Aspects of CFTs on Real Projective Space
PPrepared for submission to JHEP
Aspects of CFTs on Real Projective Space
Simone Giombi a , Himanshu Khanchandani a , Xinan Zhou b a Department of Physics, Princeton University, Princeton, NJ 08544, USA b Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA
Abstract:
We present an analytic study of conformal field theories on the real projectivespace RP d , focusing on the two-point functions of scalar operators. Due to the partiallybroken conformal symmetry, these are non-trivial functions of a conformal cross ratio andare constrained to obey a crossing equation. After reviewing basic facts about the structureof correlators on RP d , we study a simple holographic setup which captures the essentialfeatures of boundary correlators on RP d . The analysis is based on calculations of Wittendiagrams on the quotient space AdS d +1 / Z , and leads to an analytic approach to two-pointfunctions. In particular, we argue that the structure of the conformal block decompositionof the exchange Witten diagrams suggests a natural basis of analytic functionals, whoseaction on the conformal blocks turns the crossing equation into certain sum rules. We testthis approach in the canonical example of φ theory in dimension d = 4 − (cid:15) , extracting theCFT data to order (cid:15) . We also check our results by standard field theory methods, both inthe large N and (cid:15) expansions. Finally, we briefly discuss the relation of our analysis to theproblem of construction of local bulk operators in AdS/CFT. a r X i v : . [ h e p - t h ] S e p ontents RP d AdS d +1 / Z RP d CFTs 27 RP d free energy 45 Quantum field theories on non-orientable spacetime have several physical applications, andhave been studied from many different perspectives. They are an integral part of string the-ory in the description of unoriented worldsheets [1–3]. Studying theories on non-orientablemanifolds also probes the realization of time reversal symmetry [4–6] (see, e.g. , [7] fordiscussions on a refinement of electric-magnetic duality in abelian gauge theories), which– 1 –lays important roles in condensed matter physics. They also make appearances in formalstudies of quantum field theory. For example, partition functions of two dimensional CFTson non-orientable surfaces were studied in [8], where holographic connections with threedimensional geometries were explored. Their role in supersymmetric quantum field theorieswas discussed, e.g. , in [9, 10]. Recently, there has been considerable interest in studyingCFTs on real projective space – one of the simplest examples of non-orientable manifolds. This is partly in light of the modern nonperturbative conformal bootstrap (see [11, 12] forreviews) [13–16], where CFTs on real projective space provide attractive playgrounds fordeveloping and testing new techniques. Moreover, the study of such theories is also fueledby the program of constructing bulk local operators in AdS [17–22], where crosscap statesare proposed to be dual to fields inserted at a bulk point. In this paper, we continue theanalytic study of conformal field theories on real projective space, and revisit the problemfrom multiple angles.The real projective space RP d can be defined by a Z quotient of a sphere S d X = 1 , X ∈ R d +1 , with X ∼ − X . (1.1)Equivalently, since our focus is on CFT, we can perform a Weyl transformation and mapit to the flat space x µ = X µ − X d +1 , µ = 1 , . . . , d ds R d = (1 + x ) ds S d . (1.2)The real projective space is then represented as R d under the identification x µ → − x µ x (1.3)where x µ are the Cartesian coordinates on R d . Unless otherwise stated, in this paper wedenote by RP d the quotient of flat space by the inversion (1.3).Putting CFTs on RP d partially breaks the conformal symmetry and introduces newobservables. Scalar operators can have non-vanishing one-point functions (cid:104)O ∆ (cid:105) = a O (1 + x ) ∆ O . (1.4)The coefficients a O are new data that defines the CFT on real projective space, alongwith the standard operator spectrum and OPE coefficients, which remain the same as on R d . Moreover, two-point functions are no longer fixed by symmetry, but instead becomefunctions of a cross ratio η invariant under the residual conformal symmetry (cid:104)O ( x ) O ( x ) (cid:105) = G ( η )(1 + x ) ∆ (1 + x ) ∆ , (1.5)where η = ( x − x ) (1 + x )(1 + x ) . (1.6) More precisely, RP d is unorientable for d even, and orientable for d odd. – 2 –he reader might notice the breaking of conformal symmetry and the structure of correlatorsare very reminiscent of those of boundary CFTs [23, 24]. We will point out more similaritieslater in the paper. Analogous to four-point functions on R d , two-point functions of RP d CFTs obey a crossing equation because of the identification (1.3) G ( η ) = ± G (1 − η ) (1.7)where ± corresponds to the two choices O , → ± O , under the inversion. The operatorproduct expansion in the direct channel ( η → ), and in the mirror channel ( η → )allows them to be expanded in terms of conformal blocks in the respective channels. Thecrossing equation together with the conformal block decomposition then impose nontrivialconstraints on the structure of correlators. Two-point functions are therefore the primetarget for developing an analytic understanding of CFTs on real projective space.In this paper, we develop an analytic approach to two-point functions, which is universalfor RP d CFTs. However, to develop this method we will take a holographic detour. We firstlift the quotient (1.3) into the bulk as a Z quotient of AdS d +1 , and study a toy model forholography in this setup. We define the tree-level Witten diagrams in this background, andstudy their various properties. In particular, we consider in detail the two-point conformalblock decompositions of exchange Witten diagrams in the two channels. The structure ofthe conformal block decompositions suggests a natural basis for the function space of two-point correlators, which consists of special conformal blocks with discrete ‘double-trace’dimensions in both the direct and the mirror channel. The dual of this basis is a basisof analytic functionals, whose actions on a generic conformal block can be read off fromthe conformal block decomposition coefficients of the exchange Witten diagrams. Actingon the crossing equation (1.7) with the functionals allows us to extract the complete setof constraints in the form of sum rules. These sum rules are valid non-perturbatively.But they become especially simple around the mean field theory spectrum, and essentiallytrivialize the study of perturbations around mean field theory. We demonstrate the useof the analytic functionals on the model of φ theory in − (cid:15) dimensions. By solving thefunctional sum rules, we obtain the one-point function coefficients to the order (cid:15) . Setting (cid:15) = 1 , we find good agreement with the numerical bootstrap estimation for the 3d Isingmodel [13].We also develop perturbative field theory approaches to study CFTs on RP d , withthe O ( N ) vector model being our main example. We study the two-point function ofthe fundamental scalar φ of the O ( N ) model both in the large N expansion in arbitrarydimension, and in the (cid:15) expansion in d = 4 − (cid:15) dimensions. In d = 4 − (cid:15) , instead of usingthe usual loop expansions, we exploit the fact that φ satisfies an equation of motion. Theequation of motion implies a differential equation obeyed by the two-point function, whichcan be solved in perturbation theory to order (cid:15) . The essential idea of using equationsof motion to obtain CFT data was described in [25] and here we extend it to CFTs on RP d . The field theory calculations provide an independent test of the results obtained fromanalytic functionals.We also discuss other interesting features of RP d CFT. We point out a two-term ‘di-mensional reduction’ formula for conformal blocks, which expresses a conformal block in– 3 – − dimensions as the sum of two d dimensional conformal blocks with shifted conformaldimensions. An analogous five-term relation was found in [26] for CFT four-point functionsin R d , and was shown to be a consequence of the Parisi-Sourlas supersymmetry [27]. Theappearance of the dimensional reduction relation therefore suggests a possible extensionof the Parisi-Sourlas supersymmetry to real projective space. Moreover, following [28],we show that the dimensional reduction formula for conformal blocks can be extended toexchange Witten diagrams as well.The setup of our toy model for holography is also closely related to the Hamilton-Kabat-Lifschytz-Lowe (HKLL) approach for constructing local bulk operators [29–31]. Thetwo problems have the same partially broken conformal symmetry. We will make a fewcomments on how our results are relevant in the bulk reconstruction problem. In particu-lar, we point out that the bulk reconstruction of the bulk-boundary-boundary three-pointfunction can be reformulated as a conformal bootstrap problem, which can be solved usingour functionals.The rest of the paper is organized as follows. In Section 2, we review the kinematics of RP d CFT using the embedding formalism. We set up the holography toy model on the Z quotient of AdS in Section 3, and define various Witten diagrams. In Section 4 we performa detailed analysis of the tree-level two-point exchange Witten diagrams: we evaluate themin a closed form and study their conformal block decompositions. In Section 5 we studythe dimensional reduction of conformal blocks and exchange Witten diagrams. We developa functional method to RP d CFTs in Section 6, and use the method in a few perturbativeapplications. We also present a complementary field theory approach using the equation ofmotion method. The relation to bulk reconstruction is discussed in Section 7. We concludein Section 8 with a brief discussion of future directions. In Appendix A, we compute thefree energy on RP d = S d / Z for O ( N ) models; the calculation has some connection to thecontent of Section 6.4, but is mostly independent from the main text of the paper and canbe read separately. RP d It is convenient to introduce the embedding space which linearizes the action of the con-formal group. Let us first review the case where the space is just R d − , . For any point x µ ∈ R d − , , we can represent it as a null ray P A ( A = 1 , , . . . , d + 2 ) in R d, P · P = 0 , P ∼ λ P . (2.1)Operators are defined on the space of null rays , with the scaling property O ∆ ( λP ) = λ − ∆ O ∆ ( P ) . (2.2) For simplicity we focus here on scalar operators. – 4 –or definiteness, let us choose the signature of R d, to be ( − , + , − , + , . . . , +) . We canexplicitly parameterize P A with the R d − , coordinates as P A = (cid:18) x , − x , x µ (cid:19) , (2.3)after fixing a gauge for the rescaling freedom P = 12 (1 + x ) . (2.4)The conformal group SO ( d, acts on P A as linear rotations in R d, P A → Ω AB P B . (2.5)The conformal transformation on the x µ coordinates is obtained after restoring the gaugefixing condition (2.4) by an appropriate rescaling.The inversion (1.3) can be conveniently represented in terms the embedding coordi-nates, where it flips the sign of the last d + 1 components of the embedding vector I : P → P , P a → − P a , a = 2 , , . . . , d + 2 . (2.6)To go back to (2.4) we must multiply the null vector with a factor x − , and one can easilycheck that this reproduces the transformation (1.3). Under inversion operators are identifiedby I : O ± ∆ ( x ) → ± x O ± ∆ ( x (cid:48) ) , x (cid:48) µ = − x µ x . (2.7)The insertion of a crosscap introduces a fixed time-like embedding vector N c = (1 , , , . . . , . (2.8)The residual conformal symmetry after inserting the crosscap is all the SO ( d, rotationsthat leaves N c invariant. It is useful to compare the situation with the closely related caseof CFTs with a conformal boundary. The presence of a spherical boundary centered at x = 0 with unit radius, is represented in the embedding space by introducing a space-like constant vector N B = (0 , , , , . . . , . (2.9)The conformal boundary breaks the SO ( d, conformal group down to the subgroup SO ( d − , , which consists of all the rotations in the embedding space preserving the vector N B . The embedding space formalism introduced in the last section makes it straightforward todiscuss the kinematics of CFT correlators on RP d . Correlators are constructed using the SO ( d, invariants of the embedding vectors, and must scale properly according to (2.2). For Euclidean spacetime R d the embedding space is R d +1 , , and we choose the signature to be ( − , + , + , + , . . . , +) . – 5 –et us start with one-point functions. The only invariant one can write down is ( − N c · P ) , and scaling requires the one-point function must be of the form (cid:104)O ∆ ( x ) (cid:105) = a O ( − N c · P ) ∆ = a O (1 + x ) ∆ . (2.10)Using the Weyl transformation (1.2), this implies that on the Z quotient of the sphere, theone-point functions are constant (cid:104)O ∆ ( x ) (cid:105) S d / Z = a O ∆ . (2.11)Note that under the inversion (2.7), the operator O must transform with the + sign in orderfor the one-point function to be nonvanishing. This can be clearly seen from the fact thatthe one-point function is a constant on the sphere S d / Z , and we note that antipodal pointson the sphere are identified by inversion. The choice of the − sign leads to a zero valuefor a O . More generally, it follows from the fact that the total Z charge under inversionmust be zero in a correlator. Therefore, one-point functions are completely determined bysymmetry up to a constant a O . The constant a O is a new CFT data, and encodes dynamicalinformation of CFTs on real projective space. We should also point out that only scalaroperators can obtain nonzero one-point function. Operators with spin must have vanishingone-point functions, because a nonzero one-point function is inconsistent with the residualsymmetry (a completely analogous result holds in BCFT).We now discuss two-point functions, which are the main focus of this paper. With theembedding vectors P , P and N c , we can construct a cross ratio η = ( − P · P )( − N c · P )( − N c · P ) = ( x − x ) (1 + x )(1 + x ) , (2.12)which is also invariant under the independent rescaling of each operator. In Euclideanspacetime, the cross ratio takes values in η ∈ [0 , . After extracting a kinematic factorwhich takes care of the scaling property, we can write the two-point function as a functionof the cross ratio (cid:104)O ∆ ( x ) O ∆ ( x ) (cid:105) = G ( η )( − N c · P ) ∆ ( − N c · P ) ∆ = G ( η )(1 + x ) ∆ (1 + x ) ∆ . (2.13)Here we have suppressed the parity of the operators under Z . The correlator is onlynonzero when O ∆ and O ∆ have the same parity, in order for the correlator to have a zerooverall Z charge. Moreover, because of the operator identification (2.7) the correlator G ( η ) must satisfy the following crossing equation [13] G ( η ) = ± G (1 − η ) , (2.14)where ± denotes the common parity of O ∆ and O ∆ . Here and elsewhere, the upper signrefers to the + parity and the lower sign to − parity. Also note again that under the Weyltransform (1.2), the two-point function on S d / Z just becomes G ( η ) / ∆ +∆ .