l − (cid:98) ∆) × (cid:34) F (cid:32) − (cid:98) ∆ − l , d − l − (cid:98) ∆2 ; 12 ; − w − (cid:33) − ( w − ↔ w + ) (cid:35) . (B.7)Note that in terms of two cross-ratios ξ ≡ | (cid:126)x | + ( y − y ) y y , ζ ≡ ( | (cid:126)x | + y ) y ( | (cid:126)x | + y ) y ∼ x →∞ y y , (B.8)the cross-ratios w ± can be rewritten as w ± = − (1 ± ζ ) − ( ξ + 2) ζ + ζ . (B.9)We have checked that (B.6) satisfies the Klein-Gordon equation with the correct conditions.As a further consistency check, we note that the defect channel blocks for the two-point function(2.8) can be recovered from (B.7) by setting (cid:98) ∆ = l = 0 and ˆ f ij = δ ij . B.2 Scalar blocks in the bulk channel
Next, we will be interested in the bulk conformal block expansion of (B.1). For simplicity we willconsider only the case where the third operator is a boundary scalar, while leaving the generalcase for future work. In the bulk channel we plug the φ × φ OPE (2.9) to convert (B.1) into aninfinite sum over bulk-to-boundary two-point functions (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:98) O ( ∞ ) (cid:105) = (cid:88) O k ⊂ φ × φ c φφ O k (cid:104)O k ( (cid:126)x ) (cid:98) O ( ∞ ) (cid:105) + . . . (B.10)43ith the ellipsis denoting contributions from bulk descendants, which are fixed by SO ( d + 1 , conformal symmetry. As discussed in the main text (see subsection 2.2.2), spin selection rulesand current conservation imply that the bulk operator φ is the only contribution to the r.h.s of(B.10) for generic (cid:98) ∆ not equal to the scaling dimension of J (cid:96) . In this more generic case we have (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:98) O ( ∞ ) (cid:105) = (cid:20) (cid:104) φ ( x ) (cid:98) O ( ∞ ) (cid:105) + . . . (cid:21) ≡ b φ (cid:98) O W φφ (cid:98) O φ ( (cid:126)x , y , y ) . (B.11)Note that when (cid:98) ∆ equals the scaling dimension of J (cid:96) , we should add to the previous expressionan additional contribution proportional to (cid:104) J (cid:96) ( x ) (cid:98) O ( ∞ ) (cid:105) (see e.g. the case of the displacement inC.3). It is not difficult to compute W φφ (cid:98) O φ by plugging the bulk OPE into (2.15) and resummingthe bulk descendants. Using the explicit form of the differential operator that controls the scalarexchange (see e.g. [59]), we find the following series expansion b φ (cid:98) O W φφ (cid:98) O φ ( (cid:126)x , y , y ) = b φ (cid:98) O y φ − (cid:98) ∆2 ∞ (cid:88) n =0 ( − ξ ) n n ! Γ (cid:0) d − (cid:1) Γ(2∆ φ + 2 n − (cid:98) ∆)Γ(2∆ φ − (cid:98) ∆)Γ (cid:0) n + d − (cid:1) × F (cid:18) ∆ φ + n, d − (cid:98) ∆ + 2 n −
2; 2∆ φ + 2 n ; 1 − y y (cid:19) , (B.12)where the cross-ratio ξ is defined in (2.6). There are various interesting special situations inwhich the result (B.12) produces simple closed-form formulae. In the ‘cylindrical’ configuration y = y = y the infinite sum gives a simple hypergeometric function W φφ (cid:98) O φ ( (cid:126)x , y, y ) = 1 y φ − (cid:98) ∆ 2 F (cid:32) d − (cid:98) ∆ − , d − (cid:98) ∆ −
12 ; d −
