Efficient Rules for All Conformal Blocks
Jean-François Fortin, Wen-Jie Ma, Valentina Prilepina, Witold Skiba
aa r X i v : . [ h e p - t h ] F e b Efficient Rules for All Conformal Blocks
Jean-Fran¸cois Fortin ∗ , i , Wen-Jie Ma ∗ , ii , Valentina Prilepina ∗ , iii , and Witold Skiba † , iv ∗ D´epartement de Physique, de G´enie Physique et d’Optique,Universit´e Laval, Qu´ebec, QC G1V 0A6, Canada † Department of Physics, Yale University, New Haven, CT 06520, USA
We formulate a set of general rules for computing d -dimensional four-point global conformalblocks of operators in arbitrary Lorentz representations in the context of the embedding spaceoperator product expansion formalism [1]. With these rules, the procedure for determining anyconformal block of interest is reduced to (1) identifying the relevant projection operators andtensor structures and (2) applying the conformal rules to obtain the blocks. To facilitate thebookkeeping of contributing terms, we introduce a convenient diagrammatic notation. We presentseveral concrete examples to illustrate the general procedure as well as to demonstrate and testthe explicit application of the rules. In particular, we consider four-point functions involvingscalars S and some specific irreducible representations R , namely h SSSS i , h SSSR i , h SRSR i and h SSRR i (where, when allowed, R is a vector or a fermion), and determine the correspondingblocks for all possible exchanged representations.February 2020 i [email protected] ii [email protected] iii [email protected] iv [email protected] ontents1. Introduction 22. Review of the Embedding Space OPE Method 5
3. Input Data 16
4. Three-Point Functions and Rotation Matrices 22
5. Four-Point Functions and Conformal Blocks 29 G . . . . . . . . . . . . . . . . . . . . . . . . . 36
6. Summary of Results 36
7. Examples 38 h SSSS i . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.2. h SSSR i . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.2.1. h SSSV i . . . . . . . . . . . . . . . . . . . . . . . . . 407.2.2. h SSS e i . . . . . . . . . . . . . . . . . . . . . . . . . 417.3. h SRSR i and h SSRR i . . . . . . . . . . . . . . . . . . . . . . . 427.3.1. h SV SV i and h SSV V i . . . . . . . . . . . . . . . . . . . . 427.3.2. h SF SF i and h SSF F i . . . . . . . . . . . . . . . . . . . . 48
8. Conclusion 52 . Projection Operators 53 A.1. Projection Operator in the ℓ e Irreducible Representation . . . . . . . . . 53A.2. Projection Operator in the e r + ℓ e Irreducible Representation . . . . . . . 54A.3. Projection Operators in e m + ℓ e Irreducible Representations . . . . . . . . 55
References 571. Introduction
Conformal field theories (CFTs) are special quantum field theories that enjoy an enhanced symme-try, namely invariance under the conformal group SO (2 , d ). They describe the intriguing universalphysics of critical scale invariant fixed points and also lie at the core of our understanding of thespace of all quantum field theories (QFTs). CFTs represent fixed points of renormalization groupflows and describe second order phase transitions of statistical physics systems. Strikingly, theyshed light on the structure of the space of all QFTs, furnish concrete implementations of quan-tum gravity theories via the AdS/CFT correspondence and holography, and illuminate problemsin black hole physics. It is evident that the urge for a profound understanding of the landscapeof CFTs cannot be overemphasized.In recent years, this field has experienced a veritable explosion of results, largely owing to thesuccess of the conformal bootstrap, a program which seeks to systematically apply symmetries andconsistency conditions to carve out the allowed space of CFTs. The vast bootstrap literature hasbeen summarized in several comprehensive reviews and lectures (see for example [2] and referencestherein). This profusion of progress spans a wide range of high-precision numerical results as wellas many remarkable analytic advances, in addition to contributions involving global symmetriesand higher-spin fields.An implementation of the bootstrap program calls for a determination of the complete setof so-called conformal blocks, which are the building blocks of four-point correlation functionsthat capture contributions of particular exchanged representations in the operator product expan-sion (OPE). To date, only a handful of these objects have been worked out in d >
2, due tothe challenging nature of the computations involved [3] (see also [4, 5] for earlier work). Withrenewed interest in the bootstrap, a host of novel approaches and revisions of old methods havebeen proposed, including further developments of the shadow and the weight shifting operatorformalisms, adding to the ever growing variety of methods [6–11].An alternative technique was recently suggested in [1, 12]. This method hinges on exploitingthe OPE directly in the embedding space, where the conformal group acts linearly [13]. Theembedding space OPE framework was originally proposed in [14] and later expanded further in [15].Subsequent work has established this framework on a firm footing, starting with [1, 12], wherethe formalism was fully expounded for general M -point correlation functions, and later followed2p by [16–18], where it was tested and exemplified for two-, three-, and four-point functions,respectively. In this formalism, operators in arbitrary Lorentz representations are uplifted to theembedding space in a uniform fashion by building general representations solely out of products ofspinor representations. A key advantage of this approach is that arbitrary operators, whether theyare fermions or bosons, are treated democratically, so that from the perspective of the Dynkinindices, all representations effectively look the same. The method is designed to work at thefundamental level of the OPE and therefore applies to arbitrary correlation functions. A crucialaspect is the appropriate definition of an optimal embedding space OPE differential operator,which is symmetric and traceless in the embedding space indices by construction. This featurerenders the operator exceptionally useful, due to a variety of nice properties and identities, whichenable one to readily generalize the scalar case to the tensorial ones. The action of this operator onany quantity which may potentially crop up in an arbitrary M -point function has been explicitlyworked out in [1, 12]. This computation subsequently led to the tensorial generalization of thescalar Exton- G function for M -point correlators. The reader interested in the details of thegeneral method is referred to [1, 12].It turns out that further refinement of this approach enables one to compute the conformalblocks for arbitrary four-point functions quite efficiently. The method yields the infinite towers ofblocks in a compact form. The blocks are expressed as specific linear combinations of Gegenbauerpolynomials in a special variable X , with a unique substitution rule ascribed to each polynomialpiece. Once each relevant rule is applied to its associated Gegenbauer term, we directly generatethe complete conformal block in terms of a four-point tensorial generalization of the Exton G -function. As detailed in [18], in the context of this formalism, the procedure for determininga given block comes down to (1) writing down the relevant group theoretic quantities, namelythe projection operators and tensor structures (which effectively serve as intertwiners amongthe respective external and exchanged representations), and (2) identifying the specific linearcombination of Gegenbauer polynomials along with the corresponding substitution rules for eachpiece.While this approach is complete and clearly formulated as it stands, it is rather cumber-some to apply in practice for infinite towers of exchanged quasi-primary operators in irreduciblerepresentations N m + ℓ e . In the analysis [18], it was apparent that various parts involved inthe derivation of the substitution rules recurred, suggesting that the procedure could be madecompletely systematic for any ℓ . Further, while the determination of the appropriate linear com-bination of Gegenbauer polynomials for a given case was straightforward, a systematic approachfor the identification of these combinations was lacking. Moreover, the nature of the substitutionrules themselves seemed somewhat mysterious. A careful inspection of the form of the various Scalar M -point correlation functions in the comb channel were also obtained with this method in [19]. Seealso [20] for an independent computation using AdS/CFT. e.g. h SV SV i and h SSV V i , revealed that combinations of these ruleswere related to each other and that in some cases, one could map certain rules to others via aset of integer shifts, implying a deep relationship among the different rules. However, the originof such shifts remained unclear.In this work, we seek to cast our prescription for obtaining the blocks into a systematic form.In particular, we wish to understand the underlying structure of the substitution rules as wellas how to methodically generate the relevant linear combination of Gegenbauer polynomials foreach case of interest, i.e. for any ℓ for some exchanged representation N m + ℓ e . Our generalphilosophy is to formulate the procedure in terms of parameters which depend entirely on theprojection operators and the tensor structures, so that effectively, all that needs to be done fora given case is to determine these objects. The remainder would be subsequently handled by aset of conformal rules, designed to be easy to apply for arbitrary ℓ . These conformal rules (whichare reminiscent of the Feynman rules but are non-perturbative as they lead to the exact blocks)would directly use the information about the structure of the projection operators and the tensorstructures as well as the identity of the external operators to generate the complete tower ofblocks for a given case. With this machinery in place, the calculation of the blocks would becomeessentially effortless. It is the purpose of this work to formulate these conformal rules, check theirvalidity, and demonstrate how to apply them in practice.This paper is organized as follows: Section 2 provides a brief review of the method for ( M ≤ X , coupled with associated substitution rules.In these two sections, we consider infinite towers of exchanged quasi-primary operators in someirreducible representations N m + ℓ e . We provide proofs on how to handle the universal ℓ -4ependent parts of these exchanged representations, which leads to some simple ℓ -independentrules. The reader not interested in the proofs of Sections 4 and 5 can skip directly to Section 6,where we present a summary of the results along with a dictionary of the notation. In Section 7,we illustrate how to apply these rules in practice. We analyze several interesting examples. Webegin by revisiting all the conformal blocks obtained in [18], namely h SSSS i , h SSS e i , h SV SV i and h SSV V i . We demonstrate that the application of the conformal rules allows us to effortlesslyrederive these results. The diagrammatic notation is also illustrated through these examples. Wenext proceed to treat the remaining cases of the type h SSSR i , h SRSR i and h SSRR i when R isa vector or a fermion.Finally, Section 8 concludes with a summary of the results, a preview of future work, andquestions of interest raised by this analysis, while Appendix A provides details on the projectionoperators needed to compute the various conformal blocks presented in this paper.
2. Review of the Embedding Space OPE Method
This section presents a quick review of the embedding space OPE method and its implicationsfor correlation functions up to four points. The reader interested in the details is refereed to[1, 12, 16–18].
The form of the embedding space OPE that is most convenient for the determination of M -pointcorrelation functions was found in [1, 12]. It is given by O i ( η ) O j ( η ) = ( T N i Γ)( T N j Γ) · X k N ijk X a =1 a c kij a t kij ( η · η ) p ijk · D ( d,h ijk − n a / ,n a )12 ( T N k Γ) ∗ O k ( η ) ,p ijk = 12 ( τ i + τ j − τ k ) , h ijk = −
12 ( χ i − χ j + χ k ) ,τ O = ∆ O − S O , χ O = ∆ O − ξ O , ξ O = S O − ⌊ S O ⌋ , (2.1)where ∆ O and S O are the conformal dimension and spin of the quasi-primary operator O , respec-tively, while a c kij are the OPE coefficients. The remaining quantities appearing in the OPE (2.1)are described below.The first quantity of interest here is the OPE differential operator D ( d,h ijk − n a / ,n a )12 . It is givenby D ( d,h,n ) A ··· A n ij = 1( η · η ) n D h + n ) ij η A j · · · η A n j , D ij = ( η i · η j ) ∂ j − ( d + 2 η j · ∂ j ) η i · ∂ j . The explicit action of this operator on arbitrary functions of the embedding space coordinates andcross-ratios was found in [1, 12]. A useful consequence is that in the context of the computation5f conformal blocks, the action of this operator can be taken care of by simple substitution ruleson specific quantities.The remaining objects of interest are fundamental group theoretic quantities, including theprojection operators, the half-projection operators, and the tensor structures. In the OPE (2.1),we require their embedding space analogs, and these are readily obtained from the correspond-ing position space quantities. We therefore first determine the position space objects and thentranslate them into their embedding space counterparts via some simple substitutions detailedbelow.We begin with a brief discussion of the projection and half-projection operators in positionspace. The projection operators are central ingredients in the construction of M -point correlationfunctions in the context of the present framework. The position space projectors are defined asoperators that satisfy the following properties:1. the projection property ˆ P N · ˆ P N ′ = δ N ′ N ˆ P N ,
2. the completeness relation X N | n v fixed ˆ P N = − traces ,
3. the tracelessness condition g · ˆ P N = γ · ˆ P N = ˆ P N · g = ˆ P N · γ = 0 , where n v is the total number of vector indices. They are labeled by the irreducible representations N of SO ( p, q ). An arbitrary irreducible representation of SO ( p, q ) is in turn indexed by a setof nonnegative integers, the Dynkin indices, denoted by N = { N , . . . , N r } ≡ P i N i e i , where r is the rank of the Lorentz group and e i ≡ ( e i ) j = δ ij . There exists a variety of methods forconstructing such projectors to general irreducible representations N of the Lorentz group. Someexamples include Young tableaux techniques with the birdtrack notation [7], the weight-shiftingoperator formalism [9], and an approach based on the tensor product decomposition and thedefining properties above [18]. Irrespective of the method employed, the procedure comes downto an application of group theory.We may build up the projection operators to general irreducible representations from thecorresponding operators for the defining irreducible representations. These act as building blocksfor the general operators. By properly subtracting traces and smaller irreducible representationsfrom appropriately symmetrized products of the defining projectors, we may in principle generateany projection operator of interest.The hatted projectors to defining irreducible representations in odd spacetime dimensions aregiven by( ˆ P e r ) βα = δ βα , ( ˆ P e i = r ) ν ··· ν i µ i ··· µ = δ ν [ µ · · · δ ν i µ i ] , ( ˆ P e r ) ν ··· ν r µ r ··· µ = δ ν [ µ · · · δ ν r µ r ] , P e r − ) βα = δ βα , ( ˆ P e r ) ˜ β ˜ α = δ ˜ β ˜ α , ( ˆ P e i = r − ,r ) ν ··· ν i µ i ··· µ = δ ν [ µ · · · δ ν i µ i ] , ( ˆ P e r − + e r ) ν ··· ν r − µ r − ··· µ = δ ν [ µ · · · δ ν r − µ r − ] , ( ˆ P e r − ) ν ··· ν r µ r ··· µ = 12 δ ν [ µ · · · δ ν r µ r ] + ( − r K r ! ǫ ν r ··· ν µ ··· µ r , ( ˆ P e r ) ν ··· ν r µ r ··· µ = 12 δ ν [ µ · · · δ ν r µ r ] − ( − r K r ! ǫ ν r ··· ν µ ··· µ r . Here δ ν [ µ · · · δ ν i µ i ] is the totally antisymmetric normalized product of δ νµ , while K is the pro-portionality constant in γ µ ··· µ d = K ǫ µ ··· µ d which satisfies K = ( − r + q with r the rank ofthe Lorentz group, q the signature of the Lorentz group, and ǫ ··· d = 1. These hatted projectorsoperate on the “dummy” indices that are fully contracted in expressions for correlation functions.They are in place to restrict the operators to the relevant irreducible representations.Meanwhile, the half-projection operators encode the transformation properties of operators O N in general irreducible representations N under Lorentz transformations, O N ∼ T N . Theyare aptly named, because they satisfy T N ∗ T N = ˆ P N , where the star product corresponds tocontractions of the spinor indices. These operators play the role of translating the spinor indicescarried by each operator to the dummy vector and spinor indices that need to be contracted whenconstructing correlation functions.The position space half-projectors to arbitrary irreducible representations N are given by( T N ) µ ··· µ nv δα ··· α n = (cid:16) ( T e ) N · · · ( T e r − ) N r − ( T e r ) ⌊ N r / ⌋ ( T e r ) N r − ⌊ N r / ⌋ (cid:17) µ ′ ··· µ ′ nv δ ′ α ··· α n × ( ˆ P N ) µ ··· µ nv δδ ′ µ ′ nv ··· µ ′ , (2.2)where n = 2 S = 2 P r − i =1 N i + N r is twice the spin S of the irreducible representation N , n v = P r − i =1 iN i + r ⌊ N r / ⌋ is the number of vector indices of the irreducible representation N , and δ is the spinor index which appears only if N r is odd (in odd spacetime dimensions). In (2.2),the spinor indices α , . . . , α n match the free indices on the corresponding quasi-primary operator,while the remaining indices µ , . . . , µ n v , δ are dummy indices that are contracted.Further, in (2.2) the corresponding half-projectors to the defining representations are given by( T e i = r ) µ ··· µ i αβ = 1 √ r i ! ( γ µ ··· µ i C − ) αβ , ( T e r ) βα = δ βα , ( T e r ) µ ··· µ r αβ = 1 √ r r ! ( γ µ ··· µ r C − ) αβ , (2.3)where γ µ ··· µ n = 1 n ! X σ ∈ S n ( − σ γ µ σ (1) · · · γ µ σ ( n ) , is the totally antisymmetric product of γ -matrices. Lastly, the operator ˆ P N in (2.2) contractswith the dummy indices of the half-projector. It is present to ensure projection onto the proper7rreducible representation N . We note that the definitions above extend straightforwardly to evendimensions.The final objects of central interest here are the tensor structures. These are purely grouptheoretic quantities that are entirely determined by the irreducible representations of the quasi-primary operators in question. In a three-point function hO N i O N j O N k i , the objects a t ijk serveto intertwine three irreducible representations of the Lorentz group into a symmetric tracelessrepresentation. In fact, the number N ijk of symmetric irreducible representations appearing in N i ⊗ N j ⊗ N k precisely corresponds to the number