– 6 –here are several points of interests for the two-point function on the η -plane. The firstis the limit η = 0 . Physically, it means that the two operators coincide in Euclidean space-time (or light-like separated in Lorentzian spacetime). In the limit of coinciding operators,we can use the standard OPE O ∆ ( x ) O ∆ ( x ) = δ ∆ ∆ ( x − x ) + (cid:88) k C k D [ x − x , ∂ x ] O ∆ k ( x ) (2.15)where k labels the conformal primaries, and C k is the OPE coefficient. The differentialoperator D [ x − x , ∂ x ] is completely determined by the conformal symmetry. The OPEreduces the two-point function to one-point functions which are completely determined bykinematics up to an overall constant. The contribution of each primary operator in theOPE to the two-point function can be resummed into a conformal block [20] g ∆ ( η ) = η ∆ − ∆1 − ∆22 F (cid:18) ∆ + ∆ − ∆ , ∆ + ∆ − ∆ − d η (cid:19) . (2.16)The conformal block can also be obtained more conveniently from solving the conformalCasimir equation L (cid:104)O ∆ ( P ) O ∆ ( P ) (cid:105) = − ∆(∆ − d ) (cid:104)O ∆ ( P ) O ∆ ( P ) (cid:105) , (2.17)with the boundary condition g ∆ ( η ) ∼ η ∆ − ∆1 − ∆22 , η → . (2.18)The Casimir operator L = 12 ( L AB + L AB )( L ,AB + L ,AB ) (2.19)is constructed from the SO ( d, generators L AB = P A ∂∂P B − P B ∂∂P A . (2.20)In terms of these conformal blocks, the two-point function can be written as G ( η ) = (cid:88) k µ k g ∆ k ( η ) (2.21)where µ k = a k C k . (2.22)Similarly, η = 1 is also an interesting point where one operator approaches the imageof the other operator (or the lightcone of the image). We can again apply the OPE foran operator with an image operator, which is equivalent to the original OPE thanks to(2.7). We will refer to this channel as the image channel . The two-point function can bedecomposed into the image conformal blocks ¯ g ∆ ( η ) = (1 − η ) ∆ − ∆1 − ∆22 F (cid:18) ∆ + ∆ − ∆ , ∆ + ∆ − ∆ − d − η (cid:19) . (2.23)– 7 – P d CFT BCFT d One-point function (cid:104)O ∆ ( x ) (cid:105) = a O (1+ x ) ∆ (cid:104)O ∆ ( x ) (cid:105) B = a B, O | − x | ∆ Two-point cross ratio η = ( x − x ) (1+ x )(1+ x ) ξ = ( x − x ) (1 − x )(1 − x ) Two-point function G = x ) ∆1 (1+ x ) ∆2 G ( η ) G B = | − x | ∆1 | − x | ∆2 G B ( ξ ) OPE limits η → (bulk channel) ξ → (bulk channel) η → (image channel) ξ → ∞ (boundary channel)Regge limit η → ∞ ξ → − Table 1 : A comparison of kinematics for RP d CFTs and boundary CFTs.These image conformal blocks are eigenfunctions of the image conformal Casimir equation ¯ L (cid:104)O ∆ ( P ) O ∆ ( P ) (cid:105) = − ∆(∆ − d ) (cid:104)O ∆ ( P ) O ∆ ( P ) (cid:105) , (2.24)with the boundary condition ¯ g ∆ ( η ) ∼ (1 − η ) ∆ − ∆1 − ∆22 , η → . (2.25)Here the image conformal Casimir operator ¯ L = 12 ( L AB + ¯ L AB )( L ,AB + ¯ L ,AB ) (2.26)involves the generators at the image position ¯ L AB = ¯ P A ∂∂ ¯ P B − ¯ P B ∂∂ ¯ P A (2.27)where ¯ P A is the embedding vector for the image point ¯ P A = (cid:18) x − , − x − , − x µ x (cid:19) . (2.28)In terms of the conformal blocks, the crossing equation (2.14) now reads (cid:88) k µ k ( g ∆ k ( η ) ∓ ¯ g ∆ k ( η )) = 0 . (2.29)Finally, there is another interesting point η = ∞ , which can only be reached inLorentzian signature. It turns out, as we will see in Section 3, that this limit plays asimilar role as the “Regge limit” in BCFT two-point functions. In fact, the kinematics ofboundary CFTs are intimately related to RP d CFTs. We now give a detailed comparisonwith the closely related BCFT case, and the result is summarized in Table 1. In the table and discussion below, we take the conformal boundary of the BCFT to be a unit sphere.It can be mapped to the infinite plane boundary by a conformal transformation. – 8 – ntermezzo: comparing with boundary CFTs
As we mentioned in the last section, a boundary CFT preserves the conformal symmetrythat leaves N B invariant. Up to a Wick rotation, the two systems preserve the samesymmetry group. The inversion sphere x = 1 now becomes a fixed locus in the BCFTcase, and is the location of the conformal boundary. The one-point function of an operatorinserted away from the boundary is determined by kinematics (cid:104)O ∆ ( x ) (cid:105) B = a B, O | P · N B | ∆ = a B, O | − x | ∆ . (2.30)up to a constant a B, O . The one-point coefficients a B, O are part of the new data defining aBCFT. For two operators inserted away from the boundary , we can construct a cross ratio ξ = ( − P · P )(2 N B · P )(2 N B · P ) = ( x − x ) (1 − x )(1 − x ) , (2.31)which is invariant under the residual conformal symmetry and the independent rescalingof the operators. In a Euclidean spacetime, the range of the cross ratio is ξ ∈ [0 , ∞ ) . Thetwo-point function can be written as a function of the cross ratio (cid:104)O ∆ ( x ) O ∆ ( x ) (cid:105) B = G B ( ξ ) | N B · P | ∆ | N B · P | ∆ = G B ( ξ ) | − x | ∆ | − x | ∆ . (2.32)There are three interesting points on the ξ -plane. The point ξ = 0 is known as the bulkchannel OPE limit, and should be identified with the η = 0 case where operators coincide(or light-like separated in Lorentzian signature). We can apply the OPE (2.15), and reducethe two-point function into one-point functions. The two-point function can be written asa sum of bulk channel conformal blocks [23, 24] G B ( ξ ) = (cid:88) k µ B, k g bulkB, ∆ k ( ξ ) (2.33)where µ B, k = C k a B,k and g bulkB, ∆ ( ξ ) = ξ ∆ − ∆1 − ∆22 F (cid:18) ∆ + ∆ − ∆ , ∆ + ∆ − ∆ − d − ξ (cid:19) . (2.34)Note that g bulkB, ∆ ( ξ ) can be identified with g ∆ ( η ) with the replacement ξ ↔ − η , up to anoverall normalization.The limit ξ = ∞ is known as the boundary channel limit, whereoperators inserted in the bulk are taken close to the boundary. A different OPE is involvedin this limit O ∆ ( x ) = a B, O | − x | ∆ + (cid:88) l ρ (cid:96) C [ x ] (cid:98) O ∆ l ( x ) (2.35) In order to distinguish from the real projective space case, we use the subscript B to denote objects inboundary CFTs. We will assume that the operators are inserted on the same side of the boundary. There is some formal connection between boundary CFTs and RP d CFTs by analytic continuation.For example, in two dimensions the boundary states are defined by ( L n − ¯ L − n ) |B(cid:105) = 0 while the crosscapstates are defined by ( L n − ( − n ¯ L − n ) |C(cid:105) = 0 . Here we can relate them formally by making the analyticcontinuation x → − x . This not only gives ξ → − η , but also fixes the one-point function and the kinematicfactors we extracted from the two-point functions. – 9 –hich expresses a bulk operator as a sum of operators (cid:98) O ∆ l ( x ) on the boundary. Here C [ x ] is a differential operator determined by symmetry. The boundary operator spectrum ∆ l and OPE coefficient ρ l are new CFT data. Applying the boundary OPE for each operatorreduces the two-point function to a sum of two-point functions of operators living on theboundary, and the latter is fully fixed by the residual conformal symmetry. The contributionof exchanging a boundary operator is summarized by a boundary channel conformal block g boundaryB, ∆ ( ξ ) = ξ − ∆2 F (cid:18) ∆ , ∆ − d − d, − ξ (cid:19) , (2.36)and the correlator can be decomposed in the boundary channel as G B ( ξ ) = a B, O δ + (cid:88) l ρ ,l ρ ,l g boundaryB, ∆ l ( ξ ) . (2.37)The boundary channel of BCFT two-point functions has no analogue in the real projectivespace case, because the identification (1.3) does not have any fixed point. The two channelsof OPE should lead to the same answer, and gives to a “crossing equation” (cid:88) k µ B, k g bulkB, ∆ k ( ξ ) = a B, O δ + (cid:88) l ρ ,l ρ ,l g boundaryB, ∆ l ( ξ ) . (2.38)Finally, the limit of ξ = − is known as the “Regge limit” [32]. In this limit, one operator isat the lightcone created by the image of the other operator with respect to the boundary.The Regge limit can only be reached in the Lorentzian signature, and requires analyticcontinuation from the Euclidean signature. It was proven in [32] that for any unitaryboundary CFT, the two-point function has a bounded behavior at the Regge limit, whichis controlled by the bulk channel exchange of operators with the lowest dimension.We can think of the η = ∞ limit of RP d CFTs as the ξ = − limit of BCFTs, as bothcases requires analytic continuation from the Euclidean regime. This intuition will also besupported in the next section when we study Witten diagrams, which arise in the weaklycoupled duals of RP d CFTs.
AdS d +1 / Z In this section, we study a simple toy model of holography for RP d CFTs. We extend thequotient of the boundary spacetime into the bulk to define a Z quotient of AdS space, andconsider perturbative physics on this background. This over-simplified setup is effective innature, and does not correspond to top-down models. However, it captures all the essentialkinematics which are relevant to various applications later in the paper. This setup ofquotient of AdS appeared previously, e.g. , in [8, 18–20, 22]. Here we give a detailed accountusing the embedding space formalism introduced in Section 2.1. It might also be tempting to identify η = 1 with the BCFT Regge limit, since in both cases oneoperator approaches the lightcone of (or coincides with) the image of the other operator. However, thecrossing equation (2.14) tells us that η = 1 limit is physically not any different from the η = 0 limit. – 10 –or the calculations in this section, it will be most convenient to consider the EuclideanAdS space − (cid:0) Z (cid:1) + (cid:0) Z (cid:1) + . . . + (cid:16) Z d +2 (cid:17) = − , Z > , (3.1)and analytic continue the results to the Lorentzian signature in the end. In terms of thePoincaré coordinates z = ( z , (cid:126)z ) , the embedding space vector Z A is parameterized as Z A = 1 z (cid:18) z + (cid:126)z , − z − (cid:126)z , (cid:126)z (cid:19) . (3.2)We extend the boundary inversion (2.6) into the bulk by requiring that I should act in thesame way on the AdS embedding vector. This leads to I : z → z z + (cid:126)z , (cid:126)z → − (cid:126)zz + (cid:126)z , (3.3)and defines a quotient space qAdS d +1 ≡ AdS d +1 / Z by the identification z ↔ z z + (cid:126)z , (cid:126)z ↔ − (cid:126)zz + (cid:126)z . (3.4)Note that the identification (3.4) is geometrically represented in Poincaré coordinates byan inversion with respect to the hemisphere H defined by z + z = 1 , z ≥ , (3.5)This is illustrated by Figure 1. The map (3.4) has a fixed point z = 1 , (cid:126)z = 0 , (3.6)which sits at the north pole of the inversion hemisphere H . In terms of embedding coordi-nates, the fixed point corresponds to the fixed vector N c introduced in (2.8).Now we consider a scalar field ϕ ± living on the quotient space qAdS d +1 . To describethe scalar field, it is convenient to extend the definition of the scalar field to the full AdS d +1 and impose the condition that ϕ ± ( z ) = ± ϕ ± ( I ◦ z ) . (3.7)We can define the propagators of ϕ ± on qAdS d +1 (and extended to AdS d +1 ) as follows.The bulk-to-bulk propagator H ∆ , ± BB ( Z, W ) satisfies the following equation of motion ( (cid:3) Z + ∆(∆ − d )) H ∆ , ± BB ( Z, W ) = δ d +1 ( Z, W ) ± δ d +1 ( Z, ¯ W ) , (3.8)where ¯ W denotes the image point of W under the inversion (3.3). The propagator H ∆ , ± BB ( Z, W ) can be expressed in terms of the usual AdS propagators before taking the quotient, as H ∆ , ± BB ( Z, W ) = G ∆ BB ( Z, W ) ± G ∆ BB ( Z, ¯ W ) (3.9) We have set the curvature of AdS to 1. – 11 – igure 1 : Illustration of
AdS d +1 / Z in Poincaré coordinates. A point Z inside the hemi-sphere H is identified with its inversion image ¯ Z out side of the hemisphere. The quotienthas a fixed point N c , which is the north pole of the hemisphere.where G ∆ BB ( Z, W ) satisfies ( (cid:3) Z + ∆(∆ − d )) G ∆ BB ( Z, W ) = δ d +1 ( Z, W ) , (3.10)and is given explicitly by G ∆ BB ( Z, W ) = C ∆ ,dBB ( − u ) − ∆2 F (cid:18) ∆ , ∆ − d − d + 1; u − (cid:19) , (3.11)with u = − Z · W + 12 , C ∆ ,dBB = Γ(∆)Γ(∆ − d + )(4 π ) d +12 Γ(2∆ − d + 1) . (3.12)Note that Z · W = ¯ Z · ¯ W , Z · ¯ W = ¯ Z · W , (3.13)we have the following identities for the bulk-to-bulk propagator H ∆ , ± BB ( Z, W ) = H ∆ , ± BB ( W, Z ) = H ∆ , ± BB ( ¯ Z, ¯ W ) , H ∆ , ± BB ( Z, ¯ W ) = ± H ∆ , ± BB ( Z, W ) . (3.14)The bulk-to-boundary propagator H ∆ , ± B∂ ( Z, P ) can be obtained from the limit of H ∆ , ± BB ( Z, W ) as we move the bulk point W close to the boundary. It is easy to prove that Z · ¯ P = ( (cid:126)x ) − ¯ Z · P , ¯ Z · ¯ P = ( (cid:126)x ) − Z · P . (3.15)This implies that the usual
AdS d +1 bulk-to-boundary propagator G ∆ B∂ ( Z, P ) = (cid:18) − Z · P (cid:19) ∆ = (cid:18) z z + ( (cid:126)z − (cid:126)x ) (cid:19) ∆ (3.16)transforms as G ∆ B∂ ( Z, ¯ P ) = ( (cid:126)x ) ∆ G ∆ B∂ ( ¯ Z, P ) . (3.17)– 12 –n the other hand, we recall that boundary operator receives an extra ( (cid:126)x ) − ∆ under inver-sion from (2.7), which cancels the ( (cid:126)x ) ∆ in (3.17). Therefore the qAdS d +1 bulk-to-boundarypropagator should be defined similarly to (3.9), as H ∆ , ± B∂ ( Z, P ) = G ∆ B∂ ( Z, P ) ± G ∆ B∂ ( ¯ Z, P ) , (3.18)or equivalently, H ∆ , ± B∂ ( Z, P ) = G ∆ B∂ ( Z, P ) ± ( (cid:126)x ) − ∆ G ∆ B∂ ( Z, ¯ P ) . (3.19) Having obtained bulk-to-bulk and bulk-to-boundary propagators on qAdS d +1 , we are nowready to define Witten diagrams. One-point diagram
Let us start from the one-point function. It is given by a single bulk-to-boundary propagatorwhich ends on the conformal boundary where the operator is inserted. Note that the Wittendiagram must preserve the fixed vector N c defined in (2.8). Therefore, the other end of thepropagator has to end at the inversion fixed point N c (Figure 2). It corresponds to a vertex ϕ ( N c ) localized at the fixed point. We have (cid:104)O ∆ (cid:105) = H ∆ , ± B∂ ( N c , P ) = (cid:40) (cid:126)x ) ∆ for + parity for − parity (3.20)which has the correct structure (2.10) determined by symmetry. Note that the informationof the vertex ϕ ( N c ) was not contained in the original theory before the quotient, but wasinputted into this toy model by hand. Figure 2 : The one-point function is given by a bulk-to-boundary propagator with nointegration.