12 ; − ˆ χ (cid:33) , (B.13)where we introduced a cross-ratio ˆ χ ˆ χ = | (cid:126)x | y , (B.14)which is nothing but the restriction of ξ defined in (2.6) to the ‘cylindrical’ configuration. Asexplained in appendix C.2, this result can be also derived by ‘inverting’ the boundary channelexpansion (B.6). As a final comment, we note that the series representation (B.12) yields simpleclosed-form expressions, some of which will be presented in appendix C.3, when the third operatoris of D (0) (cid:96) type. C OPE relations and bulk-to-boundary crossing
In this appendix we discuss in detail the derivation of the main results presented in section 2.44 .1 Derivation of the OPE relations
In this appendix we derive the OPE relations (2.22). The starting point is the boundary channelexpansion for the correlator (B.1), given in (2.20). Away from other operator insertions, the φ × φ OPE requires this three-point function to be analytic around x µ = x µ (recall that the identityin (2.9) decouples). In order to study this limit, it is convenient to place the two bulk operatorsat the same transverse distance i.e. y = y = y , such that the expression (2.20) simplifies asfollows: (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:98) O ( l ) ( θ, ∞ ) (cid:105) = P ( l ) (cid:107) (ˆ x , θ ) y d − − (cid:98) ∆ ˆ χ − ( d − − (cid:98) ∆) × (cid:40) b ˆ f (cid:98) O ( l ) (cid:34) F (cid:32) − l − (cid:98) ∆2 , d + l − (cid:98) ∆ −
22 ; 12 ; − χ (cid:33)(cid:35) + b b ˆ f (cid:98) O ( l ) [1 + ( − l ] ˆ χ − F (cid:32) − l − (cid:98) ∆2 , d + l − (cid:98) ∆ −
12 ; 32 ; − χ (cid:33) + b ˆ f (cid:98) O ( l ) (cid:98) ∆ + l − (cid:98) ∆ − l − d + 2) (cid:34) − F (cid:32) − (cid:98) ∆ − l , d + l − (cid:98) ∆ −
22 ; 12 ; − χ (cid:33)(cid:35)(cid:41) , (C.1)where ˆ χ is the cross-ratio defined in (B.14). In this configuration with y = y Bose symmetry(2.19) implies that this expression vanishes when l is an odd integer, so we first consider even l .We then require (C.1) to be analytic around (cid:126)x = (cid:126)x , for finite y . For generic values of d, l, (cid:98) ∆ ,the r.h.s. of (C.1) contains unphysical singularities, since (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:98) O ( l ) ( θ, ∞ ) (cid:105) ∼ ˆ χ → ( − (cid:126)x · θ ) l y d + l − − (cid:98) ∆ × (cid:34)
12 ˆ χ (cid:98) ∆ − d − l (cid:32) b ˆ f (cid:98) O ( l ) − b ˆ f (cid:98) O ( l ) ( (cid:98) ∆ + l − d − (cid:98) ∆ + l − (cid:33) + √ π Γ (cid:0) d − + l (cid:1) − l − (cid:98) ∆ ˆ χ d − + l b b ˆ f (cid:98) O ( l ) Γ (cid:16) l + (cid:98) ∆+12 (cid:17) Γ (cid:16) d + l − (cid:98) ∆ − (cid:17) + b ˆ f (cid:98) O ( l ) ( (cid:98) ∆ + l − d − (cid:98) ∆ + l −
2) + b ˆ f (cid:98) O ( l ) ( (cid:98) ∆ + l − (cid:16) l + (cid:98) ∆2 (cid:17) Γ (cid:16) d + l − (cid:98) ∆2 (cid:17) + . . . , (C.2)45p to higher powers of ˆ χ . Such unphysical singularities cancel from the r.h.s. of (C.1) preciselywhen the OPE relations (2.22) are satisfied, such that the analytic result at y = y is (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:98) O ( l ) ( θ, ∞ ) (cid:105) = b b ˆ f (cid:98) O ( l ) P ( l ) (cid:107) (ˆ x , θ ) y d − − (cid:98) ∆ × √ π Γ (cid:0) − d − l (cid:1) (cid:104) − cot (cid:16) π ( (cid:98) ∆ + l ) (cid:17) cot (cid:16) π ( d − (cid:98) ∆ + l ) (cid:17)(cid:105) d − (cid:98) ∆+ l − Γ (cid:16) − l + (cid:98) ∆2 (cid:17) Γ (cid:16) (cid:98) ∆ − d − l +42 (cid:17) × ˆ χ l/ F (cid:32) d + l − (cid:98) ∆ − , d + l − (cid:98) ∆ −