of such independent tensor structures andOPE coefficients. Moreover, for fixed N i , N j , and N k , the set of all tensor structures forms abasis for a vector space.Equivalently, these structures may be viewed as contracting four irreducible representationstogether into a singlet, with the fourth representation corresponding to the symmetric tracelessdifferential operator. As such, owing to the OPE, the corresponding embedding space quantitiescan be made to satisfy the following identity [1] a t ijk = ( ˆ P N i )( ˆ P N j )( ˆ P N k )( ˆ P n a e ) · a t ijk , (2.4)where the order of the contractions is self-evident. The purpose of this condition is to restrictthe tensor structures onto the appropriate irreducible representations for the three quasi-primaryoperators and the symmetric traceless differential operator.It is straightforward to obtain the embedding space projection operators, half-projectors, andtensor structures from their position space counterparts by making the following substitutions: g µν → A AB ≡ g AB − η A η B ( η · η ) − η B η A ( η · η ) ,ǫ µ ··· µ d → ǫ A ··· A d ≡ η · η ) η A ′ ǫ A ′ A ′ ··· A ′ d A ′ d +1 η A ′ d +1 A A d A ′ d · · · A A A ′ ,γ µ ··· µ n → Γ A ··· A n ≡ Γ A ′ ··· A ′ n A A n A ′ n · · · A A A ′ ∀ n ∈ { , . . . , r } . (2.5)By construction, these exhibit all the requisite properties ( e.g. trace, number of vector indices, etc. )to guarantee proper contraction with the corresponding irreducible representations in positionspace.In the embedding space, the appropriate counterparts of the half-projectors (2.3) are given by( T N ij Γ) ≡ √ η i · η j ) T e η i A ij ! N · · · √ r ( η i · η j ) T e rE − η i A ij · · · A ij ! N r − × √ r + 1( η i · η j ) T e rE η i A ij · · · A ij ! ⌊ N r / ⌋ √ η i · η j ) T e rE η i · Γ η j · Γ ! N r − ⌊ N r / ⌋ · ˆ P N ij , where ( T e n +1 η i A ij · · · A ij ) A ··· A n ab ≡ ( T e n +1 ) A ′ ··· A ′ n ab A A n ijA ′ n · · · A A ijA ′ η iA ′ . r → r E = r + 1, as expected.With the notation established, we now discuss the two-, three-, and four-point correlationfunctions from the perspective of the embedding space OPE. Before proceeding, we first presentthe identities (these can be proven from the identities in Appendix B of [1]) η j · Γ ˆ P N ji = η j · Γ ( A ji ) n v ˆ P N jk ( A ji ) n v , ˆ P N ij η j · Γ = ( A ji ) n v ˆ P N kj ( A ji ) n v η j · Γ , (2.6)valid for an arbitrary irreducible representation N . These identities are powerful in simplifyingthe computations of correlation functions, as we will show below. From the OPE (2.1), it is easy to see that the two-point correlation functions are given by [16] hO i ( η ) O j ( η ) i = ( T N i Γ)( T N j Γ) · λ N i c ij ˆ P N i ( η · η ) τ i = ( T N i Γ) · ( T N j Γ) λ N i c ij ( η · η ) τ i , (2.7)where the sole tensor structures are t ij = λ N i ˆ P N i → λ N i , without loss of generality.As expected, the two-point correlation functions vanish unless the quasi-primary operatorsare in irreducible representations that are contragredient-reflected with respect to each other, i.e. N i = N CRj , and their conformal dimensions are the same, i.e. τ i = τ j . Here λ N i is anormalization constant that can be set to any convenient value. Applying the OPE (2.1) on the first two quasi-primary operators and then using the form ofthe two-point functions (2.7) on the result, we find that the three-point correlation functions aregiven by [17] hO i ( η ) O j ( η ) O m ( η ) i = ( T N i Γ)( T N j Γ)( T N m Γ)( η · η ) ( τ i + τ j − χ m ) ( η · η ) ( χ i − χ j + τ m ) ( η · η ) ( − χ i + χ j + χ m ) · N ijm X a =1 a c ijm G ij | m ( a | , (2.8) In Lorentzian signature, the contragredient-reflected representation corresponds to the conjugate representation, i.e. N CR = N C . G ij | m ( a | are defined as G ij | m ( a | = λ N m ¯ J ( d,h ijm ,n a , ∆ m , N m )12;3 · a t ijm . We refer to these as the three-point conformal blocks in the OPE tensor structure basis.Further, the relevant three-point correlation function quantities are defined as a c ijm = X n a c nij c nm , a t ijm = a t m CR ij [( C − )] ξ m ( g ) n mv ( g ) n a . (2.9)We begin by considering the definition of the three-point ¯ J -function in terms of the three-pointconformal substitution [17], namely¯ J ( d,h,n, ∆ , N )12;3 = (¯ η · Γ ˆ P N · ˆ P N ¯ η · Γ) cs ≡ ¯ η · Γ ˆ P N · ˆ P N ¯ η · Γ (cid:12)(cid:12)(cid:12) ( g ) s (¯ η ) s (¯ η ) s (¯ η ) s → ( g ) s (¯ η ) s (¯ η ) s × ¯ I ( d,h − n/ − s ,n + s χ − s / s / s / , (2.10)where the three-point homogenized embedding space coordinates are¯ η Ai = ( η j · η m ) ( η i · η j ) ( η i · η m ) η Ai , (2.11)with ( i, j, m ) a cyclic permutation of (1 , , I -function is given explicitly in thenext subsection in (2.18). It turns out that the identities (2.6) allow the following simplifications¯ J ( d,h,n, ∆ , N )12;3 = (¯ η · Γ ˆ P N ( A ) n v ˆ P N ( A ) n v ¯ η · Γ) cs = (¯ η · Γ ˆ P N · ˆ P N ( A ) n v ¯ η · Γ) cs = (¯ η · Γ ˆ P N ( A ) n v ¯ η · Γ) cs = (¯ η · Γ ( A ) n v ˆ P N ( A ) n v ¯ η · Γ) cs . (2.12)To obtain the first equality here, we applied (2.6) on the second hatted projection operator. Inthe second line, the A metrics were simplified to g metrics through their contractions with thetwo hatted projection operators. In the following line, we invoked the projection property of thehatted projection operator. Lastly, in the fourth equality, (2.6) was used one more time.We next remark that if we insert the result (2.12) inside (2.8), we find that the ( A ) n mv canbe simplified to g ’s through their contractions with the hatted projection operator ˆ P N m and thehalf-projector ( T N m Γ). Then the hatted projection operator ˆ P N m can be commuted through the¯ η · Γ and contracted directly with the half-projector ( T N m Γ), effectively allowing the followingrewriting: ¯ J ( d,h,n, ∆ , N )12;3 = (¯ η · Γ ( A ) n v ¯ η · Γ) cs ≡ ¯ η · Γ ( A ) n v ¯ η · Γ | ( g ) s (¯ η ) s (¯ η ) s (¯ η ) s → ( g ) s (¯ η ) s (¯ η ) s × ¯ I ( d,h − n/ − s ,n + s χ − s / s / s / , (2.13)10hich does not depend explicitly on the irreducible representation, apart from its number of vectorindices n v . Hence, the three-point conformal blocks simplify to G ij | m ( a | = λ N m (cid:0) ( A ) n mv ¯ η · Γ ¯ η · Γ (cid:1) cs · a t ijm , (2.14)with the proper parameters for the exchanged quasi-primary operator h = h ijm , n = n a , ∆ = ∆ m ,and N = N m in the three-point conformal substitution (2.10).As discussed in [17, 18], although the OPE tensor structure basis is convenient in the contextof the OPE, it is not the simplest one to use for the construction of three-point correlationfunctions. Rather, the natural optimal basis for three-point correlators is the three-point tensorstructure basis. The two bases, indicated by ( a and [ a for the OPE basis and the three-pointbasis, respectively, can be related through rotation matrices as in G ij | m ( a | = N ijm X a ′ =1 ( R − ijm ) aa ′ G ij | m [ a ′ | , a c ijm = N ijm X a ′ =1 a ′ α ijm ( R ijm ) a ′ a , (2.15)where the a α ijm are the associated three-point function coefficients, implying N ijm X a =1 a c ijm G ij | m ( a | = N ijm X a =1 a α ijm G ij | m [ a | . (2.16)We express the three-point conformal blocks in the three-point tensor structure basis as G ij | m [ a | = ¯ η · Γ a F ijm ( A , Γ , ǫ ; A · ¯ η ) , (2.17)where it is understood that the factor ¯ η · Γ on the RHS appears only if ξ k = , i.e. if theexchanged quasi-primary operator is fermionic. In this basis, the three-point correlation functions(2.8) can be effortlessly obtained without the aid of the OPE by simply enumerating the three-point tensor structure basis { a F ijm } made from A ’s, Γ ’s, ǫ ’s and A · ¯ η ’s. Note that the n a factors of A · ¯ η in the three-point tensor structures a F ijm ( A , Γ , ǫ ; A · ¯ η ) can contractwith any index, including the ones from A , Γ , and ǫ originating from the tensor structures. The three-point tensorial function appearing in the three-point conformal substitution (2.13) wasfound in [1, 12] and is given explicitly by¯ I ( d,h,n ; p )12 = ρ ( d,h ; p ) X q ,q ,q ,q ≥ q =2 q + q + q + q = n S ( q ,q ,q ,q ) K ( d,h ; p ; q ,q ,q ,q ) , (2.18)11here the totally symmetric S -tensor, the ρ -function, and the K -function are S A ··· A ¯ q ( q ,q ,q ,q ) = g ( A A · · · g A q − A q ¯ η A q · · · ¯ η A q q × ¯ η A q q · · · ¯ η A q q q ¯ η A q q q · · · ¯ η A ¯ q )3 ,ρ ( d,h ; p ) = ( − h ( p ) h ( p + 1 − d/ h ,K ( d,h ; p ; q ,q ,q ,q ) = ( − ¯ q − q − q − q ( − ¯ q − q ¯ q ! q ! q ! q ! q ! ( − h − ¯ q ) ¯ q − q − q ( p + h ) ¯ q − q − q ( p + 1 − d/ − q − q − q . (2.19)In (2.19), ¯ q = 2 q + q + q + q = n , the total number of indices on ¯ I ( d,h,n ; p )12 . The three-pointtensorial function is totally symmetric and satisfies several convenient contiguous relations [1, 12],given by g · ¯ I ( d,h,n ; p )12 = 0 , ¯ η · ¯ I ( d,h,n ; p )12 = ¯ I ( d,h +1 ,n − p )12 , ¯ η · ¯ I ( d,h,n ; p )12 = ρ ( d, − h − n ) ¯ I ( d,h,n − p )12 , ¯ η · ¯ I ( d,h,n ; p )12 = ¯ I ( d,h +1 ,n − p − . (2.20)These will be of great utility in the determination of rotation matrices. For future convenience,we also introduce e K ( d,h ; p ; q ,q ,q ,q ) = ρ ( d,h ; p ) K ( d,h ; p ; q ,q ,q ,q ) to simplify computations. Havingconstructed the three-point functions, we now turn to the four-point correlators. Together, the OPE (2.1) and the respective results for two- and three-point correlation functionsin (2.7) and (2.8) lead to the four-point correlation functions [18] hO i ( η ) O j ( η ) O k ( η ) O l ( η ) i = ( T N i Γ)( T N j Γ)( T N k Γ)( T N l Γ)( η · η ) ( τ i − χ i + τ j + χ j ) ( η · η ) ( χ i − χ j + χ k − χ l ) ( η · η ) ( χ i − χ j − χ k + χ l ) ( η · η ) ( − χ i + χ j + τ k + τ l ) · X m N ijm X a =1 N klm X b =1 a c mij b α klm G ij | m | kl ( a | b ] , (2.21)with the (four-point) conformal blocks in the mixed basis (the simplest one, as discussed in [18])given by G ij | m | kl ( a | b ] = N klm X b ′ =1 ( − ξ m λ N m ( R klm ) bb ′ a t mij · ¯ J ( d,h ijm ,n a ,h klm ,n b , ∆ m , N m )34;21 · b ′ t klm . J -function expressed in terms of the four-point conformal substitution is¯ J ( d,h ,n ,h ,n , ∆ , N )34;21 = ( x ξ ¯ η · Γ ˆ P N · ˆ P N ¯ η · Γ(¯¯ η · Γ ˆ P N · ˆ P N ¯¯ η · Γ) cs ) cs ≡ x ξ ¯ η · Γ ˆ P N · ˆ P N ¯ η · Γ × (¯¯ η · Γ ˆ P N · ˆ P N ¯¯ η · Γ) cs (cid:12)(cid:12)(cid:12) (¯ η ) s x r x r → ¯ I ( d,h − n / − s ,n s − h r ,χ + h r , (2.22)where the three- and four-point homogeneized embedding space coordinates as well as the cross-ratios are¯¯ η A = ( η · η ) ( η · η ) ( η · η ) η A , ¯¯ η A = ( η · η ) ( η · η ) ( η · η ) η A , ¯¯ η A = ( η · η ) ( η · η ) ( η · η ) η A , ¯ η A = ( η · η ) ( η · η ) ( η · η ) η A , ¯ η A = ( η · η ) ( η · η ) ( η · η )( η · η ) η A , ¯ η A = ( η · η ) ( η · η ) ( η · η ) η A , ¯ η A = ( η · η ) ( η · η ) ( η · η ) η A ,x = ( η · η )( η · η )( η · η )( η · η ) = uv , x = ( η · η )( η · η )( η · η )( η · η ) = u, (2.23)and the four-point ¯ I -function is discussed below.As before, we now apply the identities (2.6) to transform ¯ J ( d,h ,n ,h ,n , ∆ , N )34;21 . These imply thesimplifications¯ J ( d,h ,n ,h ,n , ∆ , N )34;21 = ( x ξ ¯ η · Γ ˆ P N · ˆ P N ¯ η · Γ(¯¯ η · Γ ˆ P N ( A ) n v ˆ P N ( A ) n v ¯¯ η · Γ) cs ) cs = ( x ξ ¯ η · Γ ˆ P N · ˆ P N ¯ η · Γ(¯¯ η · Γ ˆ P N ( A ) n v ¯¯ η · Γ) cs ) cs = ( x ξ ¯ η · Γ ˆ P N · ˆ P N ¯ η · Γ(¯¯ η · Γ ( A ) n v ˆ P N ( A ) n v ¯¯ η · Γ) cs ) cs = ( x ξ ¯ η · Γ ˆ P N · ˆ P N ¯ η · Γ(¯¯ η · Γ ( A ) n v ¯¯ η · Γ) cs ) cs = ( x ξ ¯ η · Γ ( A ) n v ˆ P N ( A ) n v ˆ P N ¯ η · Γ(¯¯ η · Γ ( A ) n v ¯¯ η · Γ) cs ) cs = ( x ξ ¯ η · Γ ( A ) n v ˆ P N ¯ η · Γ(¯¯ η · Γ ( A ) n v ¯¯ η · Γ) cs ) cs = ( x ξ ¯ η · Γ ( A ) n v ¯ η · Γ ˆ P N ( A ) n v (¯¯ η · Γ ( A ) n v ¯¯ η · Γ) cs ) cs . (2.24)In the first line above, the last projection operator was replaced using (2.6). Then, half of thegenerated metrics and one of the last two projection operators were annihilated. Subsequently, We note that ¯¯ η = √ x x ¯ η , ¯¯ η = q x x ¯ η and ¯¯ η = q x x ¯ η . Moreover, it is important to realize that the three-and four-point homogeneized embedding space coordinates (2.11) and (2.23) are different. Since the former are usedin the computation of the rotation matrices while the latter appear in the four-point conformal blocks, it should beclear from the context which ones are used.
13n the third and fourth lines, the two previous steps were repeated with the last projectionoperator. Further, in the fourth and fifth lines, the same two steps were then performed on thefirst projection operator. Finally, in the last equality, the sole remaining projection operator wasmoved to the middle, between the three- and four-point conformal substitutions, and a set ofmetrics was introduced using (2.6).It is interesting to note here that contrary to (2.13), it is impossible to remove the lastprojection operator in (2.24). This is expected, since in the case of four-point correlation functions,there are no half-projectors for the exchanged quasi-primary operators, unlike for three-pointcorrelation functions. A projection operator is therefore necessary to ensure that the conformalblocks are in the appropriate irreducible representation. Given this result as well as (2.14), thefour-point conformal blocks assume the form G ij | m | kl ( a | b ] = N klm X b ′ =1 ( − ξ m λ N m a t mij ( R klm ) bb ′ · ( x ξ m ¯ η · Γ ( A ) n mv ¯ η · Γ ˆ P N m ( A ) n v (¯¯ η · Γ ( A ) n mv ¯¯ η · Γ) cs ) cs · b ′ t klm = a t mij · (( − x ) ξ m ¯ η · Γ ( A ) n mv ¯ η · Γ ˆ P N m ( A ) n mv ¯¯ η · Γ b F klm ( A , Γ , ǫ ; A · ¯¯ η )) cs = a t mij · (cid:16) ( − x ) ξ m ( A ) n mv ¯ η · Γ ¯ η · Γ ˆ P N m ¯¯ η · Γ b F klm ( A , Γ , ǫ ; A · ¯¯ η ) (cid:17) cs , (2.25)with the proper parameters for the exchanged quasi-primary operator h = h ijm , n = n a , h = h klm , n = n b , ∆ = ∆ m and N = N m in the four-point conformal substitution (2.22).In [18], we remarked that the conformal blocks feature the simplest form in the mixed basis.However, for the implementation of the conformal bootstrap, it is more convenient to work in apure tensor structure basis, e.g. the three-point basis. Following the discussion in the previoussubsections [see (2.15) and (2.16)], the conformal blocks in the pure three-point basis can bedetermined from their mixed counterparts by acting with the rotation matrices as in G ij | m | kl [ a | b ] = N ijm X a ′ =1 ( R ijm ) aa ′ G ij | m | kl ( a ′ | b ] . Clearly, in the interest of setting up the bootstrap, it is therefore necessary to compute not onlythe conformal blocks in the mixed basis but also the corresponding rotation matrices.
In [1, 12], the four-point tensorial function ¯ I ( d,h,n ; p ,p )12;34 was found to be given by¯ I ( d,h,n ; p ,p )12;34 = X q ,q ,q ,q ,q ≥ q =2 q + q + q + q + q = n S ( q ) ρ ( d,h ; p + p ) x p + p + h + q + q + q + q K ( d,h ; p ,p ; q ,q ,q ,q ,q )12;34;3 ( x ; y ) , (2.26)14ith the totally symmetric object S ( q ) defined by S A ··· A ¯ q ( q ) = g ( A A · · · g A q − A q ¯ η A q · · · ¯ η A q q · · · ¯ η A ¯ q − q · · · ¯ η A ¯ q )4 , (2.27)where ¯ q = 2 q + q + q + q + q and y = 1 − x /x . The function (2.27) is the natural extensionof (2.19) to four points.The K -function appearing in the four-point ¯ I -function is given by K ( d,h ; p ; q )12;34;3 ( x ; y ) = ( − q + q + q ( − ¯ q − q ¯ q ! q ! q ! q ! q ! q ! ( − h − ¯ q ) ¯ q − q − q ( p ) q ( p + p + h ) ¯ q − q − q ( p + p ) q + q ( p + p + 1 − d/ − q − q − q ( p ) q × K ( d +2¯ q − q ,h + q + q ; p + q ,p + q )12;34;3 ( x ; y ) , (2.28)where K ( d,h ; p ,p )12;34;3 ( x ; y ) = X n ,n ≥ ( − h ) n ( p ) n ( p + p + h ) n ( p + p ) n + n ( p + p + 1 − d/ n ( p ) n n !( n − n )! y n (cid:18) x y (cid:19) n = G ( p , p + p + h, p + p + 1 − d/ , p + p ; u/v, − /v ) , (2.29)is the usual Exton G -function G ( α, β, γ, δ ; x, y ) with appropriately shifted parameters [5]. Hence,the tensorial ¯ I -function is constructed from linear combinations of the Exton G -function. Alter-natively, we may regard it as built from the conformal block for scalar exchange in the scalarfour-point correlator.Like the three-point tensorial function, the four-point ¯ I -function (2.26) satisfies a set of con-tiguous relations [1, 12], g · ¯ I ( d,h,n ; p )12;34 = 0 , ¯ η · ¯ I ( d,h,n ; p ,p )12;34 = ¯ I ( d,h +1 ,n − p ,p )12;34 , ¯ η · ¯ I ( d,h,n ; p ,p )12;34 = ρ ( d, − h − n ) ¯ I ( d,h,n − p ,p )12;34 , ¯ η · ¯ I ( d,h,n ; p ,p )12;34 = ¯ I ( d,h +1 ,n − p − ,p )12;34 , ¯ η · ¯ I ( d,h,n ; p ,p )12;34 = ¯ I ( d,h +1 ,n − p ,p − , (2.30)which enable easy contractions. To recapitulate the above review of the embedding space OPE formalism, we summarize herethe key quantities of interest, namely the three- and four-point correlation functions. Althoughwe ultimately seek to compute four-point conformal blocks, the determination of the three-pointcorrelation functions is necessary for the extraction of the rotation matrices. As discussed above,15he latter are instrumental in allowing us to translate between the mixed basis, where the four-point conformal blocks are simplest in form, to a pure tensor structure basis, which is moreconvenient for bootstrap purposes.Reintroducing the dummy indices in the correlation functions (2.8) hO i ( η ) O j ( η ) O m ( η ) i = ( T N i Γ) { Aa } ( T N j Γ) { Bb } ( T N m Γ) { Ee } ( η · η ) ( τ i + τ j − χ m ) ( η · η ) ( χ i − χ j + τ m ) ( η · η ) ( − χ i + χ j + χ m ) · N ijm X a =1 a c ijm ( G ij | m ( a | ) { aA }{ bB }{ eE } , (2.31)and (2.21) hO i ( η ) O j ( η ) O k ( η ) O l ( η ) i = ( T N i Γ) { Aa } ( T N j Γ) { Bb } ( T N k Γ) { Cc } ( T N l Γ) { Dd } ( η · η ) ( τ i − χ i + τ j + χ j ) ( η · η ) ( χ i − χ j + χ k − χ l ) ( η · η ) ( χ i − χ j − χ k + χ l ) ( η · η ) ( − χ i + χ j + τ k + τ l ) · X m N ijm X a =1 N klm X b =1 a c mij b α klm ( G ij | m | kl ( a | b ] ) { aA }{ bB }{ cC }{ dD } , (2.32)leads to the conformal blocks (2.14)( G ij | m ( a | ) { aA }{ bB }{ eE } = λ N m (cid:16) ( A E ′ E ) n mv (¯ η · Γ ¯ η · Γ) e ′ e (cid:17) cs ( a t ijm ) { aA }{ bB }{ e ′ E ′ }{ F } , (2.33)and (2.25)( G ij | m | kl ( a | b ] ) { aA }{ bB }{ cC }{ dD } = ( a t mij ) { Ee }{ F }{ aA }{ bB } (cid:16) ( − x ) ξ m ( A E ′ E ) n mv (¯¯ η · Γ) e ′ e × (¯ η · Γ ˆ P N m ¯ η · Γ) E ′′ e ′′ e ′ E ′ ( b F klm ) { cC }{ dD }{ e ′′ E ′′ } (cid:17) cs . (2.34)Here, the explicit F -indices on the tensor structures are contracted through the OPE differentialoperator (2.1) with the implicit F -indices of the conformal substitutions cs and cs , respectively.For the remainder of this paper, we will focus on developing a set of simple and efficientrules for the determination of the rotation matrices and conformal blocks in the mixed basis forquasi-primary operators in arbitrary irreducible representations, given some input data, namelythe projection operators and tensor structures. We now turn to a discussion of this input grouptheoretic data.