Two-point contact Witten diagrams
For two-point functions, we can define the following tree-level contact Witten diagram(Figure 3) V con, , ∆ ( P , P ) = 14 H ∆ , + B∂ ( N c , P ) H ∆ , + B∂ ( N c , P ) , (3.21)– 13 –hich factorizes into the product of two one-point functions. It comes from the vertex ϕ ϕ ( N c ) which localizes at the fixed point N c . Note that it is important that bothoperators are parity even since H ∆ , − B∂ ( N c , P ) vanishes. In terms of the cross ratio, thecontact Witten diagram reads V con, , ∆ ( P , P ) = V con, , ∆ ( η )(1 + x ) ∆ (1 + x ) ∆ (3.22)where V con, , ∆ ( η ) = 1 . (3.23) Figure 3 : The two-point contact Witten diagram is given by a product of bulk-to-boundarypropagators with no integration.To define two-point contact diagram for operators with odd parity, we can add twoderivatives to the vertex. The vertex now becomes ( ∇ µ ϕ ∇ µ ϕ )( N c ) , and leads to thefollowing diagram V con, , ∆ ( P , P ) = 14 ∇ µ H ∆ , − B∂ ( Z, P ) ∇ µ H ∆ , − B∂ ( Z, P ) (cid:12)(cid:12) Z = N c , = ∇ µ G ∆ B∂ ( Z, P ) ∇ µ G ∆ B∂ ( Z, P ) (cid:12)(cid:12) Z = N c . (3.24)Using the identity ∇ µ G ∆ B∂ ( Z, P ) ∇ µ G ∆ B∂ ( Z, P ) = ∆ ∆ (cid:18) G ∆ B∂ ( Z, P ) G ∆ B∂ ( Z, P ) − x G ∆ +1 B∂ ( Z, P ) G ∆ +1 B∂ ( Z, P ) (cid:19) , (3.25)we get V con, , ∆ ( P , P ) = V con, , ∆ ( η )(1 + x ) ∆ (1 + x ) ∆ (3.26)where V con, , ∆ ( η ) = ∆ ∆ (1 − η ) . (3.27)It is clear that V con, , ∆ ( η ) is antisymmetric under η ↔ − η .– 14 –t is straightforward to generalize these contact Witten diagrams to include more deriva-tives in the vertex. In general, for a vertex with L derivatives the two-point contact Wittendiagram V con, L ∆ , ∆ ( η ) is a polynomial in η of degree L . Moreover, the contact Witten diagramshave the following behavior at η → ∞V con, L ∆ , ∆ ( η ) → η L , η → ∞ . (3.28)This is consistent with the intuition that the η → ∞ limit can be thought of as a “Reggelimit” for the two-point correlator – increasing the number of derivatives in the vertex leadsto a more divergent behavior in the correlator at large cross ratio. Two-point exchange Witten diagrams
Let us now define the two-point exchange Witten diagram (Figure 4) V exchange, ± ∆ ( P , P ) = 12 (cid:90) qAdS d +1 d d +1 Z H ∆ , + BB ( N c , Z ) H ∆ , ± B∂ ( Z, P ) H ∆ , ± B∂ ( Z, P ) (3.29)which requires the presence of the localized vertex ϕ ( N c ) for the scalar field with dualconformal dimension ∆ , and a bulk cubic vertex ϕϕ ϕ ( Z ) where ϕ , have dual dimensions ∆ , . Let us use (3.9), (3.18) and (3.19) to express the qAdS d +1 propagators in terms of thepropagators in the full AdS d +1 . Note that N c is its own image, therefore H ∆ , + BB ( N c , Z ) =2 G ∆ BB ( Z, N c ) . On the other hand, H ∆ , + BB ( N c , Z ) = 2 G ∆ BB ( ¯ Z, N c ) thanks to (3.14). Usingthis and (3.17), we can expand the product and massage the expressions such that the AdS d +1 propagators join into connected diagrams. It becomes obvious that the integralsin V exchange, ± ∆ ( P , P ) can be organized into the linear combinations of exchange diagramsdefined in the full AdS d +1 space (in particular, integrals inside the sphere and their imagesoutside combine, and extend to the full space) V exchange, ± ∆ ( P , P ) = W exchange ∆ ( P , P ) ± ( x ) − ∆ ¯ W exchange, ± ∆ ( P , P ) (3.30)where W exchange ∆ ( P , P ) = (cid:90) AdS d +1 d d +1 Z G ∆ BB ( N c , Z ) G ∆ B∂ ( Z, P ) G ∆ B∂ ( Z, P ) , (3.31)and ¯ W exchange ∆ ( P , P ) = W exchange ∆ ( P , ¯ P ) . (3.32)When written in terms of the cross ratio W exchange ∆ ( P , P ) = W exchange ∆ ( η )(1 + x ) ∆ (1 + x ) ∆ , ( x ) − ∆ ¯ W exchange ∆ ( P , P ) = ¯ W exchange ∆ ( η )(1 + x ) ∆ (1 + x ) ∆ , (3.33) W exchange ∆ and ¯ W exchange ∆ are related by ¯ W exchange ∆ ( η ) = W exchange ∆ (1 − η ) . (3.34)– 15 – igure 4 : Illustration of an exchange Witten diagram. One end of the bulk-to-bulk prop-agator is fixed at N c , while the other end is connected to the cubic vertex and integratedover. Equation of motion relations
The exchange Witten diagrams W exchange ∆ and ¯ W exchange ∆ introduced above are relatedto the the zero-derivative contact Witten diagram V con, , ∆ by the equation of motion oper-ators. More precisely, let us define EOM = L + ∆(∆ − d ) , EOM = ¯ L + ∆(∆ − d ) (3.35)where L and ¯ L are the conformal Casimir operators defined in (2.19) and (2.26). Theexchange diagrams are then related to the contact diagram via EOM [ W exchange ∆ ] = V con, , ∆ , EOM [ ¯ W exchange ∆ ] = V con, , ∆ . (3.36)To prove these relations, let us notice that the integral (3.31) is conformal invariant ( L AB + L AB + L AB ) W exchange ∆ ( P , P ; N c ) = 0 . (3.37)Here we are viewing W exchange ∆ as a function of the bulk point N c , and L AB are the AdS d +1 isometry generators at the point N c . Using (3.37) twice, we get L W exchange ∆ ( P , P ; N c ) = L W exchange ∆ ( P , P ; N c ) (3.38)where L = L AB L AB . Notice that L acts on the bulk coordinates as (cid:3) , and collapses thebulk-to-bulk propagator in (3.31) to a delta function by (3.10). Integrating over Z gives usthe contact diagram (3.21). The proof for ¯ W exchange ∆ is analogous.We will also explicitly evaluate these exchange Witten diagrams, and study their de-compositions into conformal blocks. We will delay the discussion until Section 4.– 16 – igure 5 : Illustration of a geodesic Witten diagram. The integration of the cubic vertexis restricted to the geodesic line γ connecting the two boundary insertions. We can also define a variation of the exchange Witten diagrams, which is holographicallydual to the conformal blocks (2.16), (2.23). These modified exchange Witten diagrams areknown as the geodesic Witten diagrams [33]. These objects were also considered in [34] ina different context. We define the dual of g ∆ ( η ) similarly as in (3.31) by W geo ∆ ( P , P ) = (cid:90) γ dγ G ∆ BB ( N c , γ ) G ∆ B∂ ( γ, P ) G ∆ B∂ ( γ, P ) , (3.39)where γ is the geodesic connecting the (cid:126)x and (cid:126)x on the conformal boundary. In Poincarécoordinates, the geodesic γ is just a semicircle. Instead of integrating over the whole AdS d +1 , the integration is now restricted to the geodesic only. The geodesic Witten diagramis illustrated in Figure 5. After extracting a kinematic factor W geo ∆ ( P , P ) = W geo ∆ ( η )(1 + x ) ∆ (1 + x ) ∆ , (3.40)the function W geo ∆ ( η ) is proportional to g ∆ ( η ) up to some constant factor. Similarly, wedefine ¯ W geo ∆ ( P , P ) = (cid:90) γ dγ G ∆ BB ( N c , γ ) G ∆ B∂ ( γ, P ) G ∆ B∂ ( γ, ¯ P ) , (3.41)and ( x ) − ∆ ¯ W geo ∆ ( P , P ) = ¯ W geo ∆ ( η )(1 + x ) ∆ (1 + x ) ∆ , (3.42)where γ is the geodesic connecting (cid:126)x and the image of (cid:126)x (see Figure 6). The geodesicWitten diagram ¯ W geo ∆ is dual to the image conformal block ¯ g ∆ ( η ) .To prove their equivalence, we can use the equation of motion operators introduced inthe last subsection. Let us act on the geodesic Witten diagrams with EOM and
EOM ,and use the definitions of W geo ∆ and ¯ W geo ∆ . The integrations along the geodesics preserve theconformal invariance, and allows us to apply the analysis and use the equation of motionfor the bulk-to-bulk propagator. Note however that since the geodesic lines do not pass– 17 – igure 6 : Illustration of a geodesic Witten diagram in the mirror channel. The geodesic γ now connects point 1 and the image of point 2.through the fixed point N c for generic end points (cid:126)x , (cid:126)x , the delta function is not integrated.Therefore, instead of generating contact diagrams we get EOM [ W geo ∆ ] = 0 , EOM [ ¯ W geo ∆ ] = 0 , (3.43)On the boundary side, these two equations are just the quadratic conformal Casimir equa-tions. It is also clear from the definitions (3.39), (3 . ) that W geo ∆ ( η ) and ¯ W geo ∆ ( η ) satisfythe boundary conditions (2.18), (2.25). This concludes the proof that the geodesic Wittendiagrams are the bulk dual of the conformal blocks. To conclude this section, we give a quick comparison of the above discussion with theclosely related interface/boundary CFT from the probe-brane setup [32, 35–37] (which isthe simplest Karch-Randall set up [38, 39]). As we have seen in Section 2, the kinematicsof the RP d CFT and BCFT share a lot of similarities. We now show how these kinematicalsimilarities are extended into the bulk, and also point out a number of differences.In the probe-brane setup, we choose a special slice of
AdS d space inside AdS d +1 . Thereare local degrees of freedom living on the AdS d slice, and they are coupled to the bulk fieldsin AdS d +1 . However, the AdS d brane is treated as a probe and does not back-reacts to thegeometry. In [32, 37], “straight” probe branes are considered in great detail where the AdS d slice is embedded in AdS d +1 as the restriction to z d = 0 . The setup corresponds to CFTswith a straight co-dimension 1 interface at x d = 0 . One can then use method of imagesto take only half of the AdS d +1 space with z d ≥ , and consider boundary CFTs definedon x d ≥ with Dirichlet or Neumann boundary conditions, as was done in [32]. Here weconsider a slightly modified setup where the probe brane is a hemisphere in the Poincarécoordinates of AdS d +1 . It is related to the “straight” case by a conformal mapping. Themethod of images is similar in the spherical case (one can also first apply the method in the“straight” case and then perform the conformal mapping), and will not be elaborated here.We will therefore focus only on the probe brane case where the space continues beyond theinterface. – 18 –s we pointed out in Section 2.1, the spherical boundary of a BCFT preserves the fixedembedding vector N B (2.9). In the bulk, the boundary of the BCFT is extended into thehemisphere N B · Z = 1 − z − (cid:126)z z = 0 , (3.44)which is the probe AdS d brane. This hemisphere coincides with the inversion hemisphere H defined in (3.5). Note that in the RP d CFT case the fixed vector N c corresponds to apoint in the bulk, while in the BCFT case the fixed vector N B is a normal vector defininga fixed co-dimension 1 surface.We can modify the Witten diagrams defined in Section 3.2 to define their BCFT coun-terparts, by simply integrating over the whole hemisphere H instead of localizing on thenorth pole. For example, the BCFT one-point function is defined as (cid:104)O ∆ ( P ) (cid:105) B = (cid:90) H dZ H G ∆ B∂ ( Z H , P ) ∼ | − x | ∆ , (3.45)which reproduces the correct structure (2.30). The two-point contact and exchange Wittendiagrams are respectively defined as W con, B, ∆ , ∆ ( P , P ) = (cid:90) H dZ H G ∆ B∂ ( Z H , P ) G ∆ B∂ ( Z H , P ) , (3.46) W exchange ; bulkB, ∆ ( P , P ) = (cid:90) H dZ H (cid:90) AdS d +1 dW G ∆ BB ( Z H , W ) G ∆ B∂ ( W, P ) G ∆ B∂ ( W, P ) . (3.47)It is easy to verify that a similar equation of motion identity relates the exchange Wittendiagram to the contact Witten diagram EOM [ W exchange ; bulkB, ∆ ] = W con, B, ∆ , ∆ , (3.48)by using similar arguments. The integrals (3.46), (3.47) can be mapped to the straightprobe brane integrals studied in [37], by parameterizing the hemisphere H with the followingPoincaré coordinates Z H = 1 z (cid:18) z d , , z − z d + z i )2 , − ( z − z d + z i )2 , z i (cid:19) , i = 1 , . . . , d − . (3.49)It then follows that these Witten diagrams are functions of the BCFT cross ratio ξ definedin (2.31), rather than the RP d CFT cross ratio η defined in (2.12).We can also define the geodesic Witten diagram similar to (3.39) W geo ; bulkB, ∆ ( P , P ) = (cid:90) H dZ H (cid:90) γ dγ G ∆ BB ( Z H , γ ) G ∆ B∂ ( γ, P ) G ∆ B∂ ( γ, P ) , (3.50)as was first discussed in [37]. A similar argument using the equation of motion opera-tor shows that the geodesic Witten diagram is holographically dual to the bulk channelconformal block g bulkB, ∆ ( ξ ) defined in (2.34). Here we switched to the ( − , + , − , + , . . . , +) signature for R d, . We also need to perform the Wickrotation z d → iz d to make the probe AdS d brane Euclidean in order to compare with [37]. – 19 –inally, in the probe brane setup we can define the boundary exchange Witten diagram W exchange ; boundaryB, ∆ ( P , P ) = (cid:90) H dZ , H dZ , H G ∆ , H BB ( Z , H , Z , H ) G ∆ B∂ ( Z , H , P ) G ∆ B∂ ( Z , H , P ) (3.51)where an interface field localized on H with dimension ∆ is exchanged via the AdS d propa-gator G ∆ , H BB ( Z , H , Z , H ) . These diagrams have no analogue in the RP d CFT case, since theinversion hemisphere is not a boundary and there is no extra degrees of freedom living onit.
In this section, we study in detail the properties of the two-point exchange Witten diagramsdefined in the previous section. In Section 4.1 we explicitly evaluate these diagrams. InSection 4.2 we study the conformal block decomposition of Witten diagrams in differentchannels.