12 ; d −
12 + l ; − ˆ χ (cid:33) . (C.3)When (cid:98) ∆ approaches some special integer dimensions some of the boundary blocks in the r.h.s.of (C.1) are themselves regular and (C.2) is not valid. This can happen when: • The dimension of (cid:98) O equals that of a double-twist combination of (cid:98) O and (cid:98) O (cid:98) ∆ = d + l + 2 n − , n ∈ N , (C.4) • The dimension of (cid:98) O equals that of a double-twist combination of (cid:98) O and (cid:98) O (cid:98) ∆ = d + l + 2 n, n ∈ N , (C.5) • The dimension of (cid:98) O equals that of a double-twist combination of (cid:98) O and (cid:98) O (cid:98) ∆ = d + l + 2 n − , n ∈ N . (C.6)We then analyse these special cases separately. Requiring the cancellation of any residual sin-gularity on the r.h.s. of (C.1), will again impose certain relations between the boundary OPEcoefficients. It is reassuring to see that these special cases are captured by the appropriate limitsof the general result (2.22).We now discuss the case when l is an odd integer. Starting from (2.20), we need to set ˆ f (cid:98) O ( l ) =ˆ f (cid:98) O ( l ) = 0 (as dictated by Bose symmetry) so that the three-point function is proportional to ˆ f (cid:98) O ( l ) . We then study analyticity around (cid:126)x = (cid:126)x for finite y ≡ y − y . For generic values of d, l, (cid:98) ∆ this correlator features unphysical singularities, since (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:98) O ( l ) ( θ, ∞ ) (cid:105) ∼ (cid:126)x → b b ˆ f (cid:98) O ( l ) ( − (cid:126)x · θ ) l × (cid:18) − √ π (cid:19) y (cid:98) ∆+ l − | (cid:126)x | d +2 l − Γ (cid:0) d − + l (cid:1) Γ (cid:16) l + (cid:98) ∆+12 (cid:17) Γ (cid:16) d + l − (cid:98) ∆ − (cid:17) + Γ (cid:0) − d − l (cid:1) ( y ) − d + (cid:98) ∆ − l Γ (cid:16) − l − (cid:98) ∆2 (cid:17) Γ (cid:16) − d − l + (cid:98) ∆2 (cid:17) + . . . , (C.7)46p to subleading terms. Because of the first term in the above expression, which is singular for d ≥ (for d = 3 and l = 0 the singularity is logarithmic), for generic (cid:98) ∆ we must set ˆ f (cid:98) O ( l ) = 0 .On the other hand, the boundary blocks are themselves regular and the (C.7) is not valid whenthe dimension of (cid:98) O equals (cid:98) ∆ = d + l + 2 n − , n ∈ N . (C.8)Again we see that the relations (2.22), together with the constraints from Bose symmetry (2.19),promptly capture these special cases. The analytic correlator (2.20) on the special dimensions(C.8) then reads (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:98) O ( l ) ( θ, ∞ ) (cid:105) = b b ˆ f (cid:98) O ( l ) ( − (cid:126)x · θ ) l × (cid:18) − √ π (cid:19) Γ (cid:0) − d − l (cid:1) Γ (cid:0) n + (cid:1) Γ (cid:0) − d − l − n (cid:1) y n +112 2 F (cid:18) − n − , − n ; d −
12 + l ; − w − (cid:19) . (C.9) C.2 Matching with the bulk