3. Input Data
It is apparent from (2.14) and (2.25) that the input data consists of the projection operatorsand the tensor structures. Although the tensor structures are obtained once the projection op-erators are determined, it is simpler to discuss the tensor structures first. In this section, we16ill introduce a simple basis of three-point tensor structures, which is made out of products ofa small set of allowed constituents. The projection operators are built from their correspondingirreducible representations. In this work, we will primarily focus on the projection operators forthe exchanged quasi-primary operators, which are more intricate, due to the existence of infinitetowers of exchanged quasi-primary operators in N m + ℓ e , resulting in ℓ -dependent projectors. The simplest available basis of tensor structures is the three-point basis (2.17), where tensorstructures are simply constructed from products of allowed constituents. There is also its analogfor the OPE basis. However, these two bases are not related straightforwardly [17]. Indeed, achange of basis is necessary, which calls for a computation of the rotation matrices mentionedabove.Before proceeding, let us first consider the tensor structures for N m → N m + ℓ e , with N m chosen to have N = 0, i.e. a vanishing first Dynkin index. This observation will allow us tocompute conformal blocks for infinite towers of exchanged quasi-primary operators. Indeed, ifit is possible (impossible) to exchange a quasi-primary operator in the irreducible representation N m + ℓ e for some fixed ℓ ≥ ℓ min ( ℓ < ℓ min ), then it is straightforward to conclude that all quasi-primary operators in irreducible representations N m + ℓ e with ℓ ≥ ℓ min can also be exchanged,leading to an infinite tower of exchanged quasi-primary operators with the same seed irreduciblerepresentation N m + ℓ min e . We remark here that both N m and ℓ min depend on the irreduciblerepresentations of the quasi-primary operators of interest.For exchanged quasi-primary operators in the N m + ℓ e irreducible representation, the three-point basis can therefore be separated as follows: b F kl,m + ℓ = b F kl,m + i b ( A · ¯¯ η ) ℓ − i b , a F ij,m + ℓ = a F ij,m + i a ( A · ¯ η ) ℓ − i a → a t ij,m + ℓ = a t ij,m + i a ( A ) ℓ − i a , (3.1)where the ( A · ¯¯ η ) E ′′ ib +1 · · · ( A · ¯¯ η ) E ′′ ℓ and ( A · ¯ η ) E ia +1 · · · ( A · ¯ η ) E ℓ are the symmetrized ℓ -dependent parts of the respective tensor structures. We observe that in the second line of (3.1),the OPE basis is obtained from the three-point basis by simply transforming all A · ¯ η → A with the extra index contracting with the OPE differential operator, for example( A · ¯ η ) E ia +1 · · · ( A · ¯ η ) E ℓ → A E ′ ia +1 F ia +1 · · · A E ′ ℓ F ℓ . (3.2)It follows that the OPE basis used here does not satisfy the projection property (2.4) of the mostnatural tensor structures from the OPE point of view [1, 17]. However, its simple form will beof great advantage when we determine the three- and four-point conformal blocks. We note also Therefore, i a and i b are ℓ -independent nonnegative integers, i.e. i a and i b are fixed even when ℓ → ∞ . A · ¯¯ η )’s in b F kl,m + ℓ is given by n b , while that of ( A · ¯ η )’s in a F ij,m + ℓ by n a . In (3.1), the undetermined parts of the tensor structures, i.e. a t ij,m + i a and b F kl,m + i b , are fixedby the knowledge of the specific irreducible representations of the quasi-primary operators underconsideration. In the following, we dub them the “special” parts of the tensor structures andspecify them only for particular examples with known quasi-primary operators. Since there are no half-projectors for the exchanged quasi-primary operators in the four-pointcorrelation function (2.32), the projection operator to the exchanged representation necessarilyappears explicitly in the four-point conformal blocks (2.34). As for the tensor structures, we workhere with exchanged quasi-primary operators in the N m + ℓ e infinite tower of irreducible repre-sentations. To determine the four-point blocks, it is simpler to expand the projection operatorsas ˆ P N m + ℓ e = X t A t ( d, ℓ ) ˆ Q N m + ℓ t e | t ˆ P ( ℓ − ℓ t ) e | d + d t , (3.3)where A t ( d, ℓ ) are constants dependent on d and ℓ . The sum is finite and ℓ -independent here,and the number of terms depends on the irreducible representation N m . Moreover, the tensorquantities ˆ Q N m + ℓ t e | t encode information about the special parts of the irreducible representation N m + ℓ t e , while the remaining indices are carried by shifted projection operators for some d ′ and ℓ ′ , denoted by( ˆ P ℓ ′ e | d ′ ) E ′′ ··· E ′′ ℓ E ′ ℓ ··· E ′ = ⌊ ℓ ′ / ⌋ X i =0 ( − ℓ ′ ) i i i !( − ℓ ′ + 2 − d ′ / i A E ′ E ′ A ( E ′′ E ′′ · · · A E ′ i − E ′ i A E ′′ i − E ′′ i × A E ′′ i +1 E ′ i +1 · · · A E ′′ ℓ ′ )13 E ′ ℓ ) . (3.4)It is important to notice here that the ℓ E ′ -indices in (3.3), distributed among ˆ Q N m + ℓ t e | t andˆ P ( ℓ − ℓ t ) e | d + d t , are symmetrized. The same is true of the ℓ E ′′ -indices. Furthermore, we point out thatthe shifted projection operators ˆ P ( ℓ − ℓ t ) e | d + d t (3.4) are not traceless when d t = 0.For future convenience, we include some properties of the shifted projection operators underextraction of indices. Indeed, in the computation of four-point blocks, it is often necessary toextract n ′ E E ′ -indices and n ′′ E E ′′ -indices from the shifted projection operators. These are specialindices that ultimately contract with the special parts of the tensor structures. The general form18or this extraction is( ˆ P ℓ e | d ) { E ′′ }{ E ′ } = X r, r ′ , r ′′ ≥ r +2 r ′ + r ′ + r ′ = n ′ E r +2 r ′′ + r ′′ + r ′′ = n ′′ E r ′ + r ′ + r ′ = r ′′ + r ′′ + r ′′ C ( d,ℓ ) n ′ E ,n ′′ E ( r, r ′ , r ′′ )( A E ′′ s E ′ s ) r × ( A E ′ s E ′ s ) r ′ ( A E ′ s E ′ ) r ′ ( A E ′′ E ′ s ) r ′ ( A E ′ E ′ ) r ′ × ( A E ′′ s E ′′ s ) r ′′ ( A E ′′ s E ′′ ) r ′′ ( A E ′′ s E ′ ) r ′′ ( A E ′′ E ′′ ) r ′′ × ( ˆ P [ ℓ − ( r + r ′ + r ′ + r ′ + r ′ + r ′′ + r ′′ + r ′′ + r ′′ )] e | d +2( r + r ′ + r ′′ )+ r ′ + r ′ + r ′ + r ′′ + r ′′ + r ′′ ) { E ′′ }{ E ′ } , (3.5)where it is understood that the sets of special indices { E ′ s } and { E ′′ s } and the remaining sets ofindices { E ′ } and { E ′′ } are all symmetrized independently. From the identity( ˆ P ℓ e | d ) { E ′′ }{ E ′ } = A ( E ′′ E ′ s ( ˆ P ( ℓ − e | d +2 ) { E ′′ } ) { E ′ } + ℓ − − ℓ + 2 − d/ A E ′ s ( E ′ A ( E ′′ E ′′ ( ˆ P ( ℓ − e | d +2 ) { E ′′ } ) { E ′ } ) , (3.6)which corresponds to (3.5) with n ′ E = 1 and n ′′ E = 0, as can be seen directly from (3.4), it is easyto obtain the recurrence relation C ( d,ℓ ) n ′ E +1 ,n ′′ E ( r, r ′ , r ′′ )= r ′ + 1 ℓ − n ′ E C ( d,ℓ ) n ′ E ,n ′′ E ( r, r ′ − e + e , r ′′ ) + 2( r ′ + 1) ℓ − n ′ E C ( d,ℓ ) n ′ E ,n ′′ E ( r, r ′ − e + e , r ′′ )+ r ′′ + 1 ℓ − n ′ E C ( d,ℓ ) n ′ E ,n ′′ E ( r − , r ′ , r ′′ + e ) − − ℓ + r + r ′ + r ′ + r ′ + r ′ + r ′′ + r ′′ + r ′′ + r ′′ − ℓ − n ′ E C ( d,ℓ ) n ′ E ,n ′′ E ( r, r ′ − e , r ′′ )+ ( − ℓ + r + r ′ + r ′ + r ′ + r ′ + r ′′ + r ′′ + r ′′ + r ′′ − ℓ − n ′ E )[ − ℓ + ( r ′ + r ′ + r ′ + r ′′ + r ′′ + r ′′ − / − d/ C ( d,ℓ ) n ′ E ,n ′′ E ( r, r ′ − e , r ′′ − e ) , (3.7)as well as the analog recurrence relation for double-primed quantities. With the unique nonvan-ishing boundary condition C ( d,ℓ )0 , (0 , , ) = 1, the solution to (3.7) and its double-primed analogis given by C ( d,ℓ ) n ′ E ,n ′′ E ( r, r ′ , r ′′ ) = ( − r + r ′ + r ′ + n ′ E + r ′′ + r ′′ + n ′′ E n ′ E ! n ′′ E !2 r ′ + r ′ + r ′′ + r ′′ r ! r ′ ! r ′ ! r ′ ! r ′ ! r ′′ ! r ′′ ! r ′′ ! r ′′ ! [( r ′ + r ′ − r ′ + r ′′ + r ′′ − r ′′ ) / − ℓ ) n ′ E ( − ℓ ) n ′′ E × ( − r ′ − r ′ ) r ′′ ( − r ′′ − r ′′ ) r ′ ( − ℓ ) r + r ′ + r ′ + r ′ + r ′ + r ′′ + r ′′ + r ′′ + r ′′ ( − ℓ + 2 − d/ ( r ′ + r ′ + r ′ + r ′′ + r ′′ + r ′′ ) / . (3.8)This property under extraction of indices will greatly simplify the computation of four-pointconformal blocks. 19inally, we remark that the undetermined parts of the projection operators in (3.3), denotedby A t ( d, ℓ ) and ˆ Q N m + ℓ t e | t , are fixed by the knowledge of the specific irreducible representation ofthe exchanged quasi-primary operator under consideration. This mirrors the analysis of the tensorstructures. By analogy, we also refer to them as the special parts of the projection operators andfix them once we consider specific examples of four-point conformal blocks. It turns out thatwe can define a convenient diagrammatic notation that would allow us to easily enumerate thevarious terms arising from the index separation. We discuss this next. The extraction of indices (3.5) leads to specific partitions of n ′ E and n ′′ E , given by n ′ E = r + 2 r ′ + r ′ + r ′ and n ′′ E = r + 2 r ′′ + r ′′ + r ′′ , respectively. There is also the extra condition r ′ + r ′ + r ′ = r ′′ + r ′′ + r ′′ , where the maximumvalues for r ′ and r ′′ are r ′ ≤ r ′′ + r ′′ and r ′′ ≤ r ′ + r ′ . We introduce here a bookkeeping techniqueto easily generate the appropriate partitions of n ′ E and n ′′ E that appear in the computation offour-point conformal blocks.To proceed, let us symbolize the shifted projection operator (3.4) by the vertex( ˆ P ℓ e | d ) { E ′′ }{ E ′ } = . (3.9)Here the solid line represents the metrics of the form A E ′ E ′ ; the dotted line represents themetrics of the form A E ′′ E ′′ ; and the dashed line represents the metrics of the form A E ′′ E ′ . Thechosen convention sets the E ′ E ′ -line as a solid line, the E ′′ E ′′ -line as a dotted line, and the hybrid E ′ E ′′ -line as a hybrid dashed line.We are now interested in extracting n ′ E E ′ -indices and n ′′ E E ′′ -indices, which are all denoted bythe subscript s in (3.5). These are the special indices that do not contract with the ℓ -dependentparts of the tensor structures (3.1), in contrast to the non-special indices in the shifted projectionoperator (3.5), which are denoted by E ′ and E ′′ and are contracted with the known ℓ -dependentparts of the tensor structures (3.1).With these conventions, the extraction of indices can proceed as follows. On the one hand,any special E ′ s -index can be extracted from the E ′ E ′ -line or the E ′ E ′′ -line, resulting in metricsof the form A E ′ s E ′ and A E ′′ E ′ s , respectively. On the other hand, any special E ′′ s -index canbe extracted from the E ′′ E ′′ -line or the E ′ E ′′ -line, leading to metrics of the form A E ′′ s E ′′ and A E ′′ s E ′ , respectively. The extraction of an E ′ s -index ( E ′′ s -index) is denoted by an extra external20olid (dotted) line emerging from the appropriate line of the original vertex. Moreover, to accountfor metrics with two special indices, loops are also allowed on each vertex line. Thus, metricsof the form A E ′ s E ′ s are represented by solid loops on the E ′ E ′ -line, metrics of the form A E ′′ s E ′′ s by dotted loops on the E ′′ E ′′ -line, and lastly, metrics of the form A E ′′ s E ′ s by dashed loops onthe E ′ E ′′ -line. The partitions of n ′ E and n ′′ E are therefore constructed by dressing the three linesof the original vertex (3.9) with external lines such that the number of external solid (dotted)lines add up to n ′ E ( n ′′ E ). Here, the external loops count for two lines, with the dashed loopscounting for one solid and one dotted line each. In addition, due to the extra condition, onediagram can represent several extended partitions (extended partitions include also r ′ and r ′′ ,respectively). From the extra condition, the number of extended partitions per diagram is givenby min { r ′ + r ′ , r ′′ + r ′′ } + 1, i.e. the minimum value between the number of solid external linesand loops on the vertex solid line and the number of dotted external lines and loops on the vertexdotted line, plus one. The number of extended partitions is encoded in prefactors in front of eachdiagram.Hence, the partitions of interest for the identity (3.6) are easily obtained diagrammatically,with the diagrammatic equation for the identity (3.6) written as= + . The first diagram on the RHS represents the partitions n ′ E = 0 + 2 × n ′′ E =0+2 × n ′ E = 0 + 2 × n ′′ E = 0 + 2 × r ′ and r ′′ are obtainedfrom the extra condition, which can be computed from the diagrams by counting the number ofexternal lines and loops on the original solid and dotted lines, respectively, showing that bothdiagrams correspond to only one term each, as in (3.6). Finally, the associated coefficients arecomputed directly with (3.8).As a more complicated example, the diagrammatic equation for n ′ E = 2 and n ′′ E = 1 is givenby = + + + 2 ×
21 + 2 × ++ 2 × + + . For each diagram, counting the total number of external lines and loops on the original solid lineas well as the original dotted line shows that all the diagrams correspond to one term in (3.5),apart from , , , which correspond to two terms each, with ( r ′ , r ′′ ) ∈ { (0 , , (1 , } , ( r ′ , r ′′ ) ∈ { (0 , , (1 , } , and( r ′ , r ′′ ) ∈ { (0 , , (1 , } , respectively. Again, (3.8) gives the appropriate coefficients for eachterm. Having established some convenient notation for the index separation, we next turn to thedetermination of the rotation matrices.