We will focus on the evaluation of the two-point exchange Witten diagram W exchange ∆ . Theimage diagram ¯ W exchange ∆ can be obtained from W exchange ∆ via the crossing relation (3.34).The full qAdS d +1 exchange Witten diagram V exchange, ± ∆ can then be assembled using (3.30).We first discuss the special case where W exchange ∆ can be expressed as a finite sum of contactWitten diagrams. We then give the formula for the exchange diagram when the quantumnumbers ∆ , ∆ , ∆ are generic. Note that the calculation is exactly the same as doingonly one of the two integrals in a scalar four-point exchange Witten diagram in AdS d +1 ,since once we strip away the other two bulk-to-boundary propagators the integral becomesidentical. The truncated case
Let us first consider a special case when ∆ + ∆ − ∆ ∈ Z + . We can use the vertexidentity for scalar exchange [40] (see also Appendix A of [41]) to write the integrated vertexas a finite sum of products of two bulk-to-boundary propagators with shifted conformaldimensions. The exchange diagram W exchange ∆ then becomes W exchange ∆ ( P , P ) = k max (cid:88) k = k min a k ( (cid:126)x ) k − ∆ G k +∆ − ∆ B∂ ( N c , P ) G kB∂ ( N c , P )= k max (cid:88) k = k min a k ( (cid:126)x ) k − ∆ V con, k +∆ − ∆ ,k ( P , P ) (4.1)– 20 –here k min = ∆ − ∆ + ∆ , k max = ∆ − ,a k − = ( k − ∆2 + ∆ − ∆ )( k − d + ∆2 + ∆ − ∆ )( k − k − − ∆ + ∆ ) a k ,a ∆ − = 14(∆ − − . (4.2)Written in terms of the cross ratio, the exchange Witten diagram is a polynomial of η − W exchange ∆ ( η ) = k max (cid:88) k = k min a k η k − ∆ . (4.3) The general case
In the general case, we can still express the exchange Witten diagram as an infinite sum ofcontact Witten diagrams. The integral has already been computed in Appendix C of [42].Here we review the derivation and the result in the language of RP d CFT.The main idea for evaluating the integral is to use the equation of motion relation(3.36) to write down a differential equation for the exchange Witten diagram. Written interms of the cross ratio, the equation of motion identity becomes
EOM [ W exchange ∆ ( η )] = 1 (4.4)where the differential operator acts as EOM [ G ( η )] = 4 η ( η − G (cid:48)(cid:48) ( η ) + η (4( η − + ∆ + 1) + 2 d ) G (cid:48) ( η )+ ((∆ − ∆ − ∆ )( − d + ∆ + ∆ + ∆ ) + 4∆ ∆ η ) G ( η ) . (4.5)The differential equation should be supplemented by two boundary conditions:1) From the OPE limit η → , we know W exchange ∆ ( η ) should behave as η ∆ − ∆1 − ∆22 .
2) From the definition of the integral (3.31), W exchange ∆ ( η ) has to be smooth at η = 1 (see [43]).The physical solution is a linear combination of the special solution f ( η ) = F (cid:18) , ∆ , ∆ ; ∆ + ∆ − ∆2 + 1 , ∆ + ∆ + ∆ − d η (cid:19) , (4.6)and a homogeneous solution, the conformal block g ∆ ( η ) , W exchange ∆ ( η ) = C f ( η ) + C g ∆ ( η ) . (4.7) Here we are assuming that ∆ < ∆ + ∆ such that the single-trace contribution is leading. – 21 –he coefficients C , C are given by C = − + ∆ − ∆)(∆ + ∆ + ∆ − d ) ,C = Γ( ∆+∆ − ∆ )Γ( ∆ − ∆ +∆ )Γ( − ∆+∆ +∆ )Γ( − d +∆+∆ +∆ )4Γ(∆ )Γ(∆ )Γ( − d + ∆ + 1) . (4.8)The ratio C C is precisely fixed by the condition that the solution is regular at η = 1 . Wecan also write (4.7) as two infinite series in η . Using the definition (3.22), (3.23) for thecontact Witten diagram, we can then write W exchange ∆ as two infinite sums of contact Wittendiagrams W exchange ∆ = ∞ (cid:88) i =0 ( (cid:126)x ) i P i V con, + i, ∆ + i + ∞ (cid:88) i =0 ( (cid:126)x ) ∆ − ∆1 − ∆2+2 i Q i V con, ∆+∆1 − ∆22 + i, ∆ − ∆1+∆22 + i (4.9)where the coefficients P i and Q i are given by P i = (∆ ) i (∆ ) i (∆ − ∆ − ∆ )( − d + ∆ + ∆ + ∆ ) (cid:0) − ∆+∆ +∆ +22 (cid:1) i (cid:16) − d +∆+∆ +∆ +22 (cid:17) i , (4.10)and Q i = ( − i Γ (cid:0) d − i − (cid:1) sin (cid:16) π ( d − (cid:17) Γ (cid:16) − d +∆+∆ +∆ (cid:17) π Γ( i + 1)Γ(∆ )Γ(∆ ) × Γ (cid:0) ∆ − ∆ +∆ (cid:1) Γ (cid:0) ∆+∆ − ∆ (cid:1) Γ (cid:0) − ∆+∆ +∆ (cid:1) Γ (cid:0) − ∆+∆ − ∆ +22 (cid:1) Γ (cid:0) − ∆ − ∆ +∆ +22 (cid:1) Γ (cid:0) − ∆+∆ − ∆ − i +22 (cid:1) Γ (cid:0) − ∆ − ∆ +∆ − i +22 (cid:1) . (4.11)When ∆ + ∆ − ∆ = 2 Z + the infinite sum truncates and reduces to (4.1).Finally, let us examine the behavior of the exchange Witten diagrams at η → ∞ .By using the equation of motion identities (3.36) and the behavior of the contact Wittendiagram (3.28), we find W exchange ∆ ( η ) → η − , ¯ W exchange ∆ ( η ) → η − , for η → ∞ . (4.12) We now study the conformal block decomposition of two-point Witten diagrams. We startwith a warmup case, namely the zero-derivative contact Witten diagram (3.23). It isstraightforward to show that V con, , ∆ ( η ) can be written as an infinite sum over double-traceconformal blocks V con, , ∆ ( η ) = (cid:88) n a n g ∆ d.t.n ( η ) = (cid:88) n a n ¯ g ∆ d.t.n ( η ) (4.13)– 22 –here ∆ d.t.n ≡ ∆ + ∆ + 2 n , and a n = ( − n Γ( n + ∆ )Γ( n + ∆ )Γ (cid:0) − d + n + ∆ + ∆ (cid:1) Γ(∆ )Γ(∆ )Γ( n + 1)Γ (cid:0) − d + ∆ d.t.n (cid:1) . (4.14)The exchanged operators in each channel correspond to double-trace operators of theschematic form : O (cid:3) n O : with n = 0 , , . . . . Exchange Witten diagram in the direct channel
Let us now consider the conformal block decomposition of the exchange Witten diagram(3.31) in the same channel. The Witten diagram can be written as a sum of a single-trace conformal block with dimension ∆ , dual to the exchanged field, and infinitely manydouble-trace conformal blocks W exchange ∆ ( η ) = A g ∆ ( η ) + (cid:88) n A n g ∆ d.t.n ( η ) . (4.15)The single-trace OPE coefficient can be extracted from the small η expansion of (4.7), andis associated to the term with the behavior η ∆ − ∆1 − ∆22 A = Γ (cid:0) ∆+∆ − ∆ (cid:1) Γ (cid:0) ∆ − ∆ +∆ (cid:1) Γ (cid:0) − ∆+∆ +∆ (cid:1) Γ (cid:16) − d +∆+∆ +∆ (cid:17) )Γ(∆ )Γ (cid:0) − d + ∆ + 1 (cid:1) . (4.16)To extract the double-trace OPE coefficients, we use the equation of motion identity (3.36).Note that the EOM operator annihilates the single-trace conformal block g ∆ ( η ) , whilemultiplies the double-trace conformal blocks with constants EOM [ g ∆ ( η )] = 0 , EOM [ g ∆ d.t.n ( η )] = (∆(∆ − d ) − ∆ d.t.n (∆ d.t.n − d )) g ∆ d.t.n ( η ) . (4.17)Using the conformal block decomposition (4.13) of the contact Witten diagram, we find A n = a n ∆(∆ − d ) − ∆ d.t.n (∆ d.t.n − d ) . (4.18)Using these OPE coefficients, we can further expand the conformal blocks to obtain a small η expansion for the exchange Witten diagram. This expansion can be compared with theexpansion of (4.7), which provides a consistency check of our results. By crossing symmetry,we also have ¯ W exchange ∆ ( η ) = A ¯ g ∆ ( η ) + (cid:88) n A n ¯ g ∆ d.t.n ( η ) . (4.19) Exchange Witten diagram in the crossed channel
Finally we consider the conformal block decomposition of the exchange Witten diagram inthe crossed channel. The decomposition consists of crossed channel double-trace conformalblocks only W exchange ∆ ( η ) = (cid:88) n B n ¯ g ∆ d.t.n ( η ) . (4.20)– 23 –o work out the decomposition coefficients, we will generalize the recursive techniquesdeveloped in [42]. We apply the equation of motion relation (3.36) to turn the exchangeWitten diagram into a contact Witten diagram, which has already been decomposed intothe crossed channel in (4.13). On the other hand, the action of the EOM operator on ¯ g ∆ d.t.n admits a simple three-term recursion relation EOM [¯ g ∆ d.t.n ] = µ n ¯ g ∆ d.t.n − + ν n ¯ g ∆ d.t.n + ρ n ¯ g ∆ d.t.n +1 (4.21)where µ n = (∆ + ∆ − ∆ d.t.n )( d − ∆ − ∆ + ∆ d.t.n − ,ν n = ( d − − + 2)(3 d − − − d + 2∆ − − d − + 2∆ − d − d.t.n − d − d.t.n + 2) − (cid:18) ∆ − d (cid:19) − (cid:18) ∆ − d (cid:19) + 12 (cid:18) ∆ d.t.n − d (cid:19) + d −
12 + ∆(∆ − d ) ,ρ n = (cid:16) (∆ − ∆ ) − ∆ d.t.n (cid:17) ( d − ∆ − ∆ − ∆ d.t.n )(2 d − ∆ − ∆ − ∆ d.t.n +1 )( d − d.t.n + 1)) (4.22) × ( d + ∆ − ∆ − ∆ d.t.n +1 )( d − ∆ + ∆ − ∆ d.t.n +1 )( d − d.t.n )( d − d.t.n +1 ) . Note that for n = 0 , the coefficient µ vanishes. Therefore the action of the EOM operatorpreserves the double-trace spectrum. Using (4.21), (4.7) and (4.13), we obtain the followinginhomogeneous recursion relation for the decomposition coefficients ρ n − B n − + ν n B n + µ n +1 B n +1 = a n . (4.23)For n = 0 , the equation remains the same just without the ρ n piece. The coefficients B n with n > can be recursively solved, after specifying the boundary condition B , which isextracted from the η → limit of the exchange diagram (4.7) B = W exchange ∆ ( η → (cid:0) ∆ +∆ − ∆2 (cid:1) Γ (cid:16) ∆ +∆ − d +∆2 (cid:17) Γ (cid:0) − d (cid:1) ) (cid:34) Γ (cid:0) ∆+∆ − ∆ (cid:1) Γ (cid:0) ∆ − ∆ +∆ (cid:1) Γ (∆ ) Γ (cid:16) − d +∆+∆ − ∆ (cid:17) Γ (cid:16) − d +∆ − ∆ +∆ (cid:17) − ˜ F (cid:18) − ∆ − ∆2 , − ∆ − d + ∆2 , − d − d , − d ; 1 (cid:19) (cid:35) (4.24)where ˜ F is the regularized hypergeometric function defined by ˜ F ( a , a , a ; b , b ; z ) = F ( a , a , a ; b , b ; z )Γ( b )Γ( b ) . (4.25)By crossing symmetry, the conformal block decomposition for W exchange ∆ also gives ¯ W exchange ∆ ( η ) = (cid:88) n B n g ∆ d.t.n ( η ) . (4.26)– 24 – Dimensional reduction
In [28] it was pointed out that a large class of Witten diagram recursion relations can beobtained from conformal block recursion relations, essentially by just replacing conformalblocks with the corresponding exchange Witten diagrams. This idea was demonstrated forfour-point functions in generic CFTs, and two-point functions in CFTs with boundaries.Here we generalize similar statements to two-point functions in CFTs on RP d , and we willfocus on the dimensional reduction relations. Let us recall the relation between RP d CFT conformal blocks and BCFT conformal blocksin the bulk channel (see (2.16) and (2.34)) g ∆ ( η ) = ( − ∆ − ∆1 − ∆22 g bulkB, ∆ ( − η ) . (5.1)The dimensional reduction formulae derived in [28] for g bulkB, ∆ ( ξ ) therefore can be straightfor-wardly transformed into those for g ∆ ( η ) . We have the following relation between conformalblocks in d and d − dimensions g ( d )∆ ( η ) = ∞ (cid:88) j =0 α ( d ) j (∆) g ( d − j ( η ) (5.2)where we have used the superscript to emphasize the dimensional dependence, and α ( d ) j (∆) = Γ (cid:0) j + (cid:1) (cid:0) (∆ + ∆ − ∆ ) (cid:1) j (cid:0) (∆ − ∆ + ∆ ) (cid:1) j √ πj ! (cid:0) ( − d + 2∆ + 2) (cid:1) j (cid:0) j + ( − d + 2∆ −
1) + 1 (cid:1) j . (5.3)Moreover, a conformal block in d − dimensions can be expressed in terms of only twoconformal blocks in d dimensions g ( d − ( η ) = g ( d )∆ ( η ) + β (∆) g ( d )∆+2 ( η ) (5.4)where β (∆) = − (∆ + ∆ − ∆ )(∆ − ∆ + ∆ )( d − − d − − . (5.5)Using ¯ g ( d )∆ ( η ) = g ( d )∆ (1 − η ) , we also obtain similar dimensional reduction formulae for theimage channel conformal blocks ¯ g ( d )∆ ( η ) .Let us comment that the recursion relation (5.4) is quite special, as it involves onlyfinitely many terms. In fact, the inverse relation which expresses a d dimensional conformalblock in terms of d − dimensional blocks, contains infinitely many conformal blocks.A similar relation of (5.4) was first derived in [26] for conformal blocks for four-pointfunctions in CFTs without boundaries. The identity expresses a d − dimensional conformalblock in terms of the linear combination of five conformal blocks in d dimensions. Theexistence of such a recursion was explained in terms of a OSp ( d + 1 , | Parisi-Sourlassupersymmetry [27] in a d dimensional SCFT, which upon dimensional reduction gives rise– 25 –o a non-supersymmetric CFT in d − dimensions. The relation (5.4) we wrote down hereparallels the five-term relation in [26], and therefore suggests that a similar story of Parisi-Sourlas supersymmetry and dimensional reduction can also be extended to CFTs on realprojective space. Similar to the observations in [28], the recursion relations (5.2) and (5.4) can also be ex-tended to imply relations for exchange Witten diagrams. Let us rescale the exchange Wittendiagrams such that the single-trace conformal blocks appear with unit coefficients P ∆ ( η ) = 1 A W exchange ∆ ( η ) , ¯ P ∆ ( η ) = 1 A ¯ W exchange ∆ ( η ) (5.6)where A is the coefficient of the single-trace conformal block (4.16). We claim that we havethe following dimensional reduction formulae P ( d )∆ ( η ) = ∞ (cid:88) j =0 α ( d ) j (∆) P ( d − j ( η ) , (5.7) P ( d − ( η ) = P ( d )∆ ( η ) + β (∆) P ( d )∆+2 ( η ) . (5.8)Similar relations also hold for the mirror channel exchange Witten diagram upon replacing P ( d )∆ ( η ) with ¯ P ( d )∆ ( η ) . In [28], similar Witten diagram identities were proven by using simpleMellin space arguments. Unfortunately, the same arguments cannot be used here. We notethat although a Mellin representation formalism can be developed for RP d correlators similarto the BCFT