PPreprint typeset in JHEP style - HYPER VERSION
In search of conformal theories
Abhijit Gadde
School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA
Abstract:
The conformal crossing equation puts very stringent constraints on the conformaldata. We formulate it in way that makes the conformal symmetry more transparent. Thisallows for generalization of the crossing equation to arbitrary Lie group G . Using the cross-ing equation for SU (2) as a toy model, we find infinitely many solutions to the G -crossingequation. In particular, when G is specialized to the conformal group SO ( d + 1 , a r X i v : . [ h e p - t h ] F e b ontents
1. Introduction 12. Conformal representation theory and the crossing equation 4
3. Solutions to generalized crossing equation 114. Outlook 15A. The SU (2) Racah coefficient 17
1. Introduction A d -dimensional euclidean conformal field theory is symmetric under the group SO ( d + 1 , S of conformal representations r = [∆ , (cid:96) ] where ∆ is the conformal dimension and (cid:96) = ( (cid:96) , . . . , (cid:96) (cid:98) d (cid:99) ) is the spin and thestructure constants C r ,r ,r . The structure constant is a real valued function on sym ⊗ S .The spectrum is constrained by the “unitarity bound”:∆ ∈ (cid:60) , ∆ ≥ d/ − (cid:96) = (0 , . . . , , ∆ ≥ d/ (cid:96) (cid:48) for (cid:96) = (0 , . . . , , (cid:96) (cid:48) ) , . . . (1.1)We have put the word unitarity in quotation marks because it does not refer to the unitaryrepresentations of the euclidean conformal group but rather to the unitary representations ofthe Lorentzian conformal group SO ( d,
2) analytically continued to those of SO ( d +1 , (cid:88) r C r ,r ,r C r ,r ,r F r ,r ,r ,r r ( u, v ) = (cid:88) r (cid:48) C r ,r ,r (cid:48) C r ,r ,r (cid:48) F r ,r ,r ,r r (cid:48) ( v, u ) . (1.2)where F are the so called conformal partial waves. The equation is so constraining, it isgenerally believed to be sufficient. The abstract approach to conformal field theory involvessolving the crossing equation. This approach, appropriately called the conformal bootstrap,– 1 –as pioneered in [1, 2] and recently revived in [3]. Since then there has been a lot of successtowards partially solving this equation by analytical as well as numerical methods, see [4] foran example of the state of the art results and an extensive list of references.Our approach is to formulate the crossing equation such that it depends only on thesymmetry group SO ( d + 1 , u, v )which are, in a sense, extraneous. In order to achieve the desired form, one would like touse orthogonality of conformal partial waves. So when integrated over ( u, v ) with respect toappropriate measure one side of the equation gives Kronecker delta. (cid:88) r C r ,r ,r C r ,r ,r W r ,r ,r ,r ; r,r (cid:48) = C r ,r ,r (cid:48) C r ,r ,r (cid:48) (1.3)In this equation only representations of the conformal group appear. If one identifies the grouptheoretic properties of W then the equation can be generalized to any group. Unfortunately,the conformal partial waves for physical representations do not obey an orthogonality relation.It is well known that the conformal partial waves are solutions to a differential equations [5].Such solutions enjoy orthogonality if the differential operator in question is hermitian. Thisis not the case for physical conformal representations. But for conformal representations R = [∆ , (cid:96) ] with ∆ = d/ ic, c ∈ R the differential operator does become hermitian andassociated conformal partial waves obey orthogonality. These representations are called theprincipal series representations and play an important role in the harmonic analysis on thegroup. In fact, the orthogonality of conformal partial waves of the principal series is closelyrelated to the harmonic analysis.How to hear the shape of a group? This is the question that is at the heart of harmonicanalysis on G . To be precise, the space of normalizable functions on a topological group G forms a representation under the left action of G called the regular representation. Howdoes it decompose into irreducible representations? For compact groups, the complete basisof such representations is a discrete set consisting of all irreducible representations while fornon-compact groups, as it turns out, the basis consists of representations belonging to acontinuous set and in some cases consists of an additional discrete set. The usefulness of thisconcept is immediately apparent in case of the simplest non-compact group R . The harmonicanalysis on R is simply the Fourier transform, clearly a useful notion. In the context ofconformal theories, it is useful to decompose the 4-point functions into a complete basis ofconformal partial waves. The underlying problem is actually that of the harmonic analysison the conformal group.The irreducible representations that appear in the decomposition of the regular repre-sentation of G are known as tempered. By an old theorem of Harish-Chandra [6], they aredescribed as induced unitary representations of certain distinguished subgroups. We willidentify these representations in case of the euclidean conformal group SO ( d + 1 , d they fall in a continuous set known as the principal series while for odd d they consist ofthe so called discrete series in addition to the principal one.