4. Three-Point Functions and Rotation Matrices
In this section, we use the tensor structure basis introduced above to compute rotation matrices.Initial results imply ℓ -dependent sums that must eventually be re-summed, and we show how thiscan be done in all generality with the help of an identity for hypergeometric functions. From the tensor structures (3.1), the three-point conformal blocks (2.33) can be expressed as( G ij | m + ℓ ( a | ) { aA }{ bB }{ eE } = λ N m + ℓ (cid:16) ( A E ′ E ) n mv + ℓ (¯ η · Γ ¯ η · Γ) e ′ e (cid:17) cs ( a t ij,m + ℓ ) { aA }{ bB }{ e ′ E ′ }{ F } = λ N m + ℓ ( a t ij,m + i a ) { aA }{ bB }{ e ′ E ′ }{ F } × (cid:16) ( A E ′ E ) n mv + i a ( A E ′ E ) ℓ − i a (¯ η · Γ ¯ η · Γ) e ′ e (cid:17) cs ( A E ′ F ) ℓ − i a = λ N m + ℓ ( a t ij,m + i a ) { aA }{ bB }{ e ′ E ′ }{ F } (cid:16) ( A E ′ E ) n mv + i a (¯ η · Γ ¯ η · Γ) e ′ e ( A E ′ E ) ℓ − i a (cid:17) cs ( − ¯ η E ′ ¯ η F ) ℓ − i a , since A E ′ F = g E ′ F − ¯ η E ′ ¯ η F − ¯ η E ′ ¯ η F → − ¯ η E ′ ¯ η F . (4.1)It is straightforward to see that this simplification is true. Indeed, the contraction of the term¯ η E ′ ¯ η F vanishes straightforwardly, while the contraction of g E ′ F vanishes from the definitionof the conformal substitution cast in terms of the OPE differential operator as D ( d,h,n ) F n =( η · η ) − n D h + n )12 ( η F ) n [1].Now, owing to the contraction with the half-projector ( T N m Γ) in (2.31), it is self-evident thatthe A -metrics can be simplified to A E ′ E = g E ′ E − ¯ η E ¯ η E ′ − ¯ η E ¯ η E ′ + ¯ η E ¯ η E ′ → g E ′ E − ¯ η E ¯ η E ′ + ¯ η E ¯ η E ′ . Therefore, inside the three-point conformal blocks we can expand these as( A E ′ E ) n mv + i a = X σ X r ,r ≥ (cid:18) n mv + i a r + r (cid:19)(cid:18) r + r r (cid:19) ( − r ( n mv + i a )! g E ′ σ (1) E σ (1) · · · g E ′ σ ( r E σ ( r × ¯ η E σ ( r ¯ η E ′ σ ( r · · · ¯ η E σ ( r r ¯ η E ′ σ ( r r × ¯ η E σ ( r r ¯ η E ′ σ ( r r · · · ¯ η E σ ( nmv + ia ) ¯ η E ′ σ ( nmv + ia ) , ( A ) ℓ − i a = X t ,t ≥ (cid:18) ℓ − i a t + t (cid:19)(cid:18) t + t t (cid:19) ( − t ( g E ′ E ) t (¯ η E ¯ η E ′ ) t (¯ η E ¯ η E ′ ) ℓ − i a − t − t . In the first group above, the indices are not necessarily symmetrized by the special part of thetensor structure. Hence, the expansion must take into account the different indices, which forcesa sum over all the permutations σ of the n mv + i a pairs of indices. In the second group, the indicesare all already symmetrized from their contraction with the half-projector on one side and theknown ℓ -dependent part of the tensor structure ( − ¯ η E ′ ¯ η F ) ℓ − i a on the other, thus simplifying theexpansion by allowing all the indices to be treated on an equal footing.Proceeding with the conformal substitution (2.13) by simply counting the appropriate powers,the three-point conformal blocks become( G ij | m + ℓ ( a | ) { aA }{ bB }{ eE } = λ N m + ℓ ( a t ij,m + i a ) { aA }{ bB }{ e ′ E ′ }{ F } (¯ η · ΓΓ F ) e ′ e ( − ¯ η E ′ ¯ η F ) ℓ a × X σ X r ,r ,t ,t ≥ (cid:18) n m + i a v r + r (cid:19)(cid:18) r + r r (cid:19) ( − r n m + i a v ! (cid:18) ℓ a t + t (cid:19)(cid:18) t + t t (cid:19) ( − t × g E ′ σ (1) E σ (1) · · · g E ′ σ ( r E σ ( r ¯ η E ′ σ ( r · · · ¯ η E ′ σ ( r r ( g E ′ E ) t (¯ η E ′ ) t × ¯ I ( d,h ′ − ℓ a +2 t + t ,n ′ +2 ℓ a − t − t ; p ′ − t )12 E ′ σ ( r r ··· E ′ σ ( nm + iav ) ( E ′ ) ℓa − t − t F na +2 ξm E σ ( r ··· E σ ( nm + iav ) E ℓ − ia − t , h = h ij,m + ℓ − n mv − ξ m − ℓ − n a / − i a ,n = 2 n mv + 2 ξ m + n a + 2 i a ,p = ∆ m + ℓ + n mv + ℓ,n m + i a v = n mv + i a ,ℓ a = ℓ − i a , (4.2)and h ′ = h + 2 r + r , n ′ = n − r − r , p ′ = p − r . Contracting the ℓ -dependent part of the tensor structure by using the contiguous relations (2.20)gives ( G ij | m + ℓ ( a | ) { aA }{ bB }{ eE } = λ N m + ℓ ( a t ij,m + i a ) { aA }{ bB }{ e ′ E ′ }{ F } (¯ η · ΓΓ F ) e ′ e ( − ℓ a × X σ X r ,r ,t ,t ≥ (cid:18) n m + i a v r + r (cid:19)(cid:18) r + r r (cid:19) ( − r n m + i a v ! (cid:18) ℓ a t + t (cid:19)(cid:18) t + t t (cid:19) ( − t × ρ ( d,ℓ a − t − t ; − h − n − ℓ a ) g E ′ σ (1) E σ (1) · · · g E ′ σ ( r E σ ( r ¯ η E ′ σ ( r · · · ¯ η E ′ σ ( r r × ¯ I ( d,h ′ +2 t + t ,n ′ − t ; p ′ − t )12 E ′ σ ( r r ··· E ′ σ ( nm + iav ) F na − ℓa +2 ξm E σ ( r ··· E σ ( nm + iav ) E ℓa − t (¯ η E ) t . We next consider the ¯ I -function. Denoting all of its (symmetrized) E ′ σ - and F -indices by Z -indices,we find with the help of (2.18)¯ I ( d,h ′ +2 t + t ,n ′ − t ; p ′ − t )12 Z k E k σ E ℓa − t = X q ,q ,q ,q ≥ q = n ′ − t e K ( d,h ′ +2 t + t ; p ′ − t ; q ,q ,q ,q ) S ( q ,q ,q ,q ) Z k E k σ E ℓa − t , where k = n − n m + i a v − ℓ a − r − r ,k = n m + i a v − r . We now aim to extract the E ℓ a − t indices from S ( q ) to eventually re-sum over t and t , whichare both ℓ -dependent sums, i.e. both sums grow as ℓ grows. This will simplify the computationof rotation matrices by replacing ℓ -dependent summations by ℓ -independent ones.To begin with, we note that all the E -indices (including the E σ -indices) in S ( q ) must be carriedby either metrics or ¯ η ’s, due to their contraction with the half-projector in (2.31). Moreover,since there cannot be g EE ’s in S ( q ) due to the tracelessness condition of the same half-projector,there is a minimum number of ¯ η E ’s in S ( q ) which is given by the absolute value of the number24f E -indices minus the number of Z -indices, i.e. | ℓ a − t + k − k | . Moreover, from the followingidentities (see Section 4 of [1]) S A ··· A ¯ q ( q ) = 2 q ¯ q g A ¯ q ( A S A ··· A ¯ q − )( q − e ) + X r =0 q r ¯ q ¯ η A ¯ q r S A ··· A ¯ q − ( q − e r ) = 2 q ¯ q (¯ q − g A ¯ q A ¯ q − S A ··· A ¯ q − ( q − e ) + 4 q ( q − q (¯ q − g A ¯ q ( A g | A ¯ q − | A S A ··· A ¯ q − )( q − e ) + X r =0 q q r ¯ q (¯ q −
1) ¯ η ( A ¯ q r g A ¯ q − )( A S A ··· A ¯ q − )( q − e − e r ) + X r,s =0 q r ( q s − δ rs )¯ q (¯ q −
1) ¯ η A ¯ q r ¯ η A ¯ q − s S A ··· A ¯ q − ( q − e r − e s ) , it is easy by recurrence to extract all the symmetrized E ℓ a − t indices, such as S ( q ,q ,q ,q ) Z k E k σ E ℓa − t = X s ≥ (cid:18) ℓ a − t s (cid:19) ( − s ( − q ) s ( − q ) ℓ a − s − t ( k + 1) ℓ a + k − t ( k − s + 1) k × ( g ( ZE ) s S Z k − s )( q − s ,q ,q − ℓ a + s + t ,q ) E k σ ( − ¯ η E ) ℓ a − s − t . Next, shifting q → q + s and q → q + ℓ a − s − t leads to¯ I ( d,h ′ +2 t + t ,n ′ − t ; p ′ − t )12 Z k E k σ E ℓa − t = X s ≥ X q ,q ,q ,q ≥ q = n ′ − ℓ a − s (cid:18) ℓ a − t s (cid:19) ( − s ( − q − s ) s ( − q − ℓ a + s + t ) ℓ a − s − t ( k + 1) ℓ a + k − t ( k − s + 1) k × e K ( d,h ′ +2 t + t ; p ′ − t ; q + s ,q ,q + ℓ a − s − t ,q )¯ q = n ′ − t ( g ( ZE ) s S Z k − s )( q ,q ,q ,q ) E k σ ( − ¯ η E ) ℓ a − s − t , where all the dependence on t and t in the tensorial components has been removed, and thesubscript on the e K -function is to remind us that its value of ¯ q is not the same as the one for S ( q ) anymore. With this identity, the three-point conformal blocks then assume the form( G ij | m + ℓ ( a | ) { aA }{ bB }{ eE } = λ N m + ℓ ( a t ij,m + i a ) { aA }{ bB }{ e ′ E ′ }{ F } (¯ η · ΓΓ F ) e ′ e × X σ X r ,r ,s ,t ,t ≥ ( − ℓ a + r + t + t n m + i a v ! (cid:18) n m + i a v r + r (cid:19)(cid:18) r + r r (cid:19)(cid:18) ℓ a t + t (cid:19)(cid:18) t + t t (cid:19) × X q ,q ,q ,q ≥ q = n ′ − ℓ a − s (cid:18) ℓ a − t s (cid:19) ( − s ( − q − s ) s ( − q − ℓ a + s + t ) ℓ a − s − t ( k + 1) ℓ a + k − t ( k − s + 1) k × ρ ( d,ℓ a − t − t ; − h − n − ℓ a ) e K ( d,h ′ +2 t + t ; p ′ − t ; q + s ,q ,q + ℓ a − s − t ,q )¯ q = n ′ − t × g E ′ σ (1) E σ (1) · · · g E ′ σ ( r E σ ( r ¯ η E ′ σ ( r · · · ¯ η E ′ σ ( r r ( g ( ZE ) s S Z k − s )( q ,q ,q ,q ) E k σ ( − ¯ η E ) ℓ a − s = λ N m + ℓ ( a t ij,m + i a ) { aA }{ bB }{ e ′ E ′ }{ F } (¯ η · ΓΓ F ) e ′ e X σ X r ,r ,s ≥ ( − ℓ a + r ( − ℓ a +2 s n m + i a v ! s ! (cid:18) n m + i a v r + r (cid:19)(cid:18) r + r r (cid:19) ( − n ′ ) ℓ a + s ( − h − n ) ℓ a + s ( k + 1) ℓ a + k ( k − s + 1) k × X q ,q ,q ,q ≥ q = n ′ − ℓ a − s ( − q − s ) s ( − q − ℓ a + s ) ℓ a − s ( p ′ + h + n − s − q − q ) − ℓ a − s ρ ( d,ℓ a ; − h − n − ℓ a ) e K ( d,h ′ ; p ′ ; q + s ,q ,q + ℓ a − s ,q )¯ q = n ′ − ℓ a − s Σ t × g E ′ σ (1) E σ (1) · · · g E ′ σ ( r E σ ( r ¯ η E ′ σ ( r · · · ¯ η E ′ σ ( r r ( g ( ZE ) s S Z k − s )( q ,q ,q ,q ) E k σ ( − ¯ η E ) ℓ a − s , where we have used the definitions (2.19). Here the sums over t and t are included in Σ t , whichis given explicitly byΣ t = X t ,t ≥ ( − t + t s ! (cid:18) ℓ a t + t (cid:19)(cid:18) t + t t (cid:19)(cid:18) ℓ a − t s (cid:19) × ( − p ′ + 1) t ( p ′ + h + n − s − q − q ) t ( p ′ + h ′ + 1 − d/ t + t ( h ′ + ℓ a + q + q + 1) t + t ( h + n + d/ t + t . Equipped with this result, we can finally transform the ℓ -dependent sums over t and t in Σ t into ℓ -independent sums. First of all, we shift t → t − t and rewrite the sum over t as a hypergeometric function,which leads to Σ t = X t ≥ (cid:18) ℓ a t (cid:19) ( ℓ a − s − t + 1) s ( p ′ − t ) t ( p ′ + h ′ + 1 − d/ t ( h ′ + ℓ a + q + q + 1) t ( h + n + d/ t × F " p ′ + h + n − s − q − q , ℓ a − t + 1 , − t p ′ − t , ℓ a − s − t + 1 ; 1 . We then use the well known identity F " α, β, − nγ, β − m ; 1 = ( γ − α ) n ( γ ) n F " α, − m, − nα − γ − n + 1 , β − m ; 1 , (4.3)to transform the first ℓ -dependent sum,Σ t = X t ,t ≥ (cid:18) ℓ a t (cid:19) ( ℓ a − s − t + 1) s ( − h − n + s − t + q + q ) t ( p ′ + h ′ + 1 − d/ t ( h ′ + ℓ a + q + q + 1) t ( h + n + d/ t × ( p ′ + h + n − s − q − q ) t ( − s ) t ( − t ) t ( h + n − s − q − q + 1) t ( ℓ a − s − t + 1) t t ! , where the index of summation of the new hypergeometric function is t . To further transform thesum over t , we now shift t → t + t and re-express the sum as a hypergeometric function suchthat Σ t = ( − s ( − ℓ a ) s X t ≥ ( − s ) t ( p ′ + h + n − s − q − q ) t ( p ′ + h ′ + 1 − d/ t ( h ′ + ℓ a + q + q + 1) t ( h + n + d/ t t ! Clearly, from the binomial coefficient (cid:0) ℓ a t + t (cid:1) , the sums over t and t stop at ℓ a , which grows like ℓ for large ℓ . For n and m nonnegative integers and α , β and γ arbitrary complex numbers. F " p ′ + h ′ + t + 1 − d/ , h + n − s + t − q − q + 1 , − ℓ a + s h + n + t + d/ , h ′ + ℓ a + t + q + q + 1 ; 1 . We next apply the identity (4.3) once again. This leads toΣ t = ( − s ( − ℓ a ) s X s ,t ≥ ( − s ) t ( p ′ + h + n − s − q − q ) t ( p ′ + h ′ + 1 − d/ t ( h ′ + ℓ a + q + q + 1) t ( h + n + d/ t t ! × ( − p ′ + n ′ − d ) ℓ a − s ( h + n + t + d/ ℓ a − s ( p ′ + h ′ + t + 1 − d/ s ( − q ) s ( − ℓ a + s ) s ( p ′ − n ′ − ℓ a + s + 2 − d ) s ( h ′ + ℓ a + t + q + q + 1) s s ! , (4.4)where the index of summation of the new hypergeometric function was chosen to be s . At thispoint, the two ℓ -dependent sums have been transformed into two ℓ -independent sums, and we canreturn to the three-point conformal blocks.Now, inserting (4.4) in the three-point conformal blocks, shifting q → q + s using the factthat S ( q ,q ,q ,q + s ) Z k − s E k σ = ( k − s − s + 1) k ( k − s + 1) k (¯ η ( Z ) s S ( q ,q ,q ,q ) Z k − s − s ) E k σ , since all ¯ η ’s must have Z -indices only, and finally re-summing the q ’s into an ¯ I -function (2.18),our final result becomes( G ij | m + ℓ ( a | ) { aA }{ bB }{ eE } = λ N m + ℓ ( a t ij,m + i a ) { aA }{ bB }{ e ′ E ′ }{ F } (¯ η · ΓΓ F ) e ′ e × X r ,r ,s ,s ,t ≥ ( − r + s ( − ℓ a + s − t n m + i a v ! r ! r ! s ! s ! t ! ( − n m + i a v ) r + r ( − ℓ a ) s + s ( − s ) t × ( − n + n m + i a v + ℓ a + r + r ) s + s ( − h − n − ℓ a ) ℓ a + s − t × ( − h − n − ℓ a + 1 − d/ s − t ( p − n − ℓ a + r + r + s + s + 2 − d ) ℓ a − s − s × X σ g E ′ σ (1) E σ (1) · · · g E ′ σ ( r E σ ( r ¯ η E ′ σ ( r · · · ¯ η E ′ σ ( r r ( g ( ZE ) s (¯ η Z ) s × ¯ I ( d +2 ℓ a ,h + ℓ a +2 r + r + s + t,n − ℓ a − r − r − s − s ; p − r )12 Z n − nm + iav − ℓa − r − r − s − s ) E nm + iav − r σ ( − ¯ η E ) ℓ a − s , (4.5)where the Z -indices belong to Z ∈ { E ′ σ ( r + r +1) , . . . , E ′ σ ( n m + iav ) , F n a − ℓ a +2 ξ m } , and the different parameters are defined in (4.2). Here, the notation was chosen such that r and r represent the number of non-symmetrized free metrics and ¯ η ’s, respectively, while s and s represent the number of free symmetrized ( i.e. with Z -indices) metrics and ¯ η ’s, respectively.27learly, from (4.5), we have the following bounds on the different indices of summation:0 ≤ r + r ≤ n m + i a v , ≤ s + s ≤ n m + i a v + 2 ξ m + n a − ℓ a − r − r , ≤ t ≤ s , which are all ℓ -independent, as desired. It is now straightforward to find the rotation matrix from (4.5). First, we contract the remaining A E ′ F → − ¯ η E ′ ¯ η F [see (4.1)] from the special part of the tensor structure a t ij,m + i a , using thecontiguous relations (2.20) when appropriate. Then, we simply expand the ¯ I -function and contractthe remaining factors from the special part of the tensor structure. Finally, we replace all free ¯ η ’sby A · ¯ η with the appropriate sign [as in (3.2)]. Equipped with this result, we can determinethe rotation matrix from the relation G ij | m + ℓ ( a | = N ij,m + ℓ X a ′ =1 ( R − ij,m + ℓ ) aa ′ ¯ η · Γ a ′ F ij,m + ℓ ( A , Γ , ǫ ; A · ¯ η )= N ij,m + ℓ X a ′ =1 ( R − ij,m + ℓ ) aa ′ ¯ η · Γ a ′ F ij,m + i a ′ ( A , Γ , ǫ ; A · ¯ η )( A · ¯ η ) ℓ − i a ′ , (4.6)using the symmetry properties of the irreducible representations of the three quasi-primary oper-ators under consideration to match with the three-point tensor structure basis.It is also possible to first expand (4.5) and then contract with the tensor structure. With thedefinition a κ ij | m + ℓ ( q ,r ,r ,s ,s ,t ) = λ N m + ℓ e ( − ξ m + n a − ℓ + i a − r − r − s − q − q − q ( − h ij,m + ℓ + n a / − q × (2 n mv + 2 ξ m + n a − ℓ + 3 i a − r − r − s − s )!( n mv + i a )! q ! q ! q ! q ! r ! r ! s ! s ! t ! ( − ℓ + i a ) s + s × ( − n mv − i a ) r + r ( − n mv − ξ m − n a + ℓ − i a + r + r ) s + s ( − s ) t × ( − h ij,m + ℓ − n a / − d/ s − t ( − h ij,m + ℓ − n a / ℓ − i a + s − t + q + q + q × (∆ m + ℓ − n mv − ξ m − n a − i a + r + r + s + s + 2 − d ) ℓ − i a − s − s × (∆ m + ℓ + n mv + ℓ − r ) h ij,m + ℓ + n a / − ℓ + i a − s + t − q − q × (∆ m + ℓ + n mv + i a − r − q − q − q + 1 − d/ h ij,m + ℓ + n a / − ℓ + i a − s + t − q − q , (4.7)the three-point conformal blocks (4.5) take on the form( G ij | m + ℓ ( a | ) { aA }{ bB }{ eE } = ( a t ij,m + i a ) { aA }{ bB }{ e ′ E ′ }{ F } (¯ η · ΓΓ F ) e ′ e × X r ,r ,s ,s ,t ≥ X q ,q ,q ,q ≥ a κ ij | m + ℓ ( q ,r ,r ,s ,s ,t ) X σ g E ′ σ (1) E σ (1) · · · g E ′ σ ( r E σ ( r ¯ η E ′ σ ( r · · · ¯ η E ′ σ ( r r ( g ( ZE ) s (¯ η Z ) s × S ( q ,q ,q ,q ) Z n − nm + iav − ℓa − r − r − s − s ) E nm + iav − r σ ( − ¯ η E ) ℓ a − s , where 2 q + q + q + q = 2 n mv + 2 ξ m + n a − ℓ + 3 i a − r − r − s − s . Thus, each element of the rotation matrix under consideration may be conveniently expressed asa sum of κ ’s (4.7) with suitable shifts.We observe that in order to obtain the four-point conformal blocks in the three-point basis, itis also necessary to invert the rotation matrix computed from (4.6). Despite the fact that the sizeof the rotation matrices is ℓ -dependent, it turns out that one can invert them without worryingabout their ℓ -dependent size.Surprisingly, the determination of the rotation matrix is the most contrived calculation in ourquest for the four-point conformal blocks. We next consider the blocks themselves.