case [37], it is not suitable for holographic correlators. To see this, we recallthat contact Witten diagrams are polynomials of the cross ratio, and their Mellin transformare ill-defined. Nevertheless, we can still prove (5.7) and (5.8) in position space by usingthe conformal block decomposition in the direct channel.Let us denote the decomposition of the exchange Witten diagrams as P ( d )∆ ( η ) = g ( d )∆ ( η ) + ∞ (cid:88) n =0 µ ( d ) n (∆) g ( d )∆ d.t.n ( η ) (5.9)where µ ( d ) n (∆) = A n A in relation to (4.15). Substituting this decomposition into (5.8),we find that the single-trace conformal blocks on both sides cancel thanks to (5.4). Thedouble-trace conformal blocks must also match, provided ∞ (cid:88) n =0 µ ( d − n (∆) g ( d − d.t.n ( η ) = ∞ (cid:88) n =0 µ ( d ) n (∆) g ( d )∆ d.t.n ( η ) + β (∆) ∞ (cid:88) n =0 µ ( d ) n (∆ + 2) g ( d )∆ d.t.n ( η ) . (5.10)We can use (5.4) again to turn g ( d − d.t.n ( η ) into g ( d )∆ d.t.n ( η ) , and arrive at the following condition µ ( d − n (∆) + β (∆ n − ) µ ( d − n − (∆) = µ ( d ) n (∆) + β (∆) µ ( d ) n (∆ + 2) . (5.11)This identity can be straightforwardly verified, using the explicit expressions for µ ( d ) n (∆) and β (∆) . – 26 –e can proceed similarly for the relation (5.7). The single-trace operators again dropout because of the conformal block recursion relation. The double-trace coefficients needto be constrained, and the condition reads (cid:88) m + j = n µ ( d ) m (∆) α ( d ) j (∆ m ) = ∞ (cid:88) k =0 α ( d ) k (∆) µ ( d − n (∆ + 2 k ) . (5.12)The infinite sum on the r.h.s. makes it difficult to check analytically, we can neverthelessnumerically check this identity.Another nontrivial crosscheck is to use (5.7) twice to reproduce (5.8). It is not difficultto find P ( d )∆ ( η ) = (cid:88) j,k α ( d ) j (∆) α ( d − k (∆ + 2 j ) P ( d − j +2 k ( η )= ∞ (cid:88) n =0 (cid:0) ∆+∆ − ∆ (cid:1) n (cid:0) ∆ − ∆ +∆ (cid:1) n (cid:0) − d + ∆ + 1 (cid:1) n P ( d − n ( η ) . (5.13)Using this identity in (5.8), one can straightforwardly verify that the relation is valid. RP d CFTs
In this section, we present an analytic bootstrap approach for studying CFTs on RP d . Partof our discussions forms a close analogy of the analysis for BCFT two-point functions in [32](see also the related works [44–47]). In Section 6.1 we argue that the double-trace conformalblocks in both the bulk channel and the mirror channels form a complete basis for two-pointcorrelators. Their duals give a basis for analytic functionals, and we will explicitly constructtheir actions using exchange Witten diagrams. In Section 6.2 we give another constructionof the functionals from the dispersion relation. We apply these analytic functionals inSection 6.3, where we obtain the (cid:15) -expansion result of RP d ϕ theory to (cid:15) order. We alsoperform an independent field theory check of our results in Section 6.4. The study of Witten diagrams in Section 4 motivates us to propose a natural basis fortwo-point functions. As we have seen, the exchange Witten diagrams admit the followingdecompositions in two channels W exchange ∆ ( η ) = A g ∆ ( η ) + (cid:88) n A n g ∆ d.t.n ( η ) = (cid:88) n B n ¯ g ∆ d.t.n ( η ) , (6.1) ¯ W exchange ∆ ( η ) = A ¯ g ∆ ( η ) + (cid:88) n A n ¯ g ∆ d.t.n ( η ) = (cid:88) n B n g ∆ d.t.n ( η ) . (6.2)These identities show that any conformal blocks g ∆ ( η ) , ¯ g ∆ ( η ) with generic conformal dimen-sion ∆ can be expressed as linear combinations of double-trace conformal blocks g ∆ d.t.n ( η ) , ¯ g ∆ d.t.n ( η ) in both channels. This fact, loosly speaking, implies that { g ∆ d.t.n ( η ) , ¯ g ∆ d.t.n ( η ) } forma new basis. – 27 –o phrase our statement more precisely, we need to define the space of functions U for the correlators G ( η ) which we are considering. We define U to be the space with thefollowing “Regge” behavior G ( η ) ∈ U , if |G| (cid:46) | η | − (cid:15) , when η → ∞ (6.3)where (cid:15) is an infinitesimal positive number. For example, the mean field theory two-pointfunction (cid:104) φ ± ( x ) φ ± ( x ) (cid:105) = 1(1 + x ) ∆ φ (1 + x ) ∆ φ ( η − ∆ φ ± (1 − η ) − ∆ φ ) , (6.4)belongs to this space when ∆ φ > . The conformal blocks g ∆ ( η ) , ¯ g ∆ ( η ) are also in thisspace if the external dimensions min { ∆ , ∆ } > . On the other hand, the contact Wittendiagrams are not in this space (see (3.28)). This avoids having relations among the basisvectors { g ∆ d.t.n ( η ) , ¯ g ∆ d.t.n ( η ) } , as a contact Witten diagram can be decomposed into onlydouble-trace conformal blocks in either channel. Note that in the BCFT case, it wasproven that two-point correlators in any unitary theory have a bounded Regge behaviorwhen ξ → − [32]. The proof exploits the positivity of the decomposition coefficients inthe boundary channel. By contrast, in the case at hand here of RP d CFTs, positivity is not a priori guaranteed in either channel even when the theory is unitary. The Regge behaviorrequirement (6.3) is therefore imposed by hand.We claim that the double-trace conformal blocks { g ∆ d.t.n , ¯ g ∆ d.t.n } form a basis for thespace U . A basis for the dual space U ∗ is given by the set of functionals { ω m , ¯ ω m } , definedby dualizing the double-trace basis ω m ( g ∆ d.t.n ) = δ mn , ω m (¯ g ∆ d.t.n ) = 0 , ¯ ω m ( g ∆ d.t.n ) = 0 , ¯ ω m (¯ g ∆ d.t.n ) = δ mn . (6.5)Although we do not have a general proof for this proposal (except for the d = 2 case wherewe prove in Section 6.2 from the dispersion relation), we will provide ample evidence whichsupports this conjecture.The action of the basis functionals can be read off from the conformal block decom-positions of exchange Witten diagrams. Acting on (6.1) with ω m and use the orthonormalrelation (6.5), we get ω m ( g ∆ ) = − A m A . (6.6)Acting with ¯ ω m , we find ¯ ω m ( g ∆ ) = B m A . (6.7)The action of the functionals on the image channel conformal block ¯ g ∆ ( η ) can be obtainedfrom (6.2), and is related to the action on g ∆ ( η ) by crossing symmetry ω m (¯ g ∆ ) = ¯ ω m ( g ∆ ) , ¯ ω m (¯ g ∆ ) = ω m ( g ∆ ) . (6.8)Let us consider the action of the functionals on a two-point function G ∈ U with thefollowing conformal block decomposition G ( η ) = (cid:88) k µ k g ∆ O k ( η ) = ± (cid:88) k µ k ¯ g ∆ O k ( η ) . (6.9)– 28 –pplying the basis functionals allow us to extract the complete set of constraints in termsof the sum rules (cid:88) k µ k ω n ( g ∆ O k ) ∓ (cid:88) k µ k ω n (¯ g ∆ O k ) = 0 , (6.10) (cid:88) k µ k ¯ ω n ( g ∆ O k ) ∓ (cid:88) k µ k ¯ ω n (¯ g ∆ O k ) = 0 . (6.11)The exchange Witten diagrams can also be viewed as the Polyakov-Regge blocks in thePolyakov-style bootstrap [32, 44–56], and can be used as a new decomposition basis. Interms of the rescaled exchanged Witten diagrams (5.6), two-point function in (6.9) can berewritten as G ( η ) = (cid:88) k µ k ( P ∆ O k ( η ) ± ¯ P ∆ O k ( η )) . (6.12)To prove this relation, we can express (6.1), (6.2) in terms of functional actions, and sub-stitute in (6.12) with P ∆ ( η ) = g ∆ ( η ) − (cid:88) n ω n ( g ∆ ) g ∆ d.t.n ( η ) , (6.13) ¯ P ∆ ( η ) = (cid:88) n ω n (¯ g ∆ ) g ∆ d.t.n ( η ) . (6.14)We now have G ( η ) = (cid:88) k µ k g ∆ O k ( η ) − (cid:88) k µ k (cid:18) (cid:88) n ω n ( g ∆ O k ) g ∆ d.t.n ( η ) ∓ (cid:88) n ω n (¯ g ∆ O k ) g ∆ d.t.n ( η ) (cid:19) . (6.15)Interchanging the action of the functionals and the sums, the second term becomes (cid:88) n ω n (cid:18) (cid:88) k µ k ( g ∆ O k ( η ) ∓ ¯ g ∆ O k ( η )) (cid:19) g ∆ d.t.n ( η ) , (6.16)and vanishes because of the crossing equation (6.9). We have therefore proven the equiv-alence between (6.12) and the first identity in (6.9). To prove the second identity, we justneed to decompose the Polyakov-Regge blocks in the image channel P ∆ ( η ) = (cid:88) n ¯ ω n ( g ∆ )¯ g ∆ d.t.n ( η ) , (6.17) ¯ P ∆ ( η ) = ¯ g ∆ ( η ) − (cid:88) n ¯ ω n (¯ g ∆ )¯ g ∆ d.t.n ( η ) . (6.18)The alternative decomposition (6.12) can also be taken as the starting point for analyticbootstrap. In a generic interacting CFT, we do not expect operators with precise double-trace dimensions. However, if we use the the conformal block decompositions (6.13), (6.14),(6.17), (6.18) of the Polyakov-Regge blocks, we would encounter spurious double-traceoperators in (6.12). The requirement that these spurious operators should cancel gives sumrules for the OPE coefficients, which are identical to the conditions (6.10), (6.11).– 29 – .2 Functionals from dispersion relation As was pointed in [46], analytic functionals and the dispersion relation for correlators [57](see also [58]) are closely related. In particular, the kernel of the dispersion relation can beviewed as the generating function for the kernels of the analytic functional in their integralrepresentation. Here we will demonstrate that a similar relation holds for RP d CFTs byre-deriving the basis identified in Section 6.1 and constructing the dual functionals. Forsimplicity, we will only focus on d = 2 and set ∆ = ∆ = ∆ φ . Figure 7 : An illustration of the integral contours in the dispersion relation.We start with the Cauchy’s integral formula G ( η ) = (cid:73) dζ πi G ( ζ ) ζ − η , (6.19)with a contour encircling the point ζ = η . We can deform the contour, and wrap itaround the two branch cuts [1 , ∞ ) , ( −∞ , . We will denote these two deformed contoursrespectively as C and C , as is illustrated in Figure 7. Since we have assumed that thecorrelator has the Regge behavior (6.3), we can safely drop the arc at infinity. We thereforeobtain G ( η ) = G ( η ) + G ( η ) (6.20)where G ( η ) = (cid:90) C dζ πi G ( ζ ) ζ − η = ∞ (cid:90) dζ πi Disc [ G ( ζ )] ζ − η , (6.21) G ( η ) = − (cid:90) C dζ πi G ( ζ ) ζ − η = (cid:90) −∞ dζ πi Disc [ G ( ζ )] ζ − η , (6.22)– 30 –nd Disc [ G ( ζ )] = G ( ζ + i + ) − G ( ζ − i + ) , for ζ ∈ (1 , ∞ ) , (6.23) Disc [ G ( ζ )] = G ( ζ + i + ) − G ( ζ − i + ) , for ζ ∈ ( −∞ , . (6.24)Let us define k h ( η ) = η h F ( h, h ; 2 h, η ) . (6.25)These functions satisfy the following orthonormality condition (cid:73) | η | = (cid:15) dη πi η − k x + n ( η ) k − x − m ( η ) = δ mn . (6.26)Note that for d = 2 and ∆ = ∆ = ∆ φ the double-trace conformal blocks are just g ∆ d.t.n ( η ) = η − ∆ φ k ∆ φ + n ( η ) . (6.27)We expect that the Cauchy kernel can be expanded in terms of the double-trace conformalblocks ζ − η = ∞ (cid:88) n =0 H n ( ζ ) g ∆ d.t.n ( η ) . (6.28)The coefficients H n ( ζ ) can then be extracted using the orthonormality relation (6.26) H n ( ζ ) = (cid:73) | η | = (cid:15) dη πi η ∆ φ − ζ − η k − ∆ φ − n ( η ) . (6.29)The integral is simple to perform and gives H n ( ζ ) = ( − − n (∆ φ ) n (2∆ φ − n n !(∆ φ − ) n ζ − F (1 , − n, φ + n −
1; ∆ φ , ∆ φ , ζ − ) . (6.30)Therefore we have shown that G ( η ) can be expanded in terms of the bulk channel double-trace conformal blocks G ( η ) = ∞ (cid:88) n =0 r n, g ∆ d.t.n ( η ) (6.31)where r n, = (cid:90) C dζ πi H n ( ζ ) G ( ζ ) . (6.32)We can perform a similar analysis for G . However, by crossing symmetry G ( η ) = ±G (1 − η ) (6.33)where ± is common parity of the two operators. It follows that G ( η ) admits an expansioninto mirror channel double-trace conformal blocks G ( η ) = ∞ (cid:88) n =0 r n, ¯ g ∆ d.t.n ( η ) (6.34)– 31 –here r n, = − (cid:90) C dζ πi H n (1 − ζ ) G ( ζ ) . (6.35)In fact the above decomposition is even valid when we do not assume crossing symmetry.To see this, we simply need to notice ζ − η = − − ζ ) − (1 − η ) = − ∞ (cid:88) n =0 H n (1 − ζ )¯ g ∆ d.t.n ( η ) . (6.36)We have now established that the double-trace conformal blocks g ∆ d.t.n and ¯ g ∆ d.t.n forma basis for functions satisfying the Regge behavior (6.3), which is captured by the decom-position G ( η ) = ∞ (cid:88) n =0 r n, g ∆ d.t.n ( η ) + ∞ (cid:88) n =0 r n, ¯ g ∆ d.t.n ( η ) . (6.37)The coefficients can then be interpreted as the actions of the dual functionals r n, = ω n [ G ( η )] , r n, = ¯ ω n [ G ( η )] . (6.38)Using their definitions (6.32), (6.35), we see that the functional kernels indeed follow fromthe dispersion relation kernel as we claimed earlier.The discussion in this section is quite similar to the CFT case discussed in Section 2of [46]. But we should notice that the basis in Section 2 of [46] does not coincide with theexpected basis from holography. Holography suggests a basis containing all conformal blockswith dimensions φ + 2 n and their derivatives with respect to the conformal dimension,while the basis from the Cauchy dispersion kernel consists of all conformal blocks withdimensions φ + n and no derivatives. This is different in the RP d CFT case. We findthat the Cauchy dispersion kernel gives exactly the same double-trace basis which we expectfrom holography.