Owing the results from the previous subsection, we are now ready to discuss the consequencesof the bulk-boundary crossing symmetry for the three-point functions (B.1).The first step is to derive the leading terms in the bulk channel expansion of the correlator(B.1). To this end, recall that the φ × φ OPE (2.9) contains a scalar φ and infinitely manyconserved currents J (cid:96) , with (cid:96) ∈ N and ∆ (cid:96) = d + (cid:96) − . The bulk-boundary two-point functionsbetween J (cid:96) and any boundary operator (cid:98) O ( l ) are further constrained by current conservation. Theoperator (cid:98) O ( l ) can appear in the bOPE of J (cid:96) if (cid:104) ∂ µ J µµ ...µ (cid:96) − (cid:96) ( (cid:126)x, y ) (cid:98) O a ...a l (0) (cid:105) = 0 . (C.10)For l < (cid:96) , this condition is satisfied only if (cid:98) ∆ = ∆ (cid:96) , so that (cid:98) O is a protected boundary primary.On the other hand, for l = (cid:96) , conservation is automatically ensured with no extra conditions on (cid:98) ∆ , so (cid:98) O is unprotected. This is of course compatible with the Ward identities (2.23).We now plug the φ × φ OPE into (B.1), impose the selection rules from conservation in orderto figure out which bulk primary can couple to (cid:98) O ( l ) and finally compare to the boundary channelexpansion. We conclude that: • When (cid:98) ∆ (cid:54) = d + (cid:96) − and l is odd, (cid:98) O ( l ) cannot couple to any bulk operator in the φ × φ , andthe three-point function must vanish. This perfectly matches with the expectations fromthe boundary channel. Indeed, (cid:104) ( ∂J (cid:96) ) (cid:98) O (cid:96) (cid:105) = (cid:104) J (cid:96) − (cid:98) O (cid:96) (cid:105) must vanish since J (cid:96) − does not contain any spin (cid:96) component in its bOPE. When (cid:98) ∆ (cid:54) = d + (cid:96) − and l is even, (cid:98) O ( l ) can only couple to a spin l bulk current J l (or to φ when l = 0 ). This is consistent with what we expect from the the boundary channel,where we are left with only one unknown OPE coefficient ˆ f (cid:98) O ( l ) . In either case, from theleading bulk OPE we have (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) (cid:98) O ( l ) ( θ, ∞ ) (cid:105) ∼ (cid:126)x → c φφl b J l (cid:98) O ( l ) C J l P ( l ) (cid:107) (ˆ x , θ ) ˆ χ l/ y d − − (cid:98) ∆ + . . . , (C.11)and, after comparing to (C.3) we find c φφl b J l (cid:98) O ( l ) C J l = b b ˆ f (cid:98) O ( l ) √ π Γ (cid:0) − d − l (cid:1) (cid:104) − cot (cid:16) π ( (cid:98) ∆ + l ) (cid:17) cot (cid:16) π ( d − (cid:98) ∆ + l ) (cid:17)(cid:105) d − − (cid:98) ∆+ l Γ (cid:16) − l + (cid:98) ∆2 (cid:17) Γ (cid:16) (cid:98) ∆ − l − d +42 (cid:17) . (C.12)The result for a scalar ( l = 0 ) operator (cid:98) O is simply obtained from the former by setting c φφ = C J = 1 and b J (cid:98) O (0) ≡ b φ (cid:98) O .We can use the result above in order to re-interpret the expression for (B.1) obtained usingthe boundary OPE in terms of the bulk OPE, and compute the corresponding bulk block.This procedure is unambiguous, since in both channels there is just one undetermined OPEcoefficient. In practice, we solve (C.12) for ˆ f (cid:98) O ( l ) and plug the result into (C.3). We find W φφ (cid:98) O ( l ) J l ( (cid:126)x , y, y )= P ( l ) (cid:107) (ˆ x , θ ) ˆ χ l/ y φ − (cid:98) ∆ 2 F (cid:32) d + l − (cid:98) ∆ − , d + l − (cid:98) ∆ −