5. Four-Point Functions and Conformal Blocks
In this section, we use the tensor structures and the projection operators in their general formto compute the most general four-point conformal blocks in terms of the special case-dependentparts. Just like for the rotation matrices, our goal is to complete the ℓ -dependent computationsand express the final result purely in terms of the special parts of the tensor structures andthe projection operators. With such a result, the determination of the infinite towers of confor-mal blocks would be reduced to simple manipulations of the ℓ -independent special parts underconsideration. Using the tensor structures (3.1) and the projection operators (3.3) in (2.34) leads to the conformalblocks( G ij | m + ℓ | kl ( a | b ] ) { aA }{ bB }{ cC }{ dD } = X t A t ( d, ℓ )( a t ,m + i a ij ) { Ee }{ F }{ aA }{ bB } ( − ¯ η E ¯ η F ) ℓ − i a (cid:16) ( − x ) ξ m ( A E ′ E ) n mv + ℓ (¯¯ η · Γ) e ′ e × (¯ η · Γ ˆ Q N m + ℓ t e | t ˆ P ( ℓ − ℓ t ) e | d + d t ¯ η · Γ) E ′′ e ′′ e ′ E ′ ( b F kl,m + i b ) { cC }{ dD }{ e ′′ E ′′ } [( A · ¯¯ η ) E ′′ ] ℓ − i b (cid:17) cs . Here the property (4.1), which is also true for A EF by the same logical argument, allowed us tosubstitute A EF → − ¯ η E ¯ η F in the previous equation. Clearly, the metrics A EF from the specialpart of the tensor structure can also be simplified with the help of (4.1), i.e. A EF → − ¯ η E ¯ η F .29ur goal now is to manipulate the projection operator such that the symmetrizations on the ℓE ′ - and E ′′ -indices may be removed. Clearly, since the ℓ -dependent parts of the tensor structureshave ℓ − i a E -indices and ℓ − i b E ′′ -indices symmetrized, respectively, it is only necessary to extract i a E ′ -indices and i b E ′′ -indices from the symmetrizations. This is accomplished by a simple doubleexpansion, leading to(¯ η · Γ ˆ Q N m + ℓ t e | t ¯ η · Γ) ( E ′′ ℓt | E ′′ nmv e ′′ | e ′ E ′ nmv ( E ′ ℓt ( ˆ P ( ℓ − ℓ t ) e | d + d t ) E ′′ ℓ − ℓt ) E ′ ℓ − ℓt ) = X j a ,j b ≥ (cid:18) i a j a (cid:19)(cid:18) i b j b (cid:19) ( − ℓ t ) i a − j a ( − ℓ t ) i b − j b ( − ℓ + ℓ t ) j a ( − ℓ + ℓ t ) j b ( − ℓ ) i a ( − ℓ ) i b × (¯ η · Γ ˆ Q N m + ℓ t e | t ¯ η · Γ) ( E ′′ s ib − jb ( E ′′ ℓt − ib + jb | E ′′ nmv e ′′ | e ′ E ′ nmv ( E ′ ℓt − ia + ja ( E ′ sia − ja × X r, r ′ , r ′′ ≥ r +2 r ′ + r ′ + r ′ = j a r +2 r ′′ + r ′′ + r ′′ = j b r ′ + r ′ + r ′ = r ′′ + r ′′ + r ′′ C ( d + d t ,ℓ − ℓ t ) j a ,j b ( r, r ′ , r ′′ )( A E ′′ s E ′ s ) r × ( A E ′ s E ′ s ) r ′ ( A E ′ s E ′ ) r ′ ( A E ′′ E ′ s ) ) r ′ ( A E ′ E ′ ) r ′ × ( A E ′′ s E ′′ s ) r ′′ ( A E ′′ s E ′′ ) r ′′ ( A E ′′ s )13 E ′ ) r ′′ ( A E ′′ E ′′ ) r ′′ × ( ˆ P [ ℓ − ℓ t − ( r + r ′ + r ′ + r ′ + r ′ + r ′′ + r ′′ + r ′′ + r ′′ )] e | d + d t +2( r + r ′ + r ′′ )+ r ′ + r ′ + r ′ + r ′′ + r ′′ + r ′′ ) { E ′′ } ) { E ′ } ) , where we used (3.5). We note here that apart from the indices for N m appearing in the specialparts of the projection operator, all the i a special E ′ s -indices are symmetrized together. The sameis true for the i b special E ′′ s -indices, the ℓ − i a E ′ -indices, and the ℓ − i b E ′′ -indices.Upon substituting this result into the four-point block, we may remove the two explicit sym-metrizations on the ℓ − i a E ′ - and ℓ − i b E ′′ -indices and contract the ℓ -dependent part of thethree-point tensor structure straightforwardly. This gives( G ij | m + ℓ | kl ( a | b ] ) { aA }{ bB }{ cC }{ dD } = X t A t ( d, ℓ ) X j a ,j b ≥ (cid:18) i a j a (cid:19)(cid:18) i b j b (cid:19) ( − ℓ t ) i a − j a ( − ℓ t ) i b − j b ( − ℓ + ℓ t ) j a ( − ℓ + ℓ t ) j b ( − ℓ ) i a ( − ℓ ) i b × X r, r ′ , r ′′ ≥ r +2 r ′ + r ′ + r ′ = j a r +2 r ′′ + r ′′ + r ′′ = j b r ′ + r ′ + r ′ = r ′′ + r ′′ + r ′′ ( − r ′′ C ( d + d t ,ℓ − ℓ t ) j a ,j b ( r, r ′ , r ′′ )( a t ,m + i a ij ) { Ee }{ F }{ aA }{ bB } ( − ¯ η E ¯ η F ) ℓ − i a × (cid:16) ( − x ) ξ m ( A E ′′ s E s ) r ( A E s E s ) r ′ ( A EE s ) r ′ ( −A · ¯¯ η E s ) r ′ ( A EE ) r ′ × (¯¯ η · Γ ¯ η · Γ ( A ) n mv + ℓ t ˆ Q N m + ℓ t e | t ¯ η · Γ) ( E ′′ s ib − jb | E ′′ ℓt − ib + jb E ′′ nmv e ′′ || eE nmv E ℓt − ia + ja | E ia − jas ) × ( A E ′′ s E ′′ s ) r ′′ (¯¯ η E ′′ s ) r ′′ ( A E ′′ s )123 E ) r ′′ ( b F kl,m + i b ) { cC }{ dD }{ e ′′ E ′′ } [( A · ¯¯ η ) E ′′ ] ℓ t − i b + j b (( A ) ℓ ′ ˆ P ℓ ′ e | d ′ ( A · ¯¯ η ) ℓ ′ ) { E } (cid:17) cs , where d ′ = d + d t + 2( r + r ′ + r ′′ ) + r ′ + r ′ + r ′ + r ′′ + r ′′ + r ′′ ,ℓ ′ = ℓ − ℓ t − ( r + r ′ + r ′ + r ′ + r ′ + r ′′ + r ′′ + r ′′ + r ′′ ) . (5.1)Since none of the special E s -indices are contracted with − ¯ η E ¯ η F and none of the special E ′′ s -indices are contracted with A · ¯¯ η E ′′ , we can make the replacements A E s E s → g E s E s , A E ′′ s E ′′ s → g E ′′ s E ′′ s . Moreover, all the non-special ℓ -dependent E -indices, i.e. all E -indices (except for the E n mv indiceson the special part of the projection operator), must contract with − ¯ η E ¯ η F . Hence, we may replace A EE s → − ¯ η E ¯ η E s , A EE → − η E ¯ η E . With these simplifications, the four-point conformal blocks assume the form( G ij | m + ℓ | kl ( a | b ] ) { aA }{ bB }{ cC }{ dD } = X t A t ( d, ℓ ) X j a ,j b ≥ (cid:18) i a j a (cid:19)(cid:18) i b j b (cid:19) ( − ℓ t ) i a − j a ( − ℓ t ) i b − j b ( − ℓ + ℓ t ) j a ( − ℓ + ℓ t ) j b ( − ℓ ) i a ( − ℓ ) i b × X r, r ′ , r ′′ ≥ r +2 r ′ + r ′ + r ′ = j a r +2 r ′′ + r ′′ + r ′′ = j b r ′ + r ′ + r ′ = r ′′ + r ′′ + r ′′ ( − ℓ +2 ξ m − i a + r ′ + r ′ ( − r ′ + r ′′ C ( d + d t ,ℓ − ℓ t ) j a ,j b ( r, r ′ , r ′′ ) × ( a t ,m + i a ij ) { Ee }{ F }{ aA }{ bB } (¯ η E ) ℓ − i a − r ′ − r ′ (¯ η F ) ℓ − i a (cid:16) ( x ) ξ m ( x x ) ξ m × (¯ η · Γ ¯ η · Γ ( A ) n mv + ℓ t ˆ Q N m + ℓ t e | t ¯ η · Γ) ( E ′′ s ib − jb | E ′′ ℓt − ib + jb E ′′ nmv e ′′ | eE nmv E ℓt − ia + ja ( E ia − jas × ( A E ′′ s E s ) r ( g E s E s ) r ′ (¯ η E s ) r ′ ( A · ¯¯ η E s ) ) r ′ (¯ η E ) r ′ × ( g E ′′ s E ′′ s ) r ′′ (¯¯ η E ′′ s ) r ′′ ( A E ′′ s )123 E ) r ′′ ( b F kl,m + i b ) { cC }{ dD }{ e ′′ E ′′ } [( A · ¯¯ η ) E ′′ ] ℓ t − i b + j b × (( A ) ℓ ′ ˆ P ℓ ′ e | d ′ ( A · ¯¯ η ) ℓ ′ ) { E } (cid:17) cs . We now extract all the allowed factors from the conformal substitution to rewrite the result as( G ij | m + ℓ | kl ( a | b ] ) { aA }{ bB }{ cC }{ dD } As a reminder, both sets of indices originate from the ℓ e part of the projection operator for the exchangedquasi-primary operator. They are considered special because they were not symmetrized with the remaining ℓ e indices in the tensor structures. Hence, they cannot possibly contract with the symmetrized part of their respectivetensor structures. X t A t ( d, ℓ ) X j a ,j b ≥ (cid:18) i a j a (cid:19)(cid:18) i b j b (cid:19) ( − ℓ t ) i a − j a ( − ℓ t ) i b − j b ( − ℓ + ℓ t ) j a ( − ℓ + ℓ t ) j b ( − ℓ ) i a ( − ℓ ) i b × X r, r ′ , r ′′ ≥ r +2 r ′ + r ′ + r ′ = j a r +2 r ′′ + r ′′ + r ′′ = j b r ′ + r ′ + r ′ = r ′′ + r ′′ + r ′′ ( − ℓ +2 ξ m − i a + r ′ + r ′ ( − ℓ − ℓ t − j a − j b + r + r ′ + r ′′ C ( d + d t ,ℓ − ℓ t ) j a ,j b ( r, r ′ , r ′′ ) × ( a t ,m + i a ij ) { Ee }{ F }{ aA }{ bB } ( g E s E s ) r ′ (¯ η E ) ℓ − i a − r ′ − r ′ (¯ η F ) ℓ − i a × (cid:16) ( x ) − r ′ +2 ξ m ( x x ) ( r ′ + r ′′ + ℓ t + j b + n b − ℓ ) / ξ m × (¯ η F ′′ ) ξ m (( A ) n mv + ℓ t ) E ′ sia − ja E ′ ℓt − ia + ja E ′ nmv E nmv E ℓt − ia + ja E ia − jas × ( A E ′′ s E s ) r (¯ η E s ) r ′ ( A E ′′ E s ) r ′ (¯ η E ) r ′ (¯ η E ′′ s ) r ′′ ( A E ′′ s E ) r ′′ (¯ η F ′′ ) n b − ℓ + i b (¯ η F ′′ ) ℓ t − i b + j b × (( A ) ℓ ′ ˆ P ℓ ′ e | d ′ (( − − A · ¯¯ η ) ℓ ′ ) { E } (cid:17) cs × (Γ F ′′ ¯ η · Γ ˆ Q N m + ℓ t e | t Γ F ′′ ) E ′′ s ib − jb E ′′ ℓt − ib + jb E ′′ nmv e ′′ eE ′ nmv E ′ ℓt − ia + ja E ′ ia − jas × ( g E ′′ s E ′′ s ) r ′′ ( b t kl,m + i b ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } ( A E ′′ F ′′ ) ℓ t − i b + j b (¯ η E ′′ ) r ′ , where n b − ℓ + i b is the number of ( A · ¯¯ η )’s in the three-point tensor structure b F kl,m + i b . Also,note that the latter was changed to the associated OPE tensor structure b t kl,m + i b to allow itsextraction outside the conformal substitution cs . Moreover, we stop explicitly symmetrizing overthe special indices to avoid cluttering the computation too much.Equipped with this form, we now proceed with the conformal substitution CS ′ = (¯ η E ) ℓ − i a − r ′ − r ′ (¯ η F ) ℓ − i a (cid:16) ( x ) − r ′ +2 ξ m ( x x ) ( r ′ + r ′′ + ℓ t + j b + n b − ℓ ) / ξ m × (¯ η F ′′ ) ξ m (( A ) n mv + ℓ t ) E ′ sia − ja E ′ ℓt − ia + ja E ′ nmv E nmv E ℓt − ia + ja E ia − jas × ( A E ′′ s E s ) r (¯ η E s ) r ′ ( A E ′′ E s ) r ′ (¯ η E ) r ′ (¯ η E ′′ s ) r ′′ ( A E ′′ s E ) r ′′ (¯ η F ′′ ) n b − ℓ + i b (¯ η F ′′ ) ℓ t − i b + j b × (( A ) ℓ ′ ˆ P ℓ ′ e | d ′ (( − − A · ¯¯ η ) ℓ ′ ) { E } (cid:17) cs . We begin by analyzing the shifted projection operator. We find that for fixed d ′ and ℓ ′ , thecontracted shifted projection operator behaves as(( A ) ℓ ′ ˆ P ℓ ′ e | d ′ (( − − A · ¯¯ η ) ℓ ′ ) { E } = ⌊ ℓ ′ / ⌋ X i =0 ( − ℓ ′ ) i i i !( − ℓ ′ + 2 − d ′ / i (¯ η E ¯ η E ) i (cid:18) [(¯ η − ¯ η ) x − (¯ η − ¯ η ) x ] E √ x x (cid:19) ℓ ′ − i , as all the E -indices are ultimately contracted with (¯ η E ) ℓ ′ outside the conformal substitution.Assuming for a moment that the remaining factors in the conformal substitution are set to one,32e expect the conformal substitution to give CS ′′ = (¯ η E ) ℓ ′ (¯ η F ) ℓ − i a (cid:16) (( A ) ℓ ′ ˆ P ℓ ′ e | d ′ (( − − A · ¯¯ η ) ℓ ′ ) { E } (cid:17) cs = (¯ η E ) ℓ ′ (¯ η F ) ℓ − i a × ⌊ ℓ ′ / ⌋ X i =0 ( − ℓ ′ ) i i i !( − ℓ ′ + 2 − d ′ / i (¯ η E ¯ η E ) i (cid:18) [(¯ η − ¯ η ) x − (¯ η − ¯ η ) x ] E √ x x (cid:19) ℓ ′ − i cs = ℓ ′ !2 ℓ ′ ( d ′ / − ℓ ′ (cid:16) C ( d ′ / − ℓ ′ ( X ) (cid:17) s ′′ . We encounter here the Gegenbauer polynomials C ( λ ) n ( X ) in the variable X = ( α − α ) x − ( α − α ) x , (5.2)with the substitution s ′′ : α s α s α s x r x r → G ij | m + ℓ | kl ( ℓ ′ − ℓ,n a − ℓ,n a + ℓ ′ − ℓ +2 i a , , F na − ℓ + ia . (5.3)The variable (5.2) and substitution (5.3) were introduced to allow the re-summation into theGegenbauer polynomials, with ⌊ ℓ ′ / ⌋ X i =0 ( − ℓ ′ ) i i i !( − ℓ ′ + 2 − d ′ / i X ℓ ′ − i = ℓ ′ !2 ℓ ′ ( d ′ / − ℓ ′ C ( d ′ / − ℓ ′ ( X ) . Indeed, since in the expansion of ([(¯ η − ¯ η ) x − (¯ η − ¯ η ) x ] E ) ℓ ′ − i we have s + s + s = r + r = ℓ ′ − i , we can replace i by either one of i = ℓ ′ − ( s + s + s )2 , i = ℓ ′ − ( r + r )2 . Hence, in the conformal substition cs there arenumber of ¯ η ’s = i + s = ℓ ′ + s − s − s , number of x ’s = − ℓ ′ i + r = r − r , number of x ’s = − ℓ ′ i + r = − r − r , which straightforwardly lead to the substitution (5.3) for the variable X (5.2), with the quantity G ij | m + ℓ | kl ( n ,n ,n ,n ,n ) A ··· A n = ρ ( d, ( ℓ + s − s − s + n ) / − h ijm − ( ℓ + n ) / x − s x − s × ¯ I ( d,h ijm − ( s − s − s + n ) / ,n ; − h klm +( r − r + n ) / ,χ m + h klm − ( r − r + n ) / A ··· A n . (5.4)33ere, G is the quantity that naturally appears in the conformal substitutions (2.34) [18]. It turnsout to have some interesting properties, as we will discuss shortly.As for the remaining factors inside the conformal substitution, we find that these can be easilytaken into account by simply noting that the conformal substitution depends only on the powersof ¯ η , x and x . Moreover, since powers add up under multiplication of factors and these powersappear directly inside (5.4), a direct consequence is the fundamental property G ij | m + ℓ | kl ( n ,n ,n ,n ,n ) A n G ij | m + ℓ | kl ( m ,m ,m ,m ,m ) B m = G ij | m + ℓ | kl ( n + m ,n + m ,n + m ,n + m ,n + m ) A n B m , (5.5)which is understood as long as the definition of G in terms of the ¯ I -function (5.4) is not useduntil there is only one G per term.Hence, the initial conformal substitution may be rewritten as CS ′ = ℓ ′ !2 ℓ ′ ( d ′ / − ℓ ′ (¯ η E ) ℓ − i a − r ′ − r ′ − ℓ ′ (cid:16) C ( d ′ / − ℓ ′ ( X ) (cid:17) s ′ , with the new substitution s ′ : α s α s α s x r x r → ( S n mv + ℓ t ) E ′ sia − ja E ′ ℓt − ia + ja E ′ nmv E nmv E ℓt − ia + ja ( E ia − jas ( S E ′′ s E s ) r ( S E ′′ E s ) r ′ ( S E ′′ s E ) r ′′ × (cid:16) G ij | m + ℓ | kl ( ℓ ′ − ℓ,n a − ℓ,n ′ ,n ′ ,n ′ ) (cid:17) E ′′ s r ′′ F ′′ ξm F ′′ nb − ℓ + ib F ′′ ℓt − ib + jb F na − ℓ + ia E r ′ E r ′ s , (5.6)with n ′ = n a + 2 n b − ℓ + 2 i a − j a + 2 j b + ℓ t + r ′ + 2 r ′′ − r ′′ + 8 ξ m ,n ′ = n b − ℓ + j b + ℓ t − r ′ + r ′′ + 6 ξ m ,n ′ = ℓ − n b − j b − ℓ t − r ′ − r ′′ − ξ m . (5.7)Owing to the fundamental property (5.5), the new quantity S BA = g BA G ij | m + ℓ | kl (0 , , , , − G ij | m + ℓ | kl (0 , , , , A ¯ η B − ¯ η A ( G ij | m + ℓ | kl (0 , , , , ) B + ( G ij | m + ℓ | kl (0 , , , , ) BA , (5.8)originates directly from the free A B A = g BA − ¯ η A ¯ η B − x ¯ η A ¯ η B + x ¯ η A ¯ η B , inside the conformal substitution.Upon combining the remaining contractions outside the conformal substitution, we arrive at34he following form for the four-point conformal blocks:( G ij | m + ℓ | kl ( a | b ] ) { aA }{ bB }{ cC }{ dD } = X t A t ( d, ℓ ) X j a ,j b ≥ (cid:18) i a j a (cid:19)(cid:18) i b j b (cid:19) ( − ℓ t ) i a − j a ( − ℓ t ) i b − j b ( − ℓ + ℓ t ) j a ( − ℓ + ℓ t ) j b ( − ℓ ) i a ( − ℓ ) i b × X r, r ′ , r ′′ ≥ r +2 r ′ + r ′ + r ′ = j a r +2 r ′′ + r ′′ + r ′′ = j b r ′ + r ′ + r ′ = r ′′ + r ′′ + r ′′ ( − ℓ − ℓ ′ − i a + r ′ + r ′ ( − r ′ + r ′′ ℓ ′ !( d ′ / − ℓ ′ C ( d + d t ,ℓ − ℓ t ) j a ,j b ( r, r ′ , r ′′ ) × (cid:16) C ( d ′ / − ℓ ′ ( X ) (cid:17) s ij | m + ℓ | kl ( a | b ) ( t,j a ,j b ,r, r ′ , r ′′ ) , (5.9)with the substitutions s ij | m + ℓ | kl ( a | b ) ( t, j a , j b , r, r ′ , r ′′ ) : α s α s α s x r x r → ( − ξ m ( a t ,m + i a ij ) { Ee }{ F }{ aA }{ bB } ( g E s E s ) r ′ ( S E ′′ s E s ) r [( S · ¯ η ) E s ] r ′ × (cid:16) G ij | m + ℓ | kl ( ℓ ′ − ℓ +2 r ′ ,n a − ℓ,n ′ ,n ′ ,n ′ ) (cid:17) E ′′ s r ′′ F ′′ ξm F ′′ nb − ℓ + ib F ′′ ℓt − ib + jb F na − ℓ + ia E r ′ s (¯ η E ) ℓ t − i a + j a × (Γ F ′′ ¯ η · Γ S n mv + ℓ t ˆ Q N m + ℓ t e | t Γ F ′′ ) E ′′ s ib − jb E ′′ ℓt − ib + jb E ′′ nmv e ′′ eE nmv E ℓt − ia + ja E ia − jas × [(¯ η · S ) E ′′ s ] r ′′ ( g E ′′ s E ′′ s ) r ′′ ( b t kl,m + i b ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } ( A E ′′ F ′′ ) ℓ t − i b + j b , (5.10)with the quantity S AB defined in (5.8), and the various parameters given in (5.7) and (5.1). Weremind the reader that the E s -indices and the E ′′ s -indices are symmetrized separately in (5.10).The form (5.9) for the four-point conformal blocks in the mixed basis features linear combina-tions of Gegenbauer polynomials in the variable X (5.2), each with a specific substitution given by(5.10). Like for the rotation matrices, all the ℓ dependence has been taken into account, and wesimply need to work with the ℓ -independent part of the projection operators and tensor structuresto generate the complete infinite tower of associated conformal blocks. Once the substitutionsare implemented, the blocks are expressed in terms of the ¯ I -functions (2.26) which are tensorialgeneralizations of the Exton G -function, as evident from (2.28) and (2.29).For each value of t appearing in the decomposition (3.3) and every j a and j b arising in thedouble expansion of the four-point conformal blocks (5.9), there are associated partitions whichcorrespond to the allowed Gegenbauer polynomials. Using the diagrammatic notation introducedabove, we may represent these partitions by diagrams, which are distinguished by their valuesof r ′ and r ′′ . It follows that the four-point conformal blocks can be represented by a set of(nonperturbative Feynman-like) diagrams, where each diagram corresponds to a set of Gegenbauerpolynomials, with their associated substitutions (5.10).Finally, we remark that the special parts of the tensor structures may sometimes be simplified.For example, except for the simplification A EF → − ¯ η E ¯ η F that we mentioned above, we can make35he substitutions A E A → g EA , A E B → g EB , A F A → g FA , A F B → g FB , A CE → g CE , A DE → g DE , A CF → g CF , A DF → g DF , due to the contractions of these metrics with the external half-projectors in (2.32). G The final substitutions (5.10) necessitate the multiplication of several G ’s together, according to(5.5), which we repeat here G ij | m + ℓ | kl ( n ,n ,n ,n ,n ) A n G ij | m + ℓ | kl ( m ,m ,m ,m ,m ) B m = G ij | m + ℓ | kl ( n + m ,n + m ,n + m ,n + m ,n + m ) A n B m , in addition to contractions with known tensorial objects. In fact, the contiguous relations (2.30)translate directly to the G ’s as g · G ij | m + ℓ | kl ( n ,n ,n ,n ,n ) = 0 , ¯ η · G ij | m + ℓ | kl ( n ,n ,n ,n ,n ) = G ij | m + ℓ | kl ( n ,n ,n − ,n ,n ) , ¯ η · G ij | m + ℓ | kl ( n ,n ,n ,n ,n ) = G ij | m + ℓ | kl ( n +2 ,n ,n ,n ,n ) , ¯ η · G ij | m + ℓ | kl ( n ,n ,n ,n ,n ) = G ij | m + ℓ | kl ( n ,n ,n − ,n − ,n ) , ¯ η · G ij | m + ℓ | kl ( n ,n ,n ,n ,n ) = G ij | m + ℓ | kl ( n ,n ,n − ,n ,n +2) . (5.11)Moreover, the quantity (5.8) often appears contracted in some specific ways that we expound herefor completeness. In particular,(¯ η · S ) B = ¯ η B G ij | m + ℓ | kl (0 , , , , − G ij | m + ℓ | kl (2 , , , , ¯ η B − x − ( G ij | m + ℓ | kl (0 , , , , ) B + ( G ij | m + ℓ | kl (2 , , , , ) B , ( S · ¯ η ) A = ¯ η A G ij | m + ℓ | kl (0 , , , , − G ij | m + ℓ | kl (0 , , , , A − ¯ η A G ij | m + ℓ | kl (0 , , , , + G ij | m + ℓ | kl (0 , , , , A , ( S · A · S T ) AB = g AB G ij | m + ℓ | kl (0 , , , , − ¯ η A G ij | m + ℓ | kl (0 , , , , B − G ij | m + ℓ | kl (0 , , , , A ¯ η B . We note that the last identity can be directly shown to hold from the original A that givesrise to S . Indeed, one has A · A · A T = A , which matches the above result after substitution.