Nonperturbative checks
Let us now perform some quick checks of the equivalence between functional actions ob-tained from the dispersion relation, and the ones obtained from Witten diagrams. Considerthe following crossing symmetric toy example of a correlation function G ( η ) = 1 (cid:112) η (1 − η ) . (6.39)We will set the external dimensions to be ∆ = ∆ = 2 . The ‘correlator’ can be decomposedinto the d = 2 conformal blocks with a spectrum ∆ = 1 + 2 n , n ∈ Z + G ( η ) = ∞ (cid:88) n =1 α n g ∆=1+2 n ( η ) α n = ( − n − n (cid:0) n − (cid:1) ! (cid:0) − n Φ (cid:0) − , , n + (cid:1) − π (cid:1) √ π ( n − (6.40)– 32 –here Φ here is the Lerch transcendent function. Note that the mean field theory double-trace operators have conformal dimensions ∆ = 4 + 2 n , n ∈ N . The OPE spectrum of theabove correlator is therefore ‘maximally’ different from the mean field spectrum, and is inthis sense a ‘nonperturbative’ check. We can act with the functionals on both sides using(6.6) and (6.38), and it leads to the following constraints ω n [ G ( η )] = (cid:90) ∞ dζ πi Disc [ G ( ζ ) H n ( ζ )] = − ∞ (cid:88) m =1 α m (cid:18) A n A (cid:19) ∆=1+2 m, ∆ =∆ =2 . (6.41)We checked numerically that this constraint is true for n = 0 , and .A more physical example is given by the d Ising model. The exact solution is knownfor this model. The σ operator, which has conformal dimension ∆ σ = 1 / , has a two pointfunction on RP d given by (we assume that σ has positive parity) [13] G σ ( η ) = (cid:112) − √ − η + (cid:112) − √ η ( η (1 − η )) / = ∞ (cid:88) n =0 ρ ,n g ∆=4 n ( η ) + ∞ (cid:88) n =1 ρ ,n g ∆=1+4 n ( η ) . (6.42)The coefficients ρ ,n and ρ ,n can be found recursively using the expansion of the correlator.Note that G σ ( η ) approaches a constant as η → ∞ , and therefore does not belong to the spaceof functions we defined. In fact, a direct application of the functionals leads to a divergentsum. So instead we perform check on G σ ( η ) /η which has an improved Regge behavior. Wewould like the new correlator to have the same operator spectrum in the conformal blockdecomposition. Therefore, we will take the external dimensions to be ∆ = ∆ = 9 / instead of / . The action of the functionals then requires ω n (cid:20) G σ ( η ) η (cid:21) = (cid:90) ∞ dζ πi Disc (cid:20) G σ ( ζ ) ζ H n ( ζ ) (cid:21) = − ∞ (cid:88) m =0 ρ ,m (cid:18) A n A (cid:19) ∆=4 m, ∆ =∆ = − ∞ (cid:88) m =0 ρ ,m (cid:18) A n A (cid:19) ∆=4 m +1 , ∆ =∆ = . (6.43)We checked this relation numerically for n = 0 , and and it holds true. In this subsection, we apply our functionals to some perturbative examples. We start bychecking the sum rules (6.10), (6.11) on the mean field theory. We then consider smallperturbations around the mean field theory and show how we can obtain the data forWilson-Fisher model on RP d using these sum rules. Note that the mean field theory two-point function has the following conformal block decomposition G ( η ) = 1 η ∆ φ ± − η ) ∆ φ = g ∆=0 ( η ) + ∞ (cid:88) n =0 µ φφn g ∆=∆ d.t.n ( η ) (6.44)where ∆ d.t.n = 2∆ φ + 2 n and the OPE coefficients are given by µ (0) φφn = ± (∆ φ ) n (2∆ φ − d + 2 n ) − n (∆ φ − d + n + 1) − n n ! . (6.45)– 33 –hese OPE coefficients are just a simple modification of the BCFT case, which can befound for instance in [24]. We use the superscript to indicate that we will soon perturbthe mean field theory solution. The sum rules then tells us that the following must be truefor all values of n − (cid:18) A n A (cid:19) ∆=0 + µ (0) φφn = ± (cid:18) B n A (cid:19) ∆=0 . (6.46)Now recall the expression of the coefficients A n and A from (4.16) and (4.18). For ∆ =∆ = ∆ φ they take a simpler form A n = ( − n (∆ φ ) n (2∆ φ − d + 2 n ) − n (∆(∆ − d ) − (2∆ φ + 2 n )(2∆ φ + 2 n − d )) n ! A = Γ (cid:0) ∆2 (cid:1) Γ (cid:0) ∆ φ − ∆2 (cid:1) Γ (cid:16) ∆ φ − ( d − ∆)2 (cid:17) φ ) Γ (cid:0) ∆ + 1 − d (cid:1) . (6.47)It is then clear that as we take ∆ → the expression for the coefficient A diverges, while A m remains finite. So the first term in the constraint equation (6.46), A m /A does notcontribute. The constraint then becomes (cid:18) B n A (cid:19) ∆=0 = ± µ (0) φφn (6.48)This constraint can be explicitly checked for n = 0 and n = 1 using the results for B n in(4.24) and (4.23). For all other values of n , note that the recursion relation (4.23) implies (cid:18) ρ n − B n − A (cid:19) ∆=0 + (cid:18) ν n B n A (cid:19) ∆=0 + (cid:18) µ n +1 B n +1 A (cid:19) ∆=0 = (cid:16) a n A (cid:17) ∆=0 = 0= ⇒ (cid:16) ρ n − µ (0) φφn − + ν n µ (0) φφn + ν n µ (0) φφn (cid:17) ∆=0 = 0 (6.49)which can be easily checked to be true for all values of n . This completes our check ofanalytic functionals for mean field theory. Wilson-Fisher model
We now consider perturbations around the mean field solution such that the above OPEcoefficients µ φφn and dimensions receive small corrections. One such perturbation is theWilson-Fisher fixed point in d = 4 − (cid:15) which is a perturbation of free field theory with ∆ (0) φ = d − . It has a Lagrangian description, which in our normalization can be writtenas S = Γ (cid:0) d − (cid:1) π d/ (cid:90) d d x (cid:18)
12 ( ∂ µ φ I ) + λ φ I φ I ) (cid:19) . (6.50)But we will not need this Lagrangian description, and we will treat it as a perturbation ofa mean field theory of N free fields. We parametrize deviations from the mean field valuesas follows µ φφn = µ (0) φφn + (cid:15) µ (1) φφn + (cid:15) µ (2) φφn , ∆ φ = d − (cid:15) γ (2) φ ∆ = ∆ d.t.n + (cid:15)γ (1) n + (cid:15) γ (2) n = 2∆ φ + 2 n + (cid:15)γ (1) n + (cid:15) γ (2) n (6.51)– 34 –here we used the well known fact that in this model, the first order correction to theanomalous dimension of φ vanishes. From (6.45), we see that for this free field value of ∆ (0) φ , µ (0) φφn = ± δ n, , which truncates the functional equations. This leaves us with a finitenumber of terms on both sides. Using (6.5) the sum rule (6.10) at order (cid:15) just becomes µ (1) φφn ∓ (cid:18) A n A (cid:19) O ( (cid:15) )∆=∆ d.t. + (cid:15)γ (1)0 = (cid:18) B n A (cid:19) O ( (cid:15) )∆=0 + (cid:18) B n A (cid:19) O ( (cid:15) )∆=∆ d.t. + (cid:15)γ (1)0 (6.52)and the superscript indicates that we pick out the order (cid:15) contribution. Expanding in (cid:15) , wecan check that ( A n /A ) term does not contribute at order (cid:15) . Also using (6.48) and (6.45),we can check that ( B n /A ) does not contribute for ∆ = 0 . As for the other term on theright hand side, it only contributes at this order for n = 0 and and using the recursionrelation (4.23), we can check that all the other values of n start contributing at order (cid:15) .This gives the following results for the CFT data µ (1) φφ = − γ (1)0 , µ (1) φφ = γ (1)0 , µ (1) φφn ≥ = 0 (6.53)This agrees with what was found in [16]. The fact that only two of the OPE coefficients arenon-zero at this order implies that the functional equations also truncate to finite terms atnext order. At next order in (cid:15) , we obtain using the sum rule µ (2) φφn ∓ (cid:18) A n A (cid:19) O ( (cid:15) )∆=∆ d.t. + (cid:15)γ (1)0 + (cid:15) γ (2)0 − µ (1) φφ (cid:18) A n A (cid:19) O ( (cid:15) )∆=∆ d.t. + (cid:15)γ (1)0 − µ (1) φφ (cid:18) A n A (cid:19) O ( (cid:15) )∆=∆ d.t. + (cid:15)γ (1)1 = (cid:18) B n A (cid:19) O ( (cid:15) )∆=0 + (cid:18) B n A (cid:19) O ( (cid:15) )∆=∆ d.t. + (cid:15)γ (1)0 + (cid:15) γ (2)0 ± µ (1) φφ (cid:18) B n A (cid:19) O ( (cid:15) )∆=∆ d.t. + (cid:15)γ (1)0 ± µ (1) φφ (cid:18) B n A (cid:19) O ( (cid:15) )∆=∆ d.t. + (cid:15)γ (1)1 . (6.54)For the identity block, using (6.48) and (6.45), it is easy to check that (cid:18) B n A (cid:19) O ( (cid:15) )∆=0 = ± γ (2) φ (Γ( n )) Γ(2 n ) , n ≥ (cid:18) B A (cid:19) O ( (cid:15) )∆=0 = 0 . (6.55)For other values of ∆ , expanding the A n -functionals is straightforward. To expand the B n -functionals, which involve hypergeometric functions, we use the package HypExp [59].We collect below the needed expansions for a few low-lying values of n (cid:18) B A (cid:19) ∆=∆ d.t. + (cid:15)γ (1)0 + (cid:15) γ (2)0 = − γ (1)0 (cid:15) − (cid:16) γ (1)0 + 2 γ (2)0 (cid:17) (cid:15) ; (cid:18) B A (cid:19) ∆=∆ d.t. + (cid:15)γ (1)1 = γ (1)1 ( π − (cid:15), (cid:18) B A (cid:19) ∆=∆ d.t. + (cid:15)γ (1)0 + (cid:15) γ (2)0 = γ (1)0 (cid:15) + (cid:16) γ (1)0 (2 γ (1)0 −
1) + 4 γ (2)0 (cid:17) (cid:15) ; (cid:18) B A (cid:19) ∆=∆ d.t. + (cid:15)γ (1)1 = − γ (1)1 (cid:15), (cid:18) B A (cid:19) ∆=∆ d.t. + (cid:15)γ (1)0 + (cid:15) γ (2)0 = − γ (1)0 ( γ (1)0 − (cid:15) ; (cid:18) B A (cid:19) ∆=∆ d.t. + (cid:15)γ (1)1 = γ (1)1 (cid:15), (cid:18) B A (cid:19) ∆=∆ d.t. + (cid:15)γ (1)0 + (cid:15) γ (2)0 = γ (1)0 ( γ (1)0 − (cid:15) ; (cid:18) B A (cid:19) ∆=∆ d.t. + (cid:15)γ (1)1 = − γ (1)1 (cid:15). (6.56)– 35 –oing to higher values of n is also straightforward by using the recursion relation (4.23).Using these expansions of coefficients, we can obtain the results of µ (2) φφn with n = 0 , , , µ (2) φφ = γ (1)0 ( γ (1)1 − − γ (2)0 ± γ (1)0 ( γ (1)0 − γ (1)1 )4 µ (2) φφ = γ (1)0 (2 γ (1)0 − γ (1)1 − γ (2)0 ∓ γ (1)0 (3 γ (1)0 + γ (1)1 − ± γ (2) φ µ (2) φφ = ± γ (1)0 ( γ (1)0 − γ (1)1 )48 − γ (1)0 ( γ (1)0 − − γ (1)0 γ (1)1 ± γ (2) φ µ (2) φφ = γ (1)0 ( γ (1)0 − γ (1)0 γ (1)1 ∓ γ (1)0 ( γ (1)0 − γ (1)1 )360 ± γ (2) φ . (6.57)The bulk data of the O ( N ) vector model at Wilson-Fisher fixed point can be found in[25, 60, 61] γ (1)0 = N + 2 N + 8 , γ (1)1 = 1 , γ (2)0 = 6( N + 2)( N + 3)( N + 8) , γ (2) φ = N + 24( N + 8) . (6.58)This then gives us the following RP d OPE coefficients to the (cid:15) order µ φφ = ± − N + 22( N + 8) (cid:15) − N + 2)(2 N + 6 ± ( N + 8))2( N + 8) (cid:15) µ φφ = N + 24( N + 8) (cid:15) − ( N + 2)4( N + 8) (cid:18)
76 + N ( N + 10)2( N + 8) ± ( N − (cid:19) (cid:15) µ φφ = N + 248( N + 8) (cid:18) ± ( N + 4) −
43 ( N − (cid:19) (cid:15) µ φφ = N + 2320( N + 8) (cid:18) ( N + 4) ∓
89 ( N + 2) ± (cid:19) (cid:15) . (6.59)We can obtain numerical estimations for OPE coefficients in the d = 3 Ising model byplugging in (cid:15) = 1 and N = 1 in the above expressions. This gives in particular µ + φφ = 0 . and µ + φφ = 0 . , which can be compared with the results obtained by the bootstrapanalysis [13]. The bootstrap result gives µ + φφ ∼ . and µ + φφ ∼ . , and is in goodagreement with our result. We can keep going and it is completely straightforward to obtainall the OPE coefficients at order (cid:15) using (6.54). On the other hand, since all the OPEcoefficients are non-zero at this order, the sum rules at the next order will contain infinitenumber of terms. The sum rules still put nontrivial constraints on the OPE coefficients,but it is not clear how the constraints can be solved analytically. Large N checks
We now provide an independent consistency check of some of these results by consideringthe large
N O ( N ) vector model and compare the results in that with the large N limit of(6.59).Let us first note that to the leading order in (cid:15) , we can use the results from (6.59) towrite the two-point function as (cid:104) φ I ( x ) φ J ( x ) (cid:105) = δ IJ G ( η )((1 + x )(1 + x )) ∆ φ (6.60)– 36 –ith G ( η ) = g ∆=0 ( η ) + µ φφ g ∆=∆ d.t. ( η ) + µ φφ g ∆=∆ d.t. ( η )= 1 η d − (cid:32) ± (cid:18) η − η (cid:19) d − ± (cid:15) ( N + 2)2( N + 8) (cid:18) η − η (cid:19) log η + (cid:15) ( N + 2)2( N + 8) log(1 − η ) (cid:33) (6.61)To develop a large N expansion, we introduce the usual Hubbard-Stratonovich auxiliaryfield σ and write down the action as S = Γ (cid:0) d − (cid:1) π d/ (cid:90) d d x (cid:18)
12 ( ∂ µ φ I ) + σφ I φ I (cid:19) (6.62)where we omitted a σ / λ term, which can be dropped at the fixed point. At leading orderin large N , we get the following equation of motion for the φ correlator ( ∇ − (cid:104) σ ( x ) (cid:105) ) (cid:104) φ I ( x ) φ J ( x ) (cid:105) = − π d/ Γ (cid:0) d − (cid:1) δ IJ δ d ( x − x ) . (6.63)At large N the scaling dimension of σ is /N , while the scaling dimension of φ is d/ − /N . We can again express these correlators as (cid:104) σ ( x ) (cid:105) = a σ (1 + x ) , (cid:104) φ I ( x ) φ J ( x ) (cid:105) = δ IJ G ( η )((1 + x )(1 + x )) ∆ φ = δ IJ G ( η )(( x − x ) ) ∆ φ . (6.64)Plugging in this general form into the equation of motion, we get η (1 − η ) dG ( η ) dη + (8(1 − η ) − d ) dG ( η ) dη − a σ G ( η ) = 0 . (6.65)This equation can be solved and the general solution is G ( η ) = C (cid:18) η − η (cid:19) d − F (cid:18) √ − a σ , − √ − a σ , − d , − η (cid:19) + C F (cid:18) √ − a σ , − √ − a σ , d , − η (cid:19) . (6.66)Recall that the crossing equation (2.14) requires G ( η ) η d − = ± G (1 − η )(1 − η ) d − (6.67)which implies that the coefficients must satisfy C C = ± Γ (cid:16) d − √ − a σ (cid:17) Γ (cid:16) d − −√ − a σ (cid:17) Γ (cid:0) − d (cid:1) π Γ (cid:0) d (cid:1) (cid:18) − sin π (cid:18) dπ (cid:19) ∓ cos (cid:18) π √ − a σ (cid:19)(cid:19) . (6.68)– 37 –he overall constant can then be fixed by demanding that the leading term in small η expansion of G is just 1. We can expand this solution in small η as G ( η ) = C (cid:18) ± ∓ a σ d − η + O ( η ) (cid:19) + C η d − π Γ (cid:0) − d (cid:1)(cid:16) sin (cid:0) πd (cid:1) ∓ cos (cid:16) π √ − a σ (cid:17)(cid:17) Γ (cid:0) d (cid:1) Γ (cid:16) − d −√ − a σ (cid:17) Γ (cid:16) − d + √ − a σ (cid:17) + O ( η ) . (6.69)The first term in the first line is the contribution from the identity operator, while thesecond term of order η is from the σ operator of dimension . The term in the second linerepresents the φ operator of dimension d − . In the large N theory, we expect the φ operator of dimension d − in the free theory to be replaced by the σ operator of dimension . This then requires us to set the term in the second line of the above equation to zero,which implies the following possible values of a σ a + σ = − ( d − d − , − ( d − d − , .... a − σ = − ( d − d − , − ( d − d − , .... (6.70)Note that for these values of a σ , the coefficient C also vanishes. Now we expect the large N theory to match with the free theory in d = 4 . This means that we must choose thevalue of a σ such that a σ = 0 as d → . This then picks out the solution for us for both + and − parity a + σ = − ( d − d − , = ⇒ G + ( η ) = 1(1 − η ) d − a − σ = − ( d − d −