– 2 – Principal series representations: [∆ , (cid:96) ] with ∆ = d + ic, c ∈ (cid:60) . • Discrete series representations (only for odd d ): ∆ = d + Z + The physical representations i.e. the representations satisfying the unitarity bound (1.1) do not appear in the decomposition of the regular representation. There are other unitaryrepresentations beside the tempered ones e.g. the complementary series: ∆ ∈ (0 , d ) but theywill not play any role in our discussion. In the rest of the paper we will only consider the caseof conformal theory in even dimensions, so the discrete series will also not appear in the restof the paper. However, the discussion can be generalized to odd dimensions by adding thediscrete series to various formulas. We denote the tempered representations by capital letter R to distinguish them from the physical representations r .We study the Clebsch-Gordan decomposition of the tensor product of two temperedrepresentations into irreducible ones. Only the tempered representations appear in the de-composition. The dynamics must be consistent with kinematics. In particular, the 3-pointfunction is non-zero only if it is allowed by the fusion rules. The associativity of 3-pointfunctions results in a crossing equation very similar to (1.2) except that the conformal repre-sentations appearing are not physical but rather are tempered. Correspondingly the sum isreplaced by an integral over the principal series with appropriate measure (along with sumover spins and, in the case of odd d , sum over the discrete series). This new “euclidean” cross-ing equation can be rightly thought to define a unitary euclidean conformal theory . Clearly,these theories are different from the physical conformal theories that are unitary in Lorentziansignature. Nevertheless, it seems plausible to generate the solution to the physical crossingequation (1.2) given an analytic solution to the euclidean crossing equation by way of residueintegration. This is achieved by deforming the contour of integration along the principal seriesrepresentations: ∆ = d + ic , and picking up poles on the positive real axis corresponding tophysical representations. We will elaborate on this later in the paper.The advantage of working with tempered representations is that one can now use theorthogonality of the conformal partial waves to write the crossing equation as (1.3) with, bynow familiar, substitution of the sum by principal series integral. The object W appearing inthe equation is nothing but the Racah coefficient of the euclidean conformal group SO ( d +1 , G . We will find infinitelymany analytical solutions to this generalized G -crossing equation. In this effort the crossingequation for the simplest nontrivial Lie group SU (2), will serve as our main guide.The rest of the paper is organized in the following way. In section 2, we will review therelevant aspects of the representation theory of SO ( d + 1 , G . In section 3 we will find the solutions of the G -crossing Also known as the 6 j -symbol when normalized more symmetrically. – 3 –quation. One of the solutions to the SU (2)-crossing equation is already known in the litera-ture. It is obtained as a result of the associativity of the so called “symplecton polynomials”.In the modern language it is understood as the associativity of the operator ring of a certaintopological quantum mechanical system with SU (2) action on the phase space. Inspired bythe underlying algebra we will construct infinitely many solutions for the general crossingequation in terms of the Racah coefficient itself. The proposed solution satisfies the crossingequation thanks to the Biedenharn-Elliot identity also known as the pentagon identity of theRacah coefficient. The solutions are in one to one correspondence with representations of G .We end with a discussion of some of the new directions opened as a result of this work. AsRacah coefficient plays a prominent role in formulation of the crossing equation as well as inits solutions, we have added an appendix containing its definition and properties.