12 ; d + 2 l −
12 ; − ˆ χ (cid:33) . (C.13)As a consistency check, note that for l = 0 the above expression reproduces the block W φφ (cid:98) O φ ( (cid:126)x , y, y ) , which was computed explicitly in (B.13). The same logic can be appliedto compute the bulk blocks at generic transverse positions y , y , starting from the boundarychannel decomposition (2.20). • When (cid:98) ∆ = d + (cid:96) − and l is even there are two cases. For (cid:96) > l , the primary (cid:98) O ( l ) can coupleto both J l and J (cid:96) . The number of undetermined bulk OPE coefficients then matches thatof the boundary ones ( ˆ f (cid:98) O ( l ) and ˆ f (cid:98) O ( l ) ). As an example, in section C.3 we explicitly solvethe bulk-to-boundary bootstrap for the case of (cid:96) = 2 with l = 0 , but similar results can beobtained for the more general case of D ( l ) (cid:96) . When (cid:96) = l the operator (cid:98) O ( l ) can only couple to J l , and this matches with the number of undetermined boundary OPE coefficients ( ˆ f (cid:98) O ( l ) ). • When (cid:98) ∆ = d + (cid:96) − and l is odd the only possible bulk contribution comes from the spin (cid:96) currents J (cid:96) . From the leading bulk OPE limit at | (cid:126)x | = 0 we have (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) V ( l ) (cid:96) ( θ, ∞ ) (cid:105) ∼ x → c φφ(cid:96) C J (cid:96) b J (cid:96) V ( l ) (cid:96) ( y ) (cid:96) − l ( (cid:126)x · θ ) l + . . . , (C.14)48here y = y − y . So, after comparing to (C.9) (note that (cid:96) − l = 2 n + 1 ) we find c φφ(cid:96) b J (cid:96) V ( l ) (cid:96) C J (cid:96) = b b ˆ f V ( l ) (cid:96) √ π Γ (cid:0) − d − l (cid:1) (cid:0) (cid:96) − l +22 (cid:1) Γ (cid:0) − d − (cid:96) − l (cid:1) . (C.15)The Ward identity (2.23) further relates the coefficient b J (cid:96) V ( l ) (cid:96) to the coefficient in the two-point function of V ( l ) (cid:96) , e.g. for the flux operator V (1)2 ≡ V (1) the coefficient in eq. (2.31).In the case of V (1) , upon plugging the value of c φφT and C T given in eq. (2.10), the result(C.15) gives precisely the first equality of eq. (2.32). The second equality is obtained uponusing that b T V (1) = 2 (cid:98) C V (1) , as dictated by the Ward identity (2.23). C.3 Displacement Ward identity
In this appendix we derive the displacement Ward identity presented in section 2.2.3. The startingpoint is the three-point function of the displacement operator D with the free bulk scalar. From(2.20) and after imposing the OPE relations (2.22), this reads (cid:104) φ ( (cid:126)x , y ) φ ( (cid:126)x , y ) D ( ∞ ) (cid:105) = y y b ˆ f D + b ˆ f D (cid:2) | (cid:126)x | − ( d − (cid:0) y + y (cid:1)(cid:3) . (C.16)We want to match this expression against the bulk OPE channel expansion. As we discussed inthe main text – see around (2.34) – this receives a contribution from the φ as well as from thestress tensor. The complete expression, i.e. including contributions from bulk descendants, is (cid:104) φ ( x ) φ ( x ) D ( ∞ ) (cid:105) = b φ D W φφ D φ ( (cid:126)x , y , y ) + c φφT C T