6) = ⇒ G − ( η ) = 1 − η (1 − η ) d − . (6.71)It can be checked that these results in d = 4 − (cid:15) agree with the large N limit of the (cid:15) expansion solution (6.61). These two point functions can be decomposed into conformalblocks of dimension n + 2 as follows G − ( η ) = 1( η (1 − η )) d − = g ∆=0 ( η ) + ∞ (cid:88) n =0 λ + n g ∆=2 n +2 ( η ) λ + n = Γ (cid:0) d (cid:1) F (cid:0) − n − , − n ; ( d − n − (cid:1) Γ( n + 2)Γ (cid:0) d − n − (cid:1) (6.72)for the + case and G − ( η ) = 1 − η ( η (1 − η )) d − = g ∆=0 ( η ) + ∞ (cid:88) n =0 λ − n g ∆=2 n +2 ( η ) λ − n = − ( − n ( d − d ( n + 2) + 8(1 + n ) + 4)Γ(1 − d + n )Γ(2 − d + n ) − d ) Γ( n + 2)Γ(2 − d + 2 n ) (6.73) In appendix A, we provide another way to check this value of a σ , where it occurs as a large N saddlepoint of the free energy. – 38 –or the − case. These coefficients can be found in a manner similar to the one used forBCFT case which can be found in [24, 62]. We want to emphasize here that unlike themean field theory, the conformal blocks appearing here have dimensions n + 2 . These OPEcoefficients can be expanded in (cid:15) in d = 4 − (cid:15) , and we find a precise match with the large N limit of (6.59). A complementary approach to using the analytic functionals, where a Lagrangian descrip-tion for the CFT is available, is to use the CFT equations of motion. The essential ideawas described for the CFT in flat space in [25] and extended to the case of BCFT in [62].Here we will use this method to fix the two-point function of the field φ in the Wilson-Fishermodel on RP d . This case is very similar to the BCFT case. Since the two-point function isonly a function of cross-ratio η on the sphere quotient S d / Z and is proportional to G ( η ) ,in this subsection it will be more convenient to ‘undo’ the Weyl transformation (1.2) andwork on the sphere quotient.The action for the φ Wilson-Fisher model, including the conformal coupling term, canbe written as S = Γ (cid:0) d − (cid:1) π d/ (cid:90) d d x (cid:18)
12 ( ∂ µ φ I ) + d ( d − φ I φ I + λ φ I φ I ) (cid:19) . (6.74)The two-point function on the sphere is (cid:104) φ I ( x ) φ J ( x ) (cid:105) = δ IJ G ( η )4 . (6.75)Let us start with the free theory with λ = 0 . The the field φ I satisfies ( ∇ − d ( d − / φ I =0 , which implies that (cid:18) √ g ∂ µ ( g µν √ g∂ ν ) − d ( d − (cid:19) G ( η ) = 0 η (1 − η ) d G dη + d (cid:18) − η (cid:19) d G dη − d ( d − G = D (2) G ( η ) = 0 . (6.76)This equation can be solved, and the general solution is G ( η ) = b (cid:32) η d − + 1(1 − η ) d − (cid:33) + b (cid:32) η d − − − η ) d − (cid:33) . (6.77)The constants are just fixed by the normalization, and we pick b = 1 , b = 0 for + parity,and vice versa for the − parity. When we include interactions, the equation of motion getsmodified to ( ∇ − d ( d − / φ I ( x ) = λφ I φ K φ K ( x ) . This implies D (2) G ( η ) = λ ∗ ( N + 2) a φ G ( η ) + O ( λ ∗ ) (6.78) Note that this is different from what we did in Section 3, where we used the equations of motion in thebulk
AdS d +1 / Z . – 39 –o leading order in λ . We can solve this perturbatively in d = 4 − (cid:15) , where this model has anon-trivial fixed point. λ ∗ is the fixed point value of the coupling and is equal to (cid:15)/ ( N + 8) in this normalization, while a φ / is the one-point function of φ on the sphere. Since thereis a factor of λ ∗ on the right hand side, we can plug in the correlators in d = 4 , and it willgive us the two-point function on the left hand side, correct to order (cid:15) . We can expand thedifferential operator and the correlator as follows G ( η ) = G ( η ) + (cid:15) G ( η ) + (cid:15) G ( η ) + O ( (cid:15) ) D (2) = D (2)0 + (cid:15)D (2)1 + O ( (cid:15) ) (6.79)where G ( η ) is just given by (6.77) with d = 4 . The equation of motion at first order in (cid:15) is D (2)0 G ( η ) = ± N + 22( N + 8) G ( η ) − D (2)1 G ( η ) . (6.80)This equation can also be solved te give G ( η ) = c η + c − η + log η η ± log(1 − η )2(1 − η ) + N + 22( N + 8) (cid:18) log(1 − η ) η ± log η − η (cid:19) . (6.81)If we fix the normalization such that G ( η ) = η − as η → , this fixes c = 0 . Also,the crossing symmetry (2.14) requires G to be either symmetric or antisymmetric under η → − η , which sets c = 0 . This order (cid:15) correlator then agrees exactly with the resultusing functionals (6.61). Now to go to the next order, note that in the two-point function φ I ( x ) φ J ( x ) , we can also apply the equation of motion to the other φ . This gives thefollowing fourth-order differential equation D (4) G ( η ) = D (2) ( D (2) G ( η )) = λ ∗ ( N + 2)16 ( a φ ( N + 2) G ( η ) + 2 G ( η ) ) . (6.82)We can again solve it perturbatively in (cid:15) by expanding D (4) = D (4)0 + (cid:15)D (4)1 + (cid:15) D (4)2 + O ( (cid:15) ) .The differential equation at O ( (cid:15) ) then becomes D (4)0 G ( η ) = ( N + 2)4( N + 8) (cid:0) ( N + 2) G ( η ) + 2 G ( η ) (cid:1) − D (4)1 G ( η ) − D (4)2 G ( η ) . (6.83)This general solution of this equation is G ( η ) = d η + d − η + d log η − η + d log(1 − η ) η − N + 24( N + 8) (cid:18) log ηη ± log(1 − η )1 − η (cid:19) + log η η ± log (1 − η )8(1 − η )+ ( N + 2) N + 8) (cid:18) log (1 − η ) η ± log ( η )1 − η (cid:19) + N + 24( N + 8) (cid:18) η ± − η (cid:19) log η log(1 − η ) . (6.84)To fix the constants, we again use the normalization and demand symmetry/antisymmetryunder η → − η . This sets d = d = 0 , and d = ± d . To fix d , we recall that the in the– 40 –irect channel, η → , the correlator should behave as G ( η ) = η − ∆ φ + µ φφ η ∆ d.t. − ∆ φ + higher orders in η = η − ∆ φ + µ (0) φφ + (cid:15) (cid:32) µ (1) φφ + γ (1)0 η (cid:33) + (cid:15) (cid:18) µ (2) φφ + ( µ (1) φφ γ (1)0 + µ (0) φφ γ (2)0 ) log η µ (0) φφ ( γ (1)0 ) log η (cid:19) + O ( η ) . (6.85)Comparing the log η terms at order (cid:15) with (6.84) then tells us µ (1) φφ γ (1)0 + µ (0) φφ γ (2)0 = 2 d − N + 22( N + 8)= ⇒ d +3 = 3( N + 2)(3 N + 14)2( N + 8) , d − = − N + 2)( N − N + 8) . (6.86)This gives us an explicit expression for the complete two-point function to order (cid:15) forthe Wilson-Fisher model on RP d . Expanding the two-point function in powers of η andextracting the OPE coefficients, it is easy to check that this agrees with the results in (6.59)which we found by using functionals. The large N limit of this solution also agrees with(6.71) in d = 4 − (cid:15) . Our discussion of RP d CFTs can also be related to the bulk reconstruction program in thelarge N limit. In this section we make a number of comments regarding the connectionwith previous works.Let us begin by noticing that inserting a local operator in Euclidean AdS d +1 has thesame effect of breaking the isometry from SO ( d + 1 , to SO ( d, , as performing the Z quotient (3.3). This is because the Z quotient selects a special fixed point N c , just asinserting a local bulk operator. However, the identification under the inversion does notfurther change the Lie algebra of the residual symmetry group, and we will not impose suchidentifications in this section. Notice that N c now is no longer a special point, becauseAdS space is homogenous. Nevertheless, we will always use the AdS isometry generators tomove the local bulk operator to N c without loss of generality, so that it is easier to make aconnection with the discussions in Section 3.We will compare the holographic objects considered in Section 3 with those arising fromthe Hamilton-Kabat-Lifschytz-Lowe (HKLL) approach for constructing local bulk operators[29–31], which is perturbative in nature in the /N expansion. In the HKLL approach,a bulk field at a point in AdS can be defined by smearing the CFT operator with the There are also intrinsically non-perturbative and state-independent developments which exploit theidentification of twisted Ishibashi states with bulk operators [17–22], and are explored most extensively intwo dimensions. In fact, twisted Ishibashi states can be considered even when the boundary spacetime doesnot have crosscap insertions. – 41 –ulk-to-boundary propagator (we do not keep track of the overall normalizations in thissection) Φ (0)∆ ( N c ) = (cid:90) dP G ∆ B∂ ( N c , P ) O ∆ ( P ) . (7.1)The bulk-boundary two-point function can be obtained by performing the above smearingin the CFT two-point function, and we get (cid:104) Φ (0)∆ ( N c ) O ∆ ( P ) (cid:105) ∝ G ∆ B∂ ( N c , P ) . (7.2)This reproduces the one-point function (3.20).However, applying (7.1) to a CFT three-point function runs into the problem of non-vanishing commutators for space-like separated operators, as the prescription is only goodfor free particles. Doing the integral, we get [63] (cid:104) Φ (0)∆ ( N c ) O ∆ ( P ) O ∆ ( P ) (cid:105) ∝ η ∆ − ∆1 − ∆22 (1 + x ) ∆ (1 + x ) ∆ F (cid:0) ∆+∆ − ∆ , ∆+∆ − ∆ ; ∆ − d + 1; η (cid:1) . (7.3)We recognize that this bulk-boundary three-point function is nothing but the conformalblock g ∆ ( η ) , which is not surprising from the symmetry point of view. The conformalblock has a branch cut starting at η = 1 where points are space-like separated. The existenceof the singularity indicates a failure of the micro-causality. Meanwhile, we recall that inSection 3.3 we found an alternative geometric representation for the conformal blocks.This gives the above three-point function an interpretation in terms of a geodesic Wittendiagram. Using this picture, we can obtain an intuitive understanding of the singularitywithout computing the integral. We note that the point η = 1 corresponds to the limitwhere one boundary point is approaching the image of the other boundary point. In thislimit, the geodesic line which connects the two boundary points goes through the fixed bulkpoint N c (see Figure 8). This creates a short distance singularity in the integral, and makesthe three-point function singular.To resolve the singularity and restore micro-causality, [64] proposed that one shouldcorrect Φ (0)∆ with infinitely many double-trace operators Φ ∆ ( N c ) = Φ (0)∆ ( N c ) + ∞ (cid:88) n =0 a n (cid:90) dP G ∆ +∆ +2 nB∂ ( N c , P ) : O ∆ O ∆ : ( P ) (7.4)where a n contains a /N suppression so that both terms contribute at the same order. Thecoefficients of the double-trace operators can be systematically determined by cancelling thebranch point singularity at η = 1 [64]. As a result, the three-point function (cid:104) Φ ∆ O ∆ O ∆ (cid:105) now contains not only the single-trace conformal block g ∆ ( η ) , but also infinitely manydouble-trace conformal blocks g ∆ +∆ +2 n ( η ) . One can imagine that all the contributionsto (cid:104) Φ ∆ O ∆ O ∆ (cid:105) have been resummed. Then, the end result of this prescription for thebulk reconstruction should coincide with the exchange Witten diagram W exchange ∆ defined One can act on it with the two-particle conformal Casimir operator on the boundary, and use theequation of motion identity for the bulk-to-boundary propagator, to show that it is an eigenfunction. – 42 – igure 8 : Illustration of the branch cut singularity in the conformal block from the geodesicWitten diagram picture. In the limit η → , the point 2 approaches the image of point1. The geodesic line connecting these two points now goes through N c , and the geodesicWitten diagram integral divergence.in (3.31). To see it, we recall that the conformal block decomposition of W exchange ∆ has thesame structure as (7.4), and W exchange ∆ is free of singularities at η = 1 . Note that there isone detail we have glossed over: there are also homogeneous solutions to a n which do nothave branch singularities. But these solutions just correspond to contact Witten diagrams,which are polynomials of the cross ratio.The above holographic reconstruction of the three-point function (cid:104) Φ ∆ O ∆ O ∆ (cid:105) can bealternatively phrased as a conformal bootstrap problem. We can ask the following questionin the spirit of the seminal work [65]: given the appearance of a single-trace operator withdimension ∆ , what is the total contribution to the field theory two-point function of O ∆ , O ∆ at order /N , dictated by the partially broken SO ( d, ⊂ SO ( d + 1 , conformalsymmetry? This