2. Conformal representation theory and the crossing equation
This section is essentially a quick review of conformal representation theory based on theexcellent reference [7]. A discussion of conformal theory usually begins with locality, inparticular with the definition of local operators defined in an ambient d -dimensional space.For our purposes, however, it is more useful to assign primary importance to the symmetrygroup. In the case of d -dimensional euclidean theory, the symmetry group is SO ( d + 1 , SO ( d + 1 ,
1) is written uniquely as g = kan where k is an element ofthe maximal compact subgroup SO ( d + 1), a is scale transformation and n is a specialconformal transformation. For general semisimple groups, this decomposition is known asthe Iwasawa decomposition and is expressed as G = KAN . Thanks to Harish-Chandra [6],it is known that the irreducible representations appearing in the harmonic decomposition arethe unitary ones induced by the subgroup
M AN where M is the commutant of A inside K andthat the inducing representation transforms trivially under N . In the case of the conformalgroup M = SO ( d ). Such a representation is a function on the coset G/M AN labeled by itstransformation under A i.e. the conformal dimension ∆ and under M i.e. the spin (cid:96) . The coset G/M AN is nothing but the familiar conformal compactification of d -dimensional euclideanspace. This is how the coordinate space emerges from abstract symmetry considerations.The above discussion can be phrased equivalently and perhaps in a more familiar fashionby starting from the euclidean space and considering representations of the little group M AN that are invariant under N . The little group is the group of transformations that fixes a point(where a local operator is supported). The advantage of the more formal approach is thatit can be generalized to other non-compact groups. As we seek to put the conformal theoryin a group theoretic framework, this is helpful for applications that we have in mind. Tosummarize, the representations of the conformal group appearing in the harmonic analysisare vectors on R d labeled by their representations [∆ , (cid:96) ] under the little group.– 4 – .1 Unitarity and completeness From the point of view of harmonic analysis, we are interested in the induced representationsof
M AN that are unitary. Before discussing unitarity let us observe that an invariant innerproduct can be defined between a representation [∆ , (cid:96) ] and [ d − ∆ , (cid:96) ]. Letting f and f becertain vectors from these representations respectively, (cid:104) f | f (cid:105) = (cid:90) dx f ( x ) · f ( x ) . (2.1)Here · stands for inner product in the finite dimensional representation of SO ( d ). It is notdifficult to verify that the inner product defined above is indeed invariant [7]. For example,under scale transformation x → λ − x : dx d → λ − d dx d , f → λ ∆ f , f → λ d − ∆ f whichkeeps the integral invariant. A similar argument works for showing invariance under specialconformal transformation. The inner product defines a map from the representation R ≡ [∆ , (cid:96) ] to R . This allows us to interpret the representation ˜ R ≡ [ d − ∆ , (cid:96) ] as the dual ofrepresentation R . In conformal field theory literature the representation ˜ R is also knownas the shadow of representation R . Note that for a given representation, the shadow canbe constructed through convolution. As an example, if f ( x ) is a scalar representation withconformal dimension ∆ then˜ f ( x (cid:48) ) = N ∆ (cid:90) d d x | x − x (cid:48) | − d ) f ( x ) , N ∆ = π d Γ(∆ − d )Γ( d − ∆)Γ(∆)Γ( d − ∆) . (2.2)The normalization N ∆ is fixed by requiring the 2-point function of f ( x ) as well as ˜ f ( x ) to beunity: (cid:104) f ( x ) f ( x (cid:48) ) (cid:105) = | x − x (cid:48) | − , (cid:104) ˜ f ( x ) ˜ f ( x (cid:48) ) (cid:105) = | x − x (cid:48) | − d ) . (2.3)To identify unitary representations we simply need notice that for ∆ = d + ic, c ∈ R , thedual of the representation [∆ , (cid:96) ] is its complex conjugate. Such representations form the socalled principal series and play an important role in the harmonic analysis. In fact, for even d ,the principal series representations are the only representations that appear in the harmonicanalysis. For odd d , in addition to the principal series representations, the so called discreteseries representations also appear in the harmonic analysis. This is due to the existence of acompact Cartan subgroup of the conformal group [6]. In this paper we will focus on the caseof even d . Most of our results can be generalized to odd d with minor modifications. On theother hand, the familiar physical representations with real ∆ satisfying the unitarity boundare not very natural from the point of the unitarity as defined above. As remarked earlier,they are analytical continuation of the unitary representations of the Lorentzian conformalgroup SO ( d, R . A repre-sentation of the translational group is given by the momentum eigenstate e πipx where p is acomplex number. Given an normalizable function f ( x ), it can be decomposed into translation– 5 –igenstates with coefficients ˆ f ( p ). We know that the Fourier transform ˆ f ( p ) is supported onlyon the real axis. In other words, the representations with real p form a complete basis forthe space of normalizable functions on R . This completeness is sometimes expressed as, f (0) = (cid:90) (cid:61) p =0 dp ˆ f ( p ) , where ˆ f ( p ) ≡ (cid:90) ∞