x µ x ν (cid:104) T µν ( x ) D ( ∞ ) (cid:105) . (C.17)The first term in the r.h.s. of the above equation is the (cid:104) φ D (cid:105) bulk block, which is computed by(B.12) W φφ D φ ( (cid:126)x , y , y ) = ( d − y + y ) − | (cid:126)x | d − . (C.18)The second term is the contribution from the bulk stress tensor and reads [5, 6] (cid:104) T µν ( x ) D ( ∞ ) (cid:105) = b T D (cid:18) δ µy δ νy − d δ µν (cid:19) , b T D = d C D d − . (C.19)Note bulk descendant operators of T µν do not enter into (C.17), since (C.19) is a constant. Onecan further use the Ward identities for the displacement operator [5, 6] to relate the bOPEcoefficient b φ D to the one-point function of φ : b φ D = − a φ d ( d − S d , S d ≡ Vol ( S d − ) = 2 π d/ Γ (cid:0) d (cid:1) . (C.20)49e can now equate (C.16) to (C.17) and solve for ˆ f D and ˆ f D . The result is ˆ f D = a φ d C T ( d − − C D c φφT S d C T ( d − S d b , ˆ f D = − a φ d C T ( d −
2) + 4 C D c φφT S d C T S d b . (C.21)The final formula (2.36) is obtained by plugging in the above expression the values (2.10) of c φφT and C T corresponding to a d -dimensional free scalar field. It is pleasant to see that the finalresult (2.36) is consistent with the Ward identity [5] (cid:90) d d − (cid:126)x (cid:104) φ ( x ) φ ( x ) D ( (cid:126)x ) (cid:105) = ( ∂ y + ∂ y ) (cid:104) φ ( x ) φ ( x ) (cid:105) . (C.22) D Crossing equations in a vectorial form
In section 4.1 we derived the system of 7 independent crossing equations (4.6). The latter canbe rewritten in a vectorial form by introducing the 7-component vectors of × matrices (cid:126)V (cid:98) ∆ ,l (cid:88) (cid:98) O ,l (cid:16) ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) (cid:17) (cid:126)V (cid:98) ∆ ,l ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) . (D.1)The explicit form of (cid:126)V (cid:98) ∆ ,l is given in (D.4). The odd l terms in the above expression are subjectedto further restrictions. Firstly, Bose symmetry implies that ˆ f (cid:98) O ( l ) = ˆ f (cid:98) O ( l ) = 0 . Secondly, theodd- l primaries must have scaling dimensions (cid:98) ∆ = d + l + 2 n − , with n ∈ N , as follows fromthe exact relations (2.22). It is then convenient to rewrite (D.1) as (cid:88) (cid:98) O ,l = even (cid:16) ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) (cid:17) (cid:126)V (cid:98) ∆ ,l ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) + (cid:88) l = odd (cid:98) ∆= d + l +2 n − n =0 , ,... ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) (cid:16) (cid:17) (cid:126)V (cid:98) ∆ ,l . (D.2)For numerical purposes it is convenient to isolate, in the crossing equations above, the contri-butions from the primaries with fixed dimensions from those in the continuum. Such specialprimaries are the identity (for which (2.22) implies ˆ f = 0 and we choose the normalization ˆ f ii ≡ ˆ f ii = 1 ), as well as the boundary modes of the bulk higher-spin currents, D ( l ) (cid:96) and V ( l +1) (cid:96) of spin l and l + 1 in the notation of section 2.2.2. Upon implementing the exact relations (2.22),50e rewrite (D.2) as follows (cid:16) (cid:17) (cid:126)V , (cid:124) (cid:123)(cid:122) (cid:125) ≡ (cid:126)V + (cid:88) (cid:98) O ,l = even ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) (cid:16) κ ( (cid:98) ∆ , l ) 1 κ ( (cid:98) ∆ , l ) (cid:17) (cid:126)V (cid:98) ∆ ,l κ ( (cid:98) ∆ , l )1 κ ( (cid:98) ∆ , l ) (cid:124) (cid:123)(cid:122) (cid:125) ≡ (cid:126)V + , (cid:98) ∆ ,l + (cid:88) l = odd (cid:98) ∆= d + l +2 n − n =0 , ,... ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) (cid:16) (cid:17) (cid:126)V (cid:98) ∆ ,l (cid:124) (cid:123)(cid:122) (cid:125) ≡ (cid:126)V − , (cid:98) ∆ ,l + (cid:88) (cid:96) ∈ N (cid:88) l<(cid:96), even (cid:98) ∆= d + (cid:96) − (cid:16) ˆ f (cid:98) O ( l ) f (cid:98) O ( l ) (cid:17) (cid:126)V (cid:98) ∆ ,l ˆ f (cid:98) O ( l ) f (cid:98) O ( l ) (cid:124) (cid:123)(cid:122) (cid:125) ≡ (cid:18) ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) (cid:19) (cid:126)V , (cid:98) ∆ ,l ˆ f (cid:98) O ( l ) ˆ f (cid:98) O ( l ) . (D.3)The first line in the expression above accounts for the identity as well as for the unprotected,even-spin operators. The second line accounts for the odd-spin operators i.e. belonging to thefamily V ( l ) (cid:96) . The third line contains even-spin protected operators i.e. belonging to the familyD ( l ) (cid:96) . The 7-component vectors (cid:126)V ± , (cid:98) ∆ ,l , (cid:126)V , (cid:98) ∆ ,l are defined in (D.5),(D.6). The quantities (cid:126)V (cid:98) ∆ ,l are7-component vectors of × matrices defined in (D.7).51 V (cid:98) ∆ ,l = F , − , (cid:98) ∆ ,l ( u, v ) 0 00 0 00 0 0 F , − , (cid:98) ∆ ,l ( u, v ) F , − , (cid:98) ∆ ,l ( u, v ) 0 F , − , (cid:98) ∆ ,l ( u, v ) 0 00 0 0 F , − , (cid:98) ∆ ,l ( u, v )0 F , − , (cid:98) ∆ ,l ( u, v ) 0 F , − , (cid:98) ∆ ,l ( u, v ) 00 0 0 F , − , (cid:98) ∆ ,l ( u, v )0 ( − s F , − , (cid:98) ∆ ,l ( u, v ) 0 F , − , (cid:98) ∆ ,l ( u, v ) 0 0 F , , (cid:98) ∆ ,l ( u, v )0 − ( − s F , , (cid:98) ∆ ,l ( u, v ) 0 F , , (cid:98) ∆ ,l ( u, v ) 0 0 , (D.4) (cid:126)V = F , − , (cid:98) ∆ ,l ( u, v ) F , − , (cid:98) ∆ ,l ( u, v )000 F , − , (cid:98) ∆ ,l ( u, v ) F , , (cid:98) ∆ ,l ( u, v ) , (cid:126)V − , (cid:98) ∆ ,l = F , − , (cid:98) ∆ ,l ( u, v ) − F , − , (cid:98) ∆ ,l ( u, v ) F , , (cid:98) ∆ ,l ( u, v ) , (D.5) (cid:126)V + , (cid:98) ∆ ,l = κ ( (cid:98) ∆ , l ) F , − , (cid:98) ∆ ,l ( u, v ) κ ( (cid:98) ∆ , l ) F , − , (cid:98) ∆ ,l ( u, v ) κ ( (cid:98) ∆ , l ) F , − , (cid:98) ∆ ,l ( u, v ) κ ( (cid:98) ∆ , l ) F , − , (cid:98) ∆ ,l ( u, v ) F , − , (cid:98) ∆ ,l ( u, v ) κ ( (cid:98) ∆ , l ) κ ( (cid:98) ∆ , l ) F , − , (cid:98) ∆ ,l ( u, v ) + F , − , (cid:98) ∆ ,l ( u, v ) κ ( (cid:98) ∆ , l ) κ ( (cid:98) ∆ , l ) F , , (cid:98) ∆ ,l ( u, v ) − F , , (cid:98) ∆ ,l ( u, v ) , (D.6)52 V , (cid:98) ∆ ,l = (cid:32) F , − , (cid:98) ∆ ,l ( u, v ) 00 0 (cid:33)(cid:32) F , − , (cid:98) ∆ ,l ( u, v ) (cid:33)(cid:32) (cid:33)(cid:32) (cid:33)(cid:32) (cid:33)(cid:32) F , − , (cid:98) ∆ ,l ( u, v ) F , − , (cid:98) ∆ ,l ( u, v ) 0 (cid:33)(cid:32) F , , (cid:98) ∆ ,l ( u, v ) F , , (cid:98) ∆ ,l ( u, v ) 0 (cid:33) . (D.7) References [1] J. L. Cardy,
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