6. Summary of Results
In this section, we summarize the main results for the three- and four-point conformal blocksderived in the previous two sections. The solutions are given for infinite towers of exchanged36uasi-primary operators in irreducible representations N m + ℓ e , with the universal ℓ -dependentpart already processed.The tensor structures (3.1) that enter the results are decomposed into a universal ℓ -dependentpart and a special part. The universal ℓ -dependent parts have ℓ − i a or ℓ − i b e indices on theexchanged quasi-primary operators contracted with some (or all) of the n a or n b free indices onthe OPE differential operators. These have already been accounted for in the results. Meanwhile,the special parts, which arise from N m + i a e and N m + i b e , respectively, appear directly in theresults and must be contracted properly for any given case under consideration. The derivation in Section 4 leads to the following form for the three-point conformal blocks [see(4.5)]( G ij | m + ℓ ( a | ) { aA }{ bB }{ eE } = λ N m + ℓ ( a t ij,m + i a ) { aA }{ bB }{ e ′ E ′ }{ F } (¯ η · ΓΓ F ) e ′ e × X r ,r ,s ,s ,t ≥ ( − r + s ( − ℓ a + s − t n m + i a v ! r ! r ! s ! s ! t ! ( − n m + i a v ) r + r ( − ℓ a ) s + s ( − s ) t × ( − n + n m + i a v + ℓ a + r + r ) s + s ( − h − n − ℓ a ) ℓ a + s − t × ( − h − n − ℓ a + 1 − d/ s − t ( p − n − ℓ a + r + r + s + s + 2 − d ) ℓ a − s − s × X σ g E ′ σ (1) E σ (1) · · · g E ′ σ ( r E σ ( r ¯ η E ′ σ ( r · · · ¯ η E ′ σ ( r r ( g ( ZE ) s (¯ η Z ) s × ¯ I ( d +2 ℓ a ,h + ℓ a +2 r + r + s + t,n − ℓ a − r − r − s − s ; p − r )12 Z n − nm + iav − ℓa − r − r − s − s ) E nm + iav − r σ ( − ¯ η E ) ℓ a − s Here the symmetrized Z -indices belong to { E ′ σ ( r + r +1) , . . . , E ′ σ ( n m + iav ) , F n a − ℓ a +2 ξ m } , and the re-maining parameters are found in (4.2). From this result, it is relatively straightforward to deter-mine the rotation matrices. From the decomposition (3.3) and the proof laid out in Section 5, the four-point conformal blocksare given by [see (5.9) and (5.10)] 37 G ij | m + ℓ | kl ( a | b ] ) { aA }{ bB }{ cC }{ dD } = X t A t ( d, ℓ ) X j a ,j b ≥ (cid:18) i a j a (cid:19)(cid:18) i b j b (cid:19) ( − ℓ t ) i a − j a ( − ℓ t ) i b − j b ( − ℓ + ℓ t ) j a ( − ℓ + ℓ t ) j b ( − ℓ ) i a ( − ℓ ) i b × X r, r ′ , r ′′ ≥ r +2 r ′ + r ′ + r ′ = j a r +2 r ′′ + r ′′ + r ′′ = j b r ′ + r ′ + r ′ = r ′′ + r ′′ + r ′′ ( − ℓ − ℓ ′ − i a + r ′ + r ′ ( − r ′ + r ′′ ℓ ′ !( d ′ / − ℓ ′ C ( d + d t ,ℓ − ℓ t ) j a ,j b ( r, r ′ , r ′′ ) × (cid:16) C ( d ′ / − ℓ ′ ( X ) (cid:17) s ij | m + ℓ | kl ( a | b ) ( t,j a ,j b ,r, r ′ , r ′′ ) s ij | m + ℓ | kl ( a | b ) ( t, j a , j b , r, r ′ , r ′′ ) : α s α s α s x r x r → ( − ξ m ( a t ,m + i a ij ) { Ee }{ F }{ aA }{ bB } ( g E s E s ) r ′ ( S E ′′ s E s ) r [( S · ¯ η ) E s ] r ′ × (cid:16) G ij | m + ℓ | kl ( ℓ ′ − ℓ +2 r ′ ,n a − ℓ,n ′ ,n ′ ,n ′ ) (cid:17) E ′′ s r ′′ F ′′ ξm F ′′ nb − ℓ + ib F ′′ ℓt − ib + jb F na − ℓ + ia E r ′ s (¯ η E ) ℓ t − i a + j a × (Γ F ′′ ¯ η · Γ S n mv + ℓ t ˆ Q N m + ℓ t e | t Γ F ′′ ) E ′′ s ib − jb E ′′ ℓt − ib + jb E ′′ nmv e ′′ eE nmv E ℓt − ia + ja E ia − jas × [(¯ η · S ) E ′′ s ] r ′′ ( g E ′′ s E ′′ s ) r ′′ ( b t kl,m + i b ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } ( A E ′′ F ′′ ) ℓ t − i b + j b The conformal blocks are thus represented by linear combinations of the Gegenbauer polynomialsin the variable X (5.2), coupled with associated substitutions. Here j a and j b are the numbersof extracted indices from the shifted projection operators appearing in the decomposition (3.3),with the remaining extracted indices appearing in the special part of the projection operator.Moreover, the indices of summation r , r ′ , and r ′′ determine how the special indices are extractedfrom the shifted projection operators, as in (3.5). Finally, the quantity S (5.8) is built out of thequantity G (5.4), which encodes the action of the OPE differential operator. The latter satisfiessome interesting properties, listed in (5.5) and (5.11). We again stress here that the special E ′ s -and E ′′ s -indices are symmetrized independently.Having established the essential results necessary for computing arbitrary four-point conformalblocks, we next apply these results to a series of examples.
7. Examples
This section makes use of the results for the three- and four-point conformal blocks in the contextof simple examples, with external quasi-primary operators in scalar, vector, and fermion irre-38ucible representations. Known blocks are compared with previously computed blocks obtainedfrom the embedding space OPE formalism in [18], and new results are compared with the liter-ature when possible. Although quite straightforward, most steps in the computations are doneexplicitly for all examples to elucidate the methods developed in the previous sections. h SSSS i For the four-point correlation function of four scalars, the exchanged quasi-primary operators arein the ℓ e irreducible representation, with the projection operator simply given by ˆ P ℓ e = ˆ P ℓ e | d [see (A.1)]. Hence, N m = with n mv = 0, and the decomposition (3.3) is straightforward withonly one term with ( d , ℓ t =1 ) = (0 , A ( d, ℓ ) = 1, and ˆ Q | = 1. Moreover, there is just a singletensor structure for both OPEs and, following (3.1) and (3.2), the forms are( b =1 F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = [( A · ¯¯ η ) E ′′ ] ℓ → ( b =1 t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = ( A E ′′ F ′′ ) ℓ , ( a =1 F ij,m + ℓ ) { aA }{ bB }{ eE } = [( A · ¯ η ) E ] ℓ → ( a =1 t ,m + ℓij ) { Ee }{ F }{ aA }{ bB } = ( A EF ) ℓ , (7.1)implying that n a =1 = n b =1 = ℓ , i a =1 = i b =1 = 0 and a =1 t mij = b =1 t klm = 1. Hence, the specialparts of the tensor structures are all trivial, and no extraction of special indices is necessary.Therefore, the sums over t , j a =1 , j b =1 , r , r ′ and r ′′ all collapse to a single term so that theconformal blocks (5.9) are G ij | m + ℓ | kl (1 | = s | = ℓ !( d/ − ℓ (cid:16) C ( d/ − ℓ ( X ) (cid:17) s | , with the substitution (5.10) s | ≡ s ij | m + ℓ | kl (1 | (1 , , , , , ) : α s α s α s x r x r → G ij | m + ℓ | kl (0 , , , , . As expected, this is the desired result [18].In the same manner, the rotation matrix can be computed straightforwardly from the three-point conformal blocks (4.5). With n mv = 0, n a =1 = ℓ and i a =1 = 0, all the sums over r , r , s , s and t in (4.5) also collapse to only one term, leading straightforwardly to G ij | m + ℓ (1 | = λ ℓ e ( − ℓ ( − h ij,m + ℓ − ℓ/ ℓ (∆ m + ℓ − ℓ + 2 − d ) ℓ ρ ( d +2 ℓ,h ij,m + ℓ − ℓ/ m + ℓ + ℓ ) ( − ¯ η E ) ℓ . Hence, from (4.6) and using the notation (4.7) the rotation matrix is( R − ij,m + ℓ ) , = a =1 κ ij | m + ℓ (0 , , , , , , , , , (7.2)as already obtained in [18]. 39t is evident that the basic rules introduced here are quite efficient, generating the rotationmatrix and the conformal blocks effortlessly. As we will see in subsequent examples, their potentialis in full display for four-point correlation functions with external quasi-primary operators innontrivial irreducible representations of the Lorentz group. h SSSR i With two scalars SS , the only possible exchanged irreducible representations (in the s -channel)are the ℓ e representations with projection operators (A.1). Hence, in this case, we need to dealwith the same decomposition (3.3) as in the previous example with h SSSS i , and also the sametensor structure a =1 t ij,m + ℓ (7.1). Consequently, the rotation matrix transforming the blocks tothe pure three-point basis is the same as for h SSSS i and is given by (7.2). We may thereforerestrict attention to the SR part of the correlation function.Representation theory implies that the only possible irreducible representations R among theremaining defining representations ( i.e. antisymmetric tensors and fermions) for the last externalquasi-primary operator must either be e or e . Since they do not have the same number ofconformal blocks, we treat them separately below. h SSSV i In the case of the four-point correlation function of three scalars and one vector, there are two34-tensor structures in d >
3, given by b = 1 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = ( A · ¯¯ η ) D [( A · ¯¯ η ) E ′′ ] ℓ → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = A DF ′′ ( A E ′′ F ′′ ) ℓ ,b = 2 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = A DE ′′ [( A · ¯¯ η ) E ′′ ] ℓ − → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = A DE ′′ ( A E ′′ F ′′ ) ℓ − , which imply that b = 1 : n b = ℓ + 1 , i b = 0 , ( b t klm ) DF ′′ = A DF ′′ ,b = 2 : n b = ℓ − , i b = 1 , ( b t kl,m +1 ) DE ′′ = A DE ′′ . From these forms, it is apparent that no indices need to be extracted for the first tensor struc-ture, while for the second tensor structure, one E ′′ -index must be extracted. These corresponddiagrammatically to b = 1 : ˆ P ℓ e | d = , = 2 : ˆ P ℓ e | d = + . Through their associated partitions, these diagrams in turn directly give the conformal blocks interms of Gegenbauer polynomials, G ij | m + ℓ | kl (1 | = ℓ !( d/ − ℓ (cid:16) C ( d/ − ℓ ( X ) (cid:17) s | , G ij | m + ℓ | kl (1 | = − ( ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | + ( ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | , as in (5.9). Moreover, contracting the special parts of the tensor structures with the specialpart of the projection operator as in (5.10) with the help of the partitions obtained through thediagrams, the associated substitutions are s | : α s α s α s x r x r → G ij | m + ℓ | kl (0 , , , , − D ,s | : α s α s α s x r x r → (¯ η · S ) D G ij | m + ℓ | kl ( − , , − , , ,s | : α s α s α s x r x r → G ij | m + ℓ | kl (0 , , , , − D . We note here that we have replaced A DF ′′ → g DF ′′ and A DE ′′ → g DE ′′ , due to their contrac-tions with the half-projectors in (2.32). Moreover, the substitutions s | and s | are exactlythe same, although they do not originate from the same contractions. h SSS e i The tensor structure for a scalar, a two-index antisymmetric tensor, and exchanged ℓ e is b = 1 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = A D E ′′ ( A · ¯¯ η ) D [( A · ¯¯ η ) E ′′ ] ℓ − → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = A D E ′′ A D F ′′ ( A E ′′ F ′′ ) ℓ − , where the D -indices are antisymmetrized once contracted with the e half-projector. Clearly, wehave b = 1 : n b = ℓ, i b = 1 , ( b t kl,m +1 ) D D E ′′ F ′′ = A D E ′′ A D F ′′ , and we must extract one E ′′ -index. Hence, the partitions are labeled diagrammatically by b = 1 : ˆ P ℓ e | d = + , G ij | m + ℓ | kl (1 | = − ( ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | + ( ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | . Using the partitions associated with each diagram and contracting the special parts togetheraccording to (5.10), we obtain the associated substitutions s | : α s α s α s x r x r → (¯ η · S ) D G ij | m + ℓ | kl ( − , , , , − D ,s | : α s α s α s x r x r → G ij | m + ℓ | kl (0 , , , , − D D . Due to the antisymmetry of the D -indices (from their contraction with the half-projector) and thefact that G is totally symmetric in its indices, the second term in the conformal blocks vanishes,and we easily get a result that matches the one in [18]. h SRSR i and h SSRR i With the conformal bootstrap in mind, we need to determine not only the conformal blocks of h SRSR i four-point correlation functions, but also the blocks of h SSRR i (the s - and t -channels).For R a defining representation which is not a scalar, there are several different cases to consider.Here we proceed in some detail for the cases R = V and R = F , and leave the blocks for R inantisymmetric tensor representations to a forthcoming work.Once again, in the h SSRR i case, the exchanged quasi-primary operators are in the ℓ e ir-reducible representation, implying a decomposition (3.3) as in the h SSSS i example, and their12-tensor structures are given by (7.1). Hence, their rotation matrices are given by (7.2), and wecan focus only on the RR side of the computation with ℓ e exchange. h SV SV i and h SSV V i For h SV SV i , there are two possible infinite towers of exchanged quasi-primary operators, ℓ e and e + ℓ e . We first consider ℓ e exchange. In this case, the tensor structures are explicitly givenby b = 1 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = ( A · ¯¯ η ) D [( A · ¯¯ η ) E ′′ ] ℓ → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = A DF ′′ ( A E ′′ F ′′ ) ℓ ,b = 2 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = A DE ′′ [( A · ¯¯ η ) E ′′ ] ℓ − → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = A DE ′′ ( A E ′′ F ′′ ) ℓ − ,a = 1 : ( a F ij,m + ℓ ) { aA }{ bB }{ eE } = ( A · ¯ η ) B [( A · ¯ η ) E ] ℓ → ( a t ,m + ℓij ) { Ee }{ F }{ aA }{ bB } = A F B ( A EF ) ℓ ,a = 2 : ( a F ij,m + ℓ ) { aA }{ bB }{ eE } = A BE [( A · ¯ η ) E ] ℓ − ( a t ,m + ℓij ) { Ee }{ F }{ aA }{ bB } = A E B ( A EF ) ℓ − , so that we have b = 1 : n b = ℓ + 1 , i b = 0 , ( b t klm ) DF ′′ = A DF ′′ ,b = 2 : n b = ℓ − , i b = 1 , ( b t kl,m +1 ) DE ′′ = A DE ′′ ,a = 1 : n a = ℓ + 1 , i a = 0 , ( a t ijm ) FB = A F B ,a = 2 : n a = ℓ − , i a = 1 , ( a t ij,m +1 ) E B = A E B . With the aid of the diagrams, we can easily extract indices to find the contributions a = 1 , b = 1 : ˆ P ℓ e | d = ,a = 1 , b = 2 : ˆ P ℓ e | d = + ,a = 2 , b = 1 : ˆ P ℓ e | d = + ,a = 2 , b = 2 : ˆ P ℓ e | d = + + + 2 × + , which lead directly to four conformal blocks expressed in terms of Gegenbauer polynomials (5.9),namely G ij | m + ℓ | kl (1 | = ℓ !( d/ − ℓ (cid:16) C ( d/ − ℓ ( X ) (cid:17) s | , G ij | m + ℓ | kl (1 | = − ( ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | + ( ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | , G ij | m + ℓ | kl (2 | = − ( ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | + ( ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | , G ij | m + ℓ | kl (2 | = ( ℓ − ℓ ( d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | − ( ℓ − ℓ ( d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | − ( ℓ − ℓ ( d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | − ( ℓ − ℓ ( d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s |