question is similar in flavor to the question asked in [66] about four-point functions in CFTs with full conformal symmetry. Since we are in the large N limit,the conditions of our question indicate that the conformal block decomposition should takethe following form G Φ OO ( η ) = µg ∆ ( η ) + ∞ (cid:88) n =0 b n g ∆ +∆ +2 n ( η )= ∞ (cid:88) n =0 c n ¯ g ∆ +∆ +2 n ( η ) (7.5)where only double-trace conformal blocks are allowed to appear in addition to the single-trace conformal block. We can view these expressions as the leading order deformation tothe mean field theory two-point function, by adding a single-trace operator. In order toget rid of the anticipated ambiguities in the double-trace operators coming from contact This requires the single-trace operator O ∆ to appear in the OPE of O ∆ × O ∆ , and also to havea nonzero one-point function. The latter is possible because we assume the conformal symmetry to bepartially broken. We emphasize again that the breaking of conformal symmetry is not due to placing thetheory on RP d (the space is still R d ), but due to the presence of a bulk local operator (to be interpretedfrom the solution to the problem). – 43 –iagrams, we should also impose a bound on the Regge behavior |G Φ OO | (cid:46) | η | − (cid:15) , when η → ∞ . (7.6)The extra homogeneous solutions with just double-trace conformal blocks can always beconveniently added back in the very end. This conformal bootstrap problem can then beeasily solved by using the analytic functionals which we introduced in Section 6.1. Applyingthe basis of functionals on (7.5), we find that b n = − µ ω n ( g ∆ ) , c n = µ ω n (¯ g ∆ ) . (7.7)Comparing with (6.13), this indicates that G Φ OO ( η ) is just proportional to the uniquelydefined Polyakov-Regge block P ∆ ( η ) , i.e. , an exchange Witten diagram with a local operatorin the bulk AdS. In this paper we performed an analytic study of CFTs on real projective space. We gave adetailed account of a toy model of holography on a Z quotient of AdS, and studied prop-erties of Witten diagrams on this background. The investigation led to a basis of analyticfunctionals dual to double-trace conformal blocks. We explicitly constructed these func-tionals from the conformal block decomposition coefficients of exchange Witten diagrams.Although the functionals stem from a toy holography model, they apply universally to RP d CFTs. In particular, we applied these functionals to study O ( N ) vector model in − (cid:15) expansion, and obtained one-point functions to order (cid:15) . We also studied in detail the large N O ( N ) vector model on RP d using independent field theory techniques, and obtainedresults that are consistent with the (cid:15) -expansion. Our work leads to a number of interestingfuture directions.An interesting extension of our work is to include fermions, and study models on realprojective space such as QCD and the Gross-Neveu model. Including fermions is alsonecessary for considering theories with supersymmetry. The case of 4d N = 4 SYM on RP has been recently considered in [10] using supersymmetric localization techniques. It willbe nice to study it using other analytic techniques, such as those developed in this paper.As we pointed out in Section 5, the existence of the two-term dimensional reductionformula for conformal blocks suggests an extension of the Parisi-Sourlas supersymmetry toreal projective space. It would be very interesting to study in detail the realization of thesymmetry in concrete models such as branched polymers, and test its equivalence with theYang-Lee critical theory on a real projective space with two dimensions less.A noticeable omission in the literature of RP d CFTs is the top-down construction oftheir holographic duals. On the other hand, theories such as N = 4 SYM are completelywell-defined on RP at weak coupling, and presumably will remain well-defined at strongcoupling as well. Finding a dual description in IIB supergravity for the strong coupling limit Similar issues with contact diagrams can also arise in the four-point function problem, and can beeliminated by imposing conditions on the Regge growth. – 44 –hould therefore be possible. It will be interesting to find such explicit backgrounds, whichwill provide the starting point for doing holographic calculations. Similarly, it would beinteresting to investigate the same question for the Vasiliev higher-spin theory and furthercheck the conjectured duality to O ( N ) vector model [67] by using the results obtained inthis paper.Related to studying the holographic duals, an interesting question to ask is whetherthere are any universal results that can be derived for double-trace deformation of CFT on RP d similar in spirit to [68–71]. We study the two-point function and free energy in thelarge N critical O ( N ) vector model, which can be obtained as a double-trace deformationof the free O ( N ) model. It would be interesting to see if some of the results are modelindependent and hold true for more general double-trace deformation. Acknowledgments
The work of S.G and H.K is supported in part by the US NSF under Grants No. PHY-1620542 and PHY-1914860. The work of X.Z. is supported in part by the Simons FoundationGrant No. 488653. A RP d free energy In this appendix, we show how to compute the RP d free energy for the critical O ( N ) vectormodel. To calculate the free energy, we need to go to a compact space, so we will do thison the Z quotient of sphere. The action on the sphere for the O ( N ) model is S = 12 (cid:90) d d x √ g (cid:18) ( ∂ µ φ I ) + d ( d − φ I φ I + σφ I φ I (cid:19) . (A.1)As we saw earlier (2.11), the one-point function of σ is just going to be a constant. At leadingorder in large N the effect of σ will only be through this one-point function, so it is just like N free massive fields on the sphere and the action becomes quadratic. To calculate the freeenergy, we need to calculate the determinant of this quadratic operator. For that purpose,we need to study the behavior of eigenfunctions of the scalar Laplacian and calculate thedegeneracies of the eigenfunctions which are odd or even under the Z quotient. Theeigenfunctions of the scalar Laplacian on a d dimensional sphere are spherical harmonics Y (cid:126)l ( (cid:126)θ ) with (cid:126)l = { l , . . . l d } satisfying | l | ≤ l ≤ . . . l d . The eigenvalues and degeneracy isgiven by (cid:3) d Y (cid:126)l ( (cid:126)θ ) = − l d ( l d + d − , dim( l d ) = Γ ( d + l d + 1)Γ ( d + 1) Γ ( l d + 1) − Γ ( d + l d − d + 1) Γ ( l d − . (A.2)The eigenfunctions Y (cid:126)l ( (cid:126)θ ) have explicit construction in terms of associated Legendre polyno-mials [72]. Under parity, they behave as Y (cid:126)l ( − (cid:126)θ ) = ( − l d Y (cid:126)l ( (cid:126)θ ) [73]. So to compute the freeenergy, we just need to sum over either even or odd values of l d depending upon whether We are using a different normalization of φ in this appendix from the rest of the paper such that thetwo-point function goes like (A.14). – 45 –e choose to identify the scalar with itself or minus itself. So for instance, for + parity, weneed to perform the sum F + ( a σ ) = N (cid:88) l d ∈ Z dim( l d ) log (cid:18) l d ( l d + d −
1) + d ( d − a σ (cid:19) (A.3)while for − parity, we have the exact same sum but over l d ∈ Z + 1 .Let’s first consider the case of free theory, when a σ = 0 . Then there is a more convenientway of writing this sum as F + (∆) = N (cid:88) l d ∈ Z dim( l d ) log Γ ( l d + d − ∆)Γ ( l d + ∆) (A.4)which goes back to the previous expression for ∆ = d/ − . Note that this sum vanishesfor ∆ = d/ . To do this sum, we can take a derivative, perform the sum and then integrateit back [74, 75] ∂F + ∂ ∆ = N (cid:20) Γ( − d )( d − (cid:18) Γ( d − ∆)Γ(1 − ∆) − Γ(∆)Γ(1 + ∆ − d ) (cid:19) + Γ(∆)Γ( d − ∆)Γ( d ) (cid:21) (A.5)where we had to use the integral representation of polygamma function to perform the sum.Similar method also works for − parity. So for any ∆ , we get F ± (∆) = N (cid:90) ∆ − d du (cid:20) − − d ) u (cid:32) Γ (cid:0) d − u (cid:1) Γ (cid:0) − u − d (cid:1) − Γ (cid:0) d + u (cid:1) Γ (cid:0) u − d (cid:1) (cid:33) ± Γ (cid:0) d + u (cid:1) Γ (cid:0) d − u (cid:1) Γ( d ) (cid:21) . (A.6)Here we are only interested in the case of ∆ = d/ − which gives F ± = − N (cid:90) du Γ (cid:0) d + u (cid:1) Γ (cid:0) d − u (cid:1) Γ( d ) (cid:32) u sin πud sin (cid:0) πd (cid:1) ± (cid:33) . (A.7)As an explicit example, in d = 3 we find F ± = N (cid:90) du π (cid:0) − u (cid:1) (2 u tan( πu ) ∓ πu )) = N (cid:18) log(2)16 − ζ (3)32 π ∓ K π (cid:19) (A.8)where K is Catalan’s constant.As a consistency check, note that adding the results for both parities gives us back thewell-known sphere free energy of a free scalar on a sphere [76].For the interacting case, the sum is hard to perform, but it can be performed if we takea derivate with respect to a σ of eq. (A.3) first ∂F + ( a σ ) ∂a σ = N (cid:88) l d ∈ Z dim( l d ) l d ( l d + d −
1) + d ( d − + a σ . (A.9)– 46 –his sum can then be performed and after some manipulations involving Hypergeometricidentities, we get ∂F + ( a σ ) ∂a σ = N − − d d ( d −
2) + a σ − N √ πd Γ(1 − d ) (cid:20) Γ (cid:18) d − − √ − a σ (cid:19) × ˜ F (cid:18) − d , − d , d + 3 − √ − a σ , − d + 7 − √ − a σ (cid:19) + Γ (cid:18) d − √ − a σ (cid:19) × ˜ F (cid:18) − d , − d , d + 3 + √ − a σ , − d + 7 + √ − a σ (cid:19) (cid:21) . (A.10)Now we should impose ∂F + ( a σ ) ∂a σ = 0 for the critical theory, because at large N we expectthe σ path integral to be dominated by the saddle point of free energy. It can be checkedthat this happens for a σ = − ( d − d − , which precisely agrees with what we found in(6.71). For the − parity, it is the same sum, but over odd integers, and it gives ∂F − ( a σ ) ∂a σ = − N − − d ( d + 1)( d − d ( d + 2) + x ) − N √ π d ( d + 1) Γ(1 − d )128 (cid:20) Γ (cid:18) d + 1 − √ − a σ (cid:19) × ˜ F (cid:18) − d , − d , d + 5 − √ − a σ , − d + 9 − √ − a σ (cid:19) + Γ (cid:18) d + 1 + √ − a σ (cid:19) × ˜ F (cid:18) − d , − d , d + 5 + √ − a σ , − d + 9 + √ − a σ (cid:19) (cid:21) . (A.11)It can again be checked that this vanishes for a σ = − ( d − d − . There is another way toarrive at this result for the free energy. The derivative of the free energy with a σ is relatedto the one point function of φ I φ I as ∂F ± ( a σ ) ∂a σ = Vol ( S d )2 (cid:104) φ I φ I (cid:105) , Vol ( S d ) = 2 π d +12 Γ (cid:0) d +12 (cid:1) (A.12)where we used the fact that the volume of S d / Z is half the volume of S d . But this one-pointfunction can also be obtained from the coincident point limit of the two-point function of φ .Since at large N , φ is equivalent to a massive free field on a sphere, the two-point functionis just the Green’s function on a sphere for a massive scalar [77] . We already encounteredthis large N two-point function in (6.66) which gives G ( η ) = C (1 − η ) d − F (cid:18) √ − a σ , − √ − a σ , − d , − η (cid:19) + C η d − F (cid:18) √ − a σ , − √ − a σ , d , − η (cid:19) . (A.13)When we perform the Z quotient of the sphere, we have to impose the crossing symmetryrequirement G ( η ) = ±G (1 − η ) which implies the relation (6.68) for the coefficients. Here The result in [77] is given in terms of the the geodesic distance µ , which is related to η as η = sin ( µ/ . – 47 –e are using a different normalization, so the overall constant can be fixed by requiring that G ( η ) = Γ (cid:0) d − (cid:1) (4 π ) d/ η d − , η → . (A.14)The one-point function of φ I φ I can then be read off from the constant piece in the small η expansion of G ( η ) . That gives ∂F ± ( a σ ) ∂a σ = ± π d +32 Γ (cid:0) d − (cid:1) Γ (cid:0) − d (cid:1) π ) d Γ (cid:0) d +12 (cid:1) Γ (cid:0) d (cid:1) Γ (cid:16) − d −√ − a σ (cid:17) Γ (cid:16) − d + √ − a σ (cid:17) (cid:16) sin (cid:0) πd (cid:1) ∓ cos (cid:16) π √ − a σ (cid:17)(cid:17) . (A.15)Note that the large N saddle point requirement of vanishing of the derivative of the freeenergy is then clearly equivalent to the requirement that the operator φ of dimension d − is replaced by operator σ of dimension , which is what we used in subsection 6.3. We werenot able to show analytically that these formulas for the derivative of free energy are thesame as (A.10) and (A.11), but we checked numerically over a range of values of d and a σ that they agree. We can also see that they also have the same zeroes as a function of a σ .We can then find the value of the free energy at the critical point by integrating theseexpressions. In d = 3 , we get F + crit = F + ( a σ = 0) + (cid:90) − ( d − d − da σ ∂F + ( a σ ) ∂a σ d = 3 −−−→ = F + ( a σ = 0) + N (16 πK − ζ (3) − π log(2))32 π = − N π ζ (3) F − crit = F − ( a σ = 0) + (cid:90) − ( d − d − da σ ∂F − ( a σ ) ∂a σ d = 3 −−−→ = F − ( a σ = 0) + N ( − πK − ζ (3) + 14 π log(2))32 π = − N π ζ (3) + N log(2)2 . (A.16)Note that both in the free and interacting theory, we get F − > F + in d = 3 . We can alsodo a similar computation in d = 4 − (cid:15)F + crit = F + ( a σ = 0) + (cid:90) (cid:15) da σ ∂F + ( a σ ) ∂a σ d = 4 − (cid:15) −−−−−→ = F + free + N (cid:15)
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