43 ( ℓ − ℓ ( d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | + ( ℓ − ℓ ( d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | . Using the tensor structures and the extended partitions in (5.10), we find the substitutions s | : α s α s α s x r x r → G ij | m + ℓ | kl (0 , , , , − BD ,s | : α s α s α s x r x r → (¯ η · S ) D G ij | m + ℓ | kl ( − , , , , B ,s | : α s α s α s x r x r → G ij | m + ℓ | kl (0 , , , , − BD ,s | : α s α s α s x r x r → ( S · ¯ η ) B G ij | m + ℓ | kl ( − , − , , , − D ,s | : α s α s α s x r x r → G ij | m + ℓ | kl ( − , − , , , − BD ,s | : α s α s α s x r x r → ( S · ¯ η ) B (¯ η · S ) D G ij | m + ℓ | kl ( − , − , − , − , − ,s | : α s α s α s x r x r → ( S · ¯ η ) B G ij | m + ℓ | kl ( − , − , , , − D ,s | : α s α s α s x r x r → (¯ η · S ) D G ij | m + ℓ | kl ( − , − , , , B ,s | : α s α s α s x r x r → G ij | m + ℓ | kl ( − , − , , , − BD ,s | : α s α s α s x r x r → G ij | m + ℓ | kl ( − , − , , , − BD ,s | : α s α s α s x r x r → S BD G ij | m + ℓ | kl ( − , − , , , , by straightforward contraction.Before proceeding to consider the remaining infinite tower of exchanged quasi-primary opera-tors, we determine the rotation matrix ℓ e . Applying (4.5), it is straightforward to get( R − ij,m + ℓ ) , = κ ij | m + ℓ (0 , , , , , , , , + κ ij | m + ℓ (0 , , , , , , , , , ( R − ij,m + ℓ ) , = κ ij | m + ℓ (0 , , , , , , , , + κ ij | m + ℓ (0 , , , , , , , , , ( R − ij,m + ℓ ) , = − κ ij | m + ℓ (0 , , , , , , , , − κ ij | m + ℓ (0 , , , , , , , , − κ ij | m + ℓ (0 , , , , , , , , , ( R − ij,m + ℓ ) , = κ ij | m + ℓ (0 , , , , , , , , − κ ij | m + ℓ (0 , , , , , , , , − κ ij | m + ℓ (0 , , , , , , , , + κ ij | m + ℓ (1 , , , , , , , , , which agrees with [18] when ℓ = 1.The remaining infinite tower of exchanged quasi-primary operators corresponds to the irre-ducible representations e + ℓ e , with the projection operators given by (A.4) with m = 2. With E and E denoting the antisymmetric indices on the projection operator, the tensor structuresare b = 1 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = A DE ′′ ( A · ¯¯ η ) E ′′ [( A · ¯¯ η ) E ′′ ] ℓ → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = A DE ′′ A E ′′ F ′′ ( A E ′′ F ′′ ) ℓ ,a = 1 : ( a F ij,m + ℓ ) { aA }{ bB }{ eE } = A BE ( A · ¯ η ) E [( A · ¯ η ) E ] ℓ ( a t ,m + ℓij ) { Ee }{ F }{ aA }{ bB } = A E B A E F ( A EF ) ℓ so that b = 1 : n b = ℓ + 1 , i b = 0 , ( b t klm ) DE ′′ E ′′ F ′′ = A DE ′′ A E ′′ F ′′ ,a = 1 : n a = ℓ + 1 , i a = 0 , ( a t ijm ) E E FB = A E B A E F . It is evident that we do not need to extract any indices, and we may therefore bypass thediagrammatic notation altogether. From (A.4), we see that there are six different contributionsinvolved in the decomposition (3.3) of the projection operator. In consequence, we have sixcontributions to the conformal blocks (5.9), namely G ij | m + ℓ | kl (1 | = 2 ℓ !( ℓ + 2)( d/ ℓ (cid:16) C ( d/ ℓ ( X ) (cid:17) s | − ℓ !( ℓ + 2)( d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | − ℓ !( ℓ + 2)( d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | + 2 ℓ !( ℓ + d/ d + ℓ − ℓ + 2)( d + ℓ − d/ ℓ (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | − ℓ !( ℓ + d/ ℓ + 2)( d + ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | + ℓ !( ℓ + 2)( d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | , with the substitutions s | : α s α s α s x r x r → − ( S · A ) B [ D (¯ η · S · A ) F ′′ ] (cid:16) G ij | m + ℓ | kl (0 , , , , − (cid:17) F ′′ ,s | : α s α s α s x r x r → −
12 (
S · A ) B [ D (¯ η · S · A ) F ′′ ] (¯ η · S · A ) F ′′ (cid:16) G ij | m + ℓ | kl ( − , , , , − (cid:17) F ′′ ,s | : α s α s α s x r x r → ( S · A ) B [ D A F ′′ ] F ′′ (cid:16) G ij | m + ℓ | kl (1 , , , , − (cid:17) F ′′ −
12 (¯ η · S · A ) [ D A F ′′ ] F ′′ (cid:16) G ij | m + ℓ | kl ( − , , , , − (cid:17) F ′′ B ,s | : α s α s α s x r x r → ( S · A ) B [ D A F ′′ ] F ′′ (cid:16) G ij | m + ℓ | kl (1 , , , , − (cid:17) F ′′ −
12 (¯ η · S · A ) [ D A F ′′ ] F ′′ (cid:16) G ij | m + ℓ | kl ( − , , , , − (cid:17) F ′′ B ,s | : α s α s α s x r x r → ( S · A ) BF ′′ (¯ η · S · A ) [ D A F ′′ ] F ′′ (cid:16) G ij | m + ℓ | kl (0 , , , , − (cid:17) F ′′ −
12 (¯ η · S · A ) [ D A F ′′ ] F ′′ (¯ η · S · A ) F ′′ (cid:16) G ij | m + ℓ | kl ( − , , , , − (cid:17) F ′′ B ,s | : α s α s α s x r x r → S · A ) B [ D (¯ η · S · A ) F ′′ ] (cid:16) G ij | m + ℓ | kl (0 , , , , − (cid:17) F ′′ , A E F → − ¯ η E ¯ η F and performed the contractions using the con-tiguous relations (5.11). For example, we used¯ η · S · A · S T · ¯ η = − G ij | m + ℓ | kl (2 , , , , , ( S · A · S T · ¯ η ) B → − G ij | m + ℓ | kl (0 , , , , B , where the replacements are warranted by the contractions with the half-projectors in (2.32).To complete our analysis of h SV SV i , we need to also compute the rotation matrix for theexchanged quasi-primary operators in the e + ℓ e representation. Using (4.5) directly with thetensor structures yields( R − ij,m + ℓ ) , = 2 κ (0 , , , , , , , , + 2 κ (0 , , , , , , , , + κ (0 , , , , , , , , − κ (0 , , , , , , , , − κ (0 , , , , , , , , + κ (0 , , , , , , , , + 2 κ (0 , , , , , , , , + 12 κ (0 , , , , , , , , − κ (0 , , , , , , , , − κ (0 , , , , , , , , + 12 κ (0 , , , , , , , , + 13 κ (0 , , , , , , , , + 12 κ (0 , , , , , , , , − κ (0 , , , , , , , , − κ (0 , , , , , , , , + κ (0 , , , , , , , , + 13 κ (0 , , , , , , , , + 2 κ (0 , , , , , , , , + 14 κ (0 , , , , , , , , + 16 κ (0 , , , , , , , , + 13 κ (0 , , , , , , , , + κ (1 , , , , , , , , + 23 κ (1 , , , , , , , , + κ (1 , , , , , , , , , where we have taken into account the antisymmetry of the pair E , E .Proceeding further, we find that the case h SSV V i is much simpler, as there is only one infinitetower of exchanged quasi-primary operators involved, namely ℓ e . The tensor structures are givenby b = 1 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = ( A · ¯¯ η ) C ( A · ¯¯ η ) D [( A · ¯¯ η ) E ′′ ] ℓ → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = A CF ′′ A DF ′′ ( A E ′′ F ′′ ) ℓ ,b = 2 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = A CD [( A · ¯¯ η ) E ′′ ] ℓ → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = A CD ( A E ′′ F ′′ ) ℓ ,b = 3 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = A CE ′′ ( A · ¯¯ η ) D [( A · ¯¯ η ) E ′′ ] ℓ − → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = A CE ′′ A DF ′′ ( A E ′′ F ′′ ) ℓ − ,b = 4 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = ( A · ¯¯ η ) C A DE ′′ [( A · ¯¯ η ) E ′′ ] ℓ − → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = A CF ′′ A DE ′′ ( A E ′′ F ′′ ) ℓ − b = 5 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = A CE ′′ A DE ′′ [( A · ¯¯ η ) E ′′ ] ℓ − → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = A CE ′′ A DE ′′ ( A E ′′ F ′′ ) ℓ − , which imply b = 1 : n b = ℓ + 2 , i b = 0 , ( b t klm ) CDF ′′ = A CF ′′ A DF ′′ , = 2 : n b = ℓ, i b = 0 , ( b t klm ) CDF ′′ = A CD ,b = 3 : n b = ℓ, i b = 1 , ( b t kl,m +1 ) CDE ′′ F ′′ = A CE ′′ A DF ′′ ,b = 4 : n b = ℓ, i b = 1 , ( b t kl,m +1 ) CDE ′′ F ′′ = A CF ′′ A DE ′′ ,b = 5 : n b = ℓ − , i b = 2 , ( b t kl,m +2 ) CDE ′′ E ′′ = A CE ′′ A DE ′′ . Given these, we find that we therefore need to extract zero, zero, one, one, and two E ′′ -indices,respectively, which results in the diagrams b ∈ { , } : ˆ P ℓ e | d = ,b ∈ { , } : ˆ P ℓ e | d = + ,b = 5 : ˆ P ℓ e | d = + + + . The associated extended partitions allow us to straightforwardly write the conformal blocks as b ∈ { , } : G ij | m + ℓ | kl (1 | b ] = ℓ !( d/ − ℓ (cid:16) C ( d/ − ℓ ( X ) (cid:17) s | b ) ,b ∈ { , } : G ij | m + ℓ | kl (1 | b ] = − ( ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | b ) + ( ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | b ) ,b = 5 : G ij | m + ℓ | kl (1 | b ] = ( ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | b ) − ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | b ) + ( ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | b ) + ( ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | b ) , using (5.9). Moreover, we can easily extract the substitution rules (5.10) for each block. We findthese to be s | : α s α s α s x r x r → G ij | m + ℓ | kl (0 , , , , − CD ,s | : α s α s α s x r x r → g CD G ij | m + ℓ | kl (0 , , , , ,s | : α s α s α s x r x r → (¯ η · S ) C G ij | m + ℓ | kl ( − , , , , − D ,s | : α s α s α s x r x r → G ij | m + ℓ | kl (0 , , , , − CD ,s | : α s α s α s x r x r → (¯ η · S ) D G ij | m + ℓ | kl ( − , , , , − C , | : α s α s α s x r x r → G ij | m + ℓ | kl (0 , , , , − CD ,s | : α s α s α s x r x r → (¯ η · S ) C (¯ η · S ) D G ij | m + ℓ | kl ( − , , − , , ,s | : α s α s α s x r x r → (¯ η · S ) ( C G ij | m + ℓ | kl ( − , , , , − D ) ,s | : α s α s α s x r x r → G ij | m + ℓ | kl (0 , , , , − CD ,s | : α s α s α s x r x r → g CD G ij | m + ℓ | kl (0 , , , , . Again, here we replaced all metrics of the type A CF ′′ by g CF ′′ without loss of generality.Although written differently, we have checked that all the conformal blocks above match theones found in [18]. h SF SF i and h SSF F i We next consider some examples involving fermions, namely h SF SF i and h SSF F i . Technically,for fermionic representations we should in principle consider odd and even dimensions separately.However, a study of the tensor structures shows that the even-dimensional case corresponds to halfof the odd-dimensional case. We may therefore restrict attention to fermions in odd dimensions inour analysis, since the even-dimensional case may be straightforwardly derived from these results.We first analyze the h SF SF i conformal blocks. For these, the sole exchanged quasi-primaryoperators are in the e r + ℓ e irreducible representation, with projection operators (A.2). Thereare thus two terms in the sum over t , each with nontrivial special parts and shifted projectionoperators.Moreover, in our simple basis, the tensor structures are given by b = 1 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = ( C − ) de ′′ [( A · ¯¯ η ) E ′′ ] ℓ → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = ( C − ) de ′′ ( A E ′′ F ′′ ) ℓ ,b = 2 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = (¯¯ η · Γ C − ) de ′′ [( A · ¯¯ η ) E ′′ ] ℓ → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = (Γ F ′′ C − ) de ′′ ( A E ′′ F ′′ ) ℓ ,a = 1 : ( a F ij,m + ℓ ) { aA }{ bB }{ eE } = ( C − ) be [( A · ¯ η ) E ] ℓ → ( a t ,m + ℓij ) { Ee }{ F }{ aA }{ bB } = δ eb ( A EF ) ℓ ,a = 2 : ( a F ij,m + ℓ ) { aA }{ bB }{ eE } = (¯ η · Γ C − ) be [( A · ¯ η ) E ] ℓ → ( a t ,m + ℓij ) { Ee }{ F }{ aA }{ bB } = (Γ F ) eb ( A EF ) ℓ , so that b = 1 : n b = ℓ, i b = 0 , ( b t klm ) de ′′ = ( C − ) de ′′ ,b = 2 : n b = ℓ + 1 , i b = 0 , ( b t klm ) de ′′ F ′′ = (Γ F ′′ C − ) de ′′ , = 1 : n a = ℓ, i a = 0 , ( a t ijm ) eb = δ eb ,a = 2 : n a = ℓ + 1 , i a = 0 , ( a t ijm ) eFb = (Γ F ) eb . Since i a = i b = 0 for all tensor structures, we do not need to extract any indices from theprojection operators in this case. Diagramatically, we therefore have ∀ a, b : A ˆ Q e r | ˆ P ℓ e | d +2 + A ˆ Q e r + e | ˆ P ( ℓ − e | d +2 = A ˆ Q e r | × + A ˆ Q e r + e | × , and the four different conformal blocks have the same form when expressed in terms of Gegenbauerpolynomials, namely ∀ a, b : G ij | m + ℓ | kl ( a | b ] = ℓ !( d/ ℓ (cid:16) C ( d/ ℓ ( X ) (cid:17) s a | b ) + ℓ !2( d/ ℓ (cid:16) C ( d/ ℓ − ( X ) (cid:17) s a | b ) , with the explicit values for A t =1 , (A.2). This is however not the case for their associated substi-tutions, which are all different due to the tensor structures and the different values of n a and n b .They are s | : α s α s α s x r x r → − (Γ F ′′ ¯ η · Γ Γ F ′′ C − T Γ ) bd (cid:16) G ij | m + ℓ | kl (0 , , , , − (cid:17) F ′′ = − F ′′ C − T Γ ) bd (cid:16) G ij | m + ℓ | kl (0 , , , , − (cid:17) F ′′ ,s | : α s α s α s x r x r → − (Γ F ′′ ¯ η · Γ ¯ η · S · Γ Γ F ′′ Γ F ′′ C − T Γ ) bd (cid:16) G ij | m + ℓ | kl ( − , , , , − (cid:17) F ′′ ,s | : α s α s α s x r x r → − (Γ F ′′ ¯ η · Γ Γ F ′′ (Γ F ′′ C − ) T ) bd (cid:16) G ij | m + ℓ | kl (0 , , , , − (cid:17) F ′′ = − F ′′ (Γ F ′′ C − ) T ) bd (cid:16) G ij | m + ℓ | kl (0 , , , , − (cid:17) F ′′ ,s | : α s α s α s x r x r → − (Γ F ′′ ¯ η · Γ ¯ η · S · Γ Γ F ′′ Γ F ′′ (Γ F ′′ C − ) T ) bd (cid:16) G ij | m + ℓ | kl ( − , , , , − (cid:17) F ′′ ,s | : α s α s α s x r x r → − (Γ F Γ F ′′ ¯ η · Γ Γ F ′′ C − T Γ ) bd (cid:16) G ij | m + ℓ | kl (0 , , , , − (cid:17) F ′′ F = − F Γ F ′′ C − T Γ ) bd (cid:16) G ij | m + ℓ | kl (0 , , , , − (cid:17) F ′′ F ,s | : α s α s α s x r x r → − (Γ F Γ F ′′ ¯ η · Γ ¯ η · S · Γ Γ F ′′ Γ F ′′ C − T Γ ) bd (cid:16) G ij | m + ℓ | kl ( − , , , , − (cid:17) F ′′ F ,s | : α s α s α s x r x r → − (Γ F Γ F ′′ ¯ η · Γ Γ F ′′ (Γ F ′′ C − )) bd (cid:16) G ij | m + ℓ | kl (0 , , , , − (cid:17) F ′′ F = − F Γ F ′′ (Γ F ′′ C − ) T ) bd (cid:16) G ij | m + ℓ | kl (0 , , , , − (cid:17) F ′′ F ,s | : α s α s α s x r x r → − (Γ F Γ F ′′ ¯ η · Γ ¯ η · S · Γ Γ F ′′ Γ F ′′ (Γ F ′′ C − ) T ) bd (cid:16) G ij | m + ℓ | kl ( − , , , , − (cid:17) F ′′ F , F ′′ -indices, as they are fullysymmetrized. Moreover, for some of the substitutions, we have simplified the result using theusual Γ-matrix algebra.Finally, by straightforward substitution in (4.5) (with r = r = 0 but possible sums on s , s and t ) we obtain the three-point conformal blocks which then lead to the rotation matrix( R − ij,m + ℓ ) , = 0 , ( R − ij,m + ℓ ) , = ( − r +11 κ ij | m + ℓ (0 , , , , , , , , , ( R − ij,m + ℓ ) , = ( − r h κ ij | m + ℓ (0 , , , , , , , , − κ ij | m + ℓ (0 , , , , , , , , − κ ij | m + ℓ (0 , , , , , , , , + κ ij | m + ℓ (0 , , , , , , , , + d κ ij | m + ℓ (0 , , , , , , , , + κ ij | m + ℓ (0 , , , , , , , , + κ ij | m + ℓ (1 , , , , , , , , i , ( R − ij,m + ℓ ) , = 0 , as in (4.6). Here r is the rank of the Lorentz group, and we have again applied the Γ-matrixalgebra to simplify the rotation matrix.We next examine h SSF F i conformal blocks. Now, there are four different tensor structuresgiven by b = 1 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = ( C − ) cd [( A · ¯¯ η ) E ′′ ] ℓ → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = ( C − ) cd ( A E ′′ F ′′ ) ℓ ,b = 2 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = (¯¯ η · Γ C − ) cd [( A · ¯¯ η ) E ′′ ] ℓ → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = (Γ F ′′ C − ) cd ( A E ′′ F ′′ ) ℓ ,b = 3 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = (Γ E ′′ C − ) cd [( A · ¯¯ η ) E ′′ ] ℓ − → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = (Γ E ′′ C − ) cd ( A E ′′ F ′′ ) ℓ − b = 4 : ( b F kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ } = (¯¯ η · Γ Γ E ′′ C − ) cd [( A · ¯¯ η ) E ′′ ] ℓ − → ( b t kl,m + ℓ ) { cC }{ dD }{ e ′′ E ′′ }{ F ′′ } = (Γ F ′′ Γ E ′′ C − ) cd ( A E ′′ F ′′ ) ℓ − , so that b = 1 : n b = ℓ, i b = 0 , ( b t klm ) cd = ( C − ) cd ,b = 2 : n b = ℓ + 1 , i b = 0 , ( b t klm ) cdF ′′ = (Γ F ′′ C − ) cd ,b = 3 : n b = ℓ − , i b = 1 , ( b t kl,m +1 ) cdE ′′ = (Γ E ′′ C − ) cd ,b = 4 : n b = ℓ, i b = 1 , ( b t kl,m +1 ) cdE ′′ F ′′ = (Γ F ′′ Γ E ′′ C − ) cd . From the form of the first two tensor structures, it is apparent that no indices need to be extractedfrom the projection operators. Meanwhile, it is necessary to extract one E ′′ -index for the last two50ensor structures. Diagrammatically, we thus have b ∈ { , } : ˆ P ℓ e | d = ,b ∈ { , } : ˆ P ℓ e | d = + , which give the conformal blocks b ∈ { , } : G ij | m + ℓ | kl (1 | b ] = ℓ !( d/ − ℓ (cid:16) C ( d/ − ℓ ( X ) (cid:17) s | b ) ,b ∈ { , } : G ij | m + ℓ | kl (1 | b ] = − ( ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | b ) + ( ℓ − d/ ℓ − (cid:16) C ( d/ ℓ − ( X ) (cid:17) s | b ) , in terms of Gegenbauer polynomials (5.9). From the partitions associated with the diagrams, thesubstitutions (5.10) for each block are easily found to be s | : α s α s α s x r x r → ( C − ) cd G ij | m + ℓ | kl (0 , , , , ,s | : α s α s α s x r x r → (Γ F ′′ C − ) cd (cid:16) G ij | m + ℓ | kl (0 , , , , − (cid:17) F ′′ ,s | : α s α s α s x r x r → (¯ η · S · Γ C − ) cd G ij | m + ℓ | kl ( − , , − , , ,s | : α s α s α s x r x r → (Γ E ′′ C − ) cd (cid:16) G ij | m + ℓ | kl (0 , , , , − (cid:17) E ′′ ,s | : α s α s α s x r x r → (Γ F ′′ ¯ η · S · Γ C − ) cd (cid:16) G ij | m + ℓ | kl ( − , , , , − (cid:17) F ′′ ,s | : α s α s α s x r x r → (Γ F ′′ Γ E ′′ C − ) cd (cid:16) G ij | m + ℓ | kl (0 , , , , − (cid:17) E ′′ F ′′ = − C − ) cd G ij | m + ℓ | kl (0 , , , , , where we simplified whenever possible.As mentioned above, the even-dimensional case may be straightforwardly derived from theabove results. Indeed, a comparison of (A.2) and (A.3) shows little difference between the pro-jection operators in odd and even dimensions. Hence, for the even-dimensional case, the form ofthe conformal blocks in terms of Gegenbauer polynomials is equivalent to the odd-dimensionalone. The same statement does not apply to the tensor structures, however. Since there are twodifferent spinor representations in even dimensions, namely F and ˜ F , not all tensor structuresexist for each of the four possible pairs of fermions F F , F ˜ F , ˜ F F and ˜ F ˜ F . An inspection of thetensor structures shows that only half of these are possible for a given fermion pair (depending onthe rank and the exchanged fermion, either the half with an even number of Γ-matrices, or thehalf with an odd number of Γ-matrices, but not both). In this way, conformal blocks for fermionsin even dimensions can be seen as the appropriate half of the conformal blocks for fermions inodd dimensions. 51 . Conclusion In this work, we have established a set of highly efficient rules for determining all possible four-point conformal blocks in terms of fundamental group theoretic quantities, namely the projectionoperators of the external and exchanged quasi-primary operators. Once known, these projectionoperators imply two sets of tensor structures, one for the left and right OPE at the origin of theconformal blocks. With the knowledge of the projection operators and the tensor structures inhand, the rules introduced here allow us to seamlessly generate any conformal block of interest.For infinite towers of exchanged quasi-primary operators in irreducible representations N m + ℓ e , the results summarized in Section 6 lead to simple conformal blocks expressed in terms oflinear combinations of Gegenbauer polynomials in a specific variable X , coupled with associatedsubstitutions. The attractive simplicity of the blocks has its origin in the embedding space OPEformalism applied in the mixed basis of tensor structures.Although the blocks feature the simplest available form in the mixed basis, it is in our bestinterest to derive their corresponding form in the pure three-point basis, given our hope of ulti-mately implementing the conformal bootstrap program. Obtaining the conformal blocks in a purebasis, either the OPE or the three-point one, necessitates the computation of rotation matrices.These are obtained from the three-point correlation functions and are summarized in Section 6.In this work, we also introduce a convenient diagrammatic notation in order to easily determinethe appropriate linear combination of Gegenbauer polynomials appearing in a specific conformalblock. The rules are quite straightforward to apply. To illustrate their utility in action, we haveapplied them explicitly across a range of examples involving quasi-primary operators in scalar,vector and fermion irreducible representations.Our results make it transparent that all one requires in order to compute conformal blocksare the projection operators for the infinite towers of exchanged irreducible representations. Wehave conveniently expressed these group theoretic objects in terms of shifted projection operatorsfor ℓ e , i.e. projection operators with an unnatural spacetime dimension. As a consequence, theseshifted projection operators are not traceless. Nevertheless, the original projection operators forthe infinite tower of irreducible representations N m + ℓ e are most directly useful in the compu-tation of conformal blocks when cast in terms of these shifted projection operators. Moreover,the shifted projection operators satisfy several interesting properties that will be described in anupcoming work.A salient feature of the form of the blocks presented in this work is the ubiquitous presence ofthe Gegenbauer polynomials. This aspect is not surprising, as we expect Gegenbauer polynomialsto appear for any tower of conformal blocks with exchanged quasi-primary operators in N m + ℓ e .The existence of such a form raises the question: Is there another closed form expression thatwe may write down, which may effectively enable us to remove the multiple finite sums arisinghere and replace them by a smaller number of sums? Motivated by the well known closed form52xpressions for the ℓ e exchange blocks in d = 2 and d = 4 spacetime dimensions in scalar four-point functions in terms of specific linear combinations of products of hypergeometric functions,we may hope to determine a suitable generalization of such expressions for conformal blocks fornontrivial Lorentz representations in arbitrary spacetime dimensions.With the rules laid out in this paper, the next logical step is to study correlation functions ofthe energy-momentum tensors. Indeed, the energy-momentum tensor is the only nontrivial localquasi-primary operator present in all CFTs. However, even when all the appropriate projectionoperators are known, it is still necessary to understand conserved currents within the context ofthe present formalism. The analysis of conserved currents in the embedding space OPE formalismwill be the subject of a forthcoming publication. Acknowledgments
The work of JFF and VP is supported by NSERC and FRQNT. The work of WJM is supportedby the Chinese Scholarship Council and in part by NSERC and FRQNT.
A. Projection Operators
In this appendix, we list the projection operators needed to compute the infinite towers of confor-mal blocks for the examples presented in Section 7. The projection operators are first expressedin terms of the usual ℓ -dependent sums over traces with coefficients related to a i ( d, ℓ ) = ( − ℓ ) i i i !( − ℓ + 2 − d/ i , as in the shifted projection operators (3.4). They are then re-expressed in terms of finite ℓ -independent sums in these same shifted projection operators (3.4), as in (3.3). A.1. Projection Operator in the ℓ e Irreducible Representation
The projection operator in the ℓ e irreducible representation is well known. It is given by( ˆ P ℓ e ) µ ′ ··· µ ′ ℓ µ ℓ ··· µ = ⌊ ℓ/ ⌋ X i =0 a i ( d, ℓ ) g ( µ µ g ( µ ′ µ ′ · · · g µ i − µ i g µ ′ i − µ ′ i g µ ′ i +1 µ i +1 · · · g µ ′ ℓ ) µ ℓ ) . Since it is already written in terms of the shifted projection operators, for our purposes the ℓ e projection operator is simply ( ˆ P ℓ e ) µ ′ ··· µ ′ ℓ µ ℓ ··· µ = ( ˆ P ℓ e d ) µ ′ ℓ µ ℓ , (A.1)since the µ -indices (and also the µ ′ -indices) are symmetrized, with t ( d t , ℓ t ) A t ( d, ℓ ) ˆ Q | t ,
0) 1 1in the form (3.3). 53 .2. Projection Operator in the e r + ℓ e Irreducible Representation
For fermionic irreducible representation representations, the projection operators depend on thespacetime dimensions.In odd dimensions, there is only one fermionic representation, given by e r , and the associated e r + ℓ e projection operator is( ˆ P e r + ℓ e ) µ ′ ··· µ ′ ℓ α ′ αµ ℓ ··· µ = ⌊ ℓ/ ⌋ X i =0 a i ( d + 2 , ℓ ) g ( µ µ g ( µ ′ µ ′ · · · g µ i − µ i g µ ′ i − µ ′ i g µ ′ i +1 µ i +1 · · · g µ ′ ℓ ) µ ℓ ) δ α ′ α + ⌊ ( ℓ − / ⌋ X i =0 ℓa i ( d + 2 , ℓ − − ℓ + 1 − d/ g ( µ µ g ( µ ′ µ ′ · · · g µ i − µ i g µ ′ i − µ ′ i g µ ′ i +1 µ i +1 · · · g µ ′ ℓ − µ ℓ − ( γ µ ℓ ) γ µ ′ ℓ ) ) α ′ α . This result is obtained by combining allowed objects (among metrics, epsilon tensors and γ -matrices) in all possible ways consistent with the symmetry properties of the irreducible represen-tation, demanding tracelessness, and enforcing the projection property ˆ P = ˆ P .It can be rewritten in terms of the shifted projection operators as( ˆ P e r + ℓ e ) µ ′ ··· µ ′ ℓ α ′ αµ ℓ ··· µ = δ α ′ α ( ˆ P ℓ e d +2 ) µ ′ ℓ µ ℓ + ℓ − ℓ + 1 − d/
2) ( γ ( µ γ ( µ ′ ) α ′ α ( ˆ P ( ℓ − e d +2 ) µ ′ ℓ − ) µ ℓ − ) . (A.2)Here, the fact that the projection operator accompanying δ α ′ α is shifted implies that the firstterm is not traceless by itself, and a second term is therefore necessary. Thus there are two terms,given by t ( d t , ℓ t ) A t ( d, ℓ ) ˆ Q | t ,
0) 1 δ α ′ α , ℓ − ℓ +1 − d/ ( γ µ γ µ ′ ) α ′ α in the decomposition (3.3) of (A.2).In even dimensions, there are two irreducible fermionic representations, namely e r − and e r .However, their respective associated projection operators e r − + ℓ e and e r + ℓ e are straightfor-wardly obtained from the equivalent projection operator in odd dimensions (A.2). Indeed, theyare given by( ˆ P e r − + ℓ e ) µ ′ ··· µ ′ ℓ α ′ αµ ℓ ··· µ = δ α ′ α ( ˆ P ℓ e d +2 ) µ ′ ℓ µ ℓ + ℓ − ℓ + 1 − d/
2) ( γ ( µ ˜ γ ( µ ′ ) α ′ α ( ˆ P ( ℓ − e d +2 ) µ ′ ℓ − ) µ ℓ − ) , ( ˆ P e r + ℓ e ) µ ′ ··· µ ′ ℓ ˜ α ′ ˜ αµ ℓ ··· µ = δ ˜ α ′ ˜ α ( ˆ P ℓ e d +2 ) µ ′ ℓ µ ℓ + ℓ − ℓ + 1 − d/
2) (˜ γ ( µ γ ( µ ′ ) ˜ α ′ ˜ α ( ˆ P ( ℓ − e d +2 ) µ ′ ℓ − ) µ ℓ − ) , (A.3)and therefore have the same expansion according to (3.3) as the one in odd dimensions.54 .3. Projection Operators in e m + ℓ e Irreducible Representations
For the m -index antisymmetric irreducible representations of the type e m + ℓ e , where in termsof Dynkin indices one has more precisely e m ∈ { e , e , . . . , e r − , e r } in odd dimensions , e m ∈ { e , e , . . . , e r − , e r − + e r } in even dimensions , for m from 2 to r (represented by 2 e r Dynkin indices) in odd dimensions and m from 2 to r − e r − + e r Dynkin indices) in even dimensions, we find that the projectionoperators are( ˆ P e m + ℓ e ) µ ′ ··· µ ′ ℓ ν ′ ··· ν ′ m ν m ··· ν µ ℓ ··· µ = ⌊ ℓ/ ⌋ X i =0 a mi g ν ′ [ ν · · · g ν ′ m ν m ] g ( µ µ g ( µ ′ µ ′ · · · g µ i − µ i g µ ′ i − µ ′ i g µ ′ i +1 µ i +1 · · · g µ ′ ℓ ) µ ℓ ) + ⌊ ( ℓ − / ⌋ X i =0 b mi g [ ν ′ [ ν · · · g ν ′ m − ν m − g ( µ ′ ν m ] g ν ′ m ]( µ g µ µ g µ ′ µ ′ · · · g µ i µ i +1 g µ ′ i µ ′ i +1 g µ ′ i +2 µ i +2 · · · g µ ′ ℓ ) µ ℓ ) + ⌊ ( ℓ − / ⌋ X i =0 c mi g [ ν ′ [ ν · · · g ν ′ m − ν m − g ν m ]( µ g ν ′ m ]( µ ′ g µ µ g µ ′ µ ′ · · · g µ i µ i +1 g µ ′ i µ ′ i +1 g µ ′ i +2 µ i +2 · · · g µ ′ ℓ ) µ ℓ ) + ⌊ ( ℓ − / ⌋ X i =0 d mi g [ ν ′ [ ν · · · g ν ′ m − ν m − g ν m − ( µ g ν ′ m − ( µ ′ g µ ′ ν m ] g ν ′ m ] µ × g µ µ g µ ′ µ ′ · · · g µ i +1 µ i +2 g µ ′ i +1 µ ′ i +2 g µ ′ i +3 µ i +3 · · · g µ ′ ℓ ) µ ℓ ) + ⌊ ( ℓ − / ⌋ X i =0 e mi g [ ν ′ [ ν · · · g ν ′ m − ν m − (cid:16) g ν m − ( µ g ν ′ m − ν m ] g ν ′ m ] µ g ( µ ′ µ ′ + g ν ′ m − ( µ ′ g ν ′ m ] ν m − g µ ′ ν m ] g ( µ µ (cid:17) × g µ µ g µ ′ µ ′ · · · g µ i +1 µ i +2 g µ ′ i +1 µ ′ i +2 g µ ′ i +3 µ i +3 · · · g µ ′ ℓ ) µ ℓ ) , with a mi = mℓ + m a i ( d + 2 , ℓ ) , b mi = ( ℓ − i ) a mi ,c mi = ( ℓ − i )[( d + ℓ − m ) i + ( m + 1) d/ m ( ℓ − − ℓ + 1 − d/ i )( d + ℓ − m ) a mi ,d mi = − m − i + 1)( − ℓ − d/ d + ℓ − m a mi +1 , e mi = − i + 1) a mi +1 . Here, the µ ℓ indices are the ℓ e symmetrized indices while the ν m indices are the e m antisym-metrized indices. The case e m = e was already found in [11]. P e m + ℓ e ) µ ′ ··· µ ′ ℓ ν ′ ··· ν ′ m ν m ··· ν µ ℓ ··· µ = mℓ + m g ν ′ [ ν · · · g ν ′ m ν m ] ( ˆ P ℓ e d +2 ) µ ′ ℓ µ ℓ + mℓℓ + m g [ ν ′ [ ν · · · g ν ′ m − ν m − g ( µ ′ ν m ] g ν ′ m ]( µ ( ˆ P ( ℓ − e d +4 ) µ ′ ℓ − ) µ ℓ − ) + mℓℓ + m g [ ν ′ [ ν · · · g ν ′ m − ν m − g ν m ]( µ g ν ′ m ]( µ ′ × (cid:20) ( ˆ P ( ℓ − e d +4 ) µ ′ ℓ − ) µ ℓ − ) − ( − ℓ − d/ d + ℓ − − ℓ + 1 − d/ d + ℓ − m ) ( ˆ P ( ℓ − e d +2 ) µ ′ ℓ − ) µ ℓ − ) (cid:21) − m ( m − ℓ ( ℓ − − ℓ − d/ ℓ + m )( − ℓ + 1 − d/ d + ℓ − m ) g [ ν ′ [ ν · · · g ν ′ m − ν m − × g ν m − ( µ g ν ′ m − ( µ ′ g µ ′ ν m ] g ν ′ m ] µ ( ˆ P ( ℓ − e d +4 ) µ ′ ℓ − ) µ ℓ − ) − mℓ ( ℓ − ℓ + m )( − ℓ + 1 − d/ g [ ν ′ [ ν · · · g ν ′ m − ν m − × (cid:16) g ν m − ( µ g ν ′ m − ν m ] g ν ′ m ] µ g ( µ ′ µ ′ + g ν ′ m − ( µ ′ g ν ′ m ] ν m − g µ ′ ν m ] g ( µµ (cid:17) ( ˆ P ( ℓ − e d +4 ) µ ′ ℓ − ) µ ℓ − ) , (A.4)which corresponds to t ( d t , ℓ t ) A t ( d, ℓ ) ˆ Q | t , mℓ + m g ν ′ [ ν · · · g ν ′ m ν m ] , mℓℓ + m g [ ν ′ [ ν · · · g ν ′ m − ν m − g µ ′ ν m ] g ν ′ m ] µ , mℓℓ + m g [ ν ′ [ ν · · · g ν ′ m − ν m − g ν m ] µ g ν ′ m ] µ ′ , − mℓ ( − ℓ − d/ d + ℓ − ℓ + m )( − ℓ +1 − d/ d + ℓ − m ) g [ ν ′ [ ν · · · g ν ′ m − ν m − g ν m ] µ g ν ′ m ] µ ′ , − m ( m − ℓ ( ℓ − − ℓ − d/ ℓ + m )( − ℓ +1 − d/ d + ℓ − m ) g [ ν ′ [ ν · · · g ν ′ m − ν m − g ν m − µ g ν ′ m − µ ′ g µ ′ ν m ] g ν ′ m ] µ , − mℓ ( ℓ − ℓ + m )( − ℓ +1 − d/ g [ ν ′ ν ··· g ν ′ m − νm − × (cid:18) g νm − µ g ν ′ m − νm ] g ν ′ m ] µ g µ ′ µ ′ + g ν ′ m − µ ′ g ν ′ m ] νm − g µ ′ νm ] g µµ (cid:19) in the decomposition (3.3).The projection operators (A.4) can be obtained directly by combining the allowed objects inthe most general way satisfying the symmetry properties of the irreducible representations, leadingto the five terms above (symmetry under the exchange of the primed and unprimed indices impliestwo contributions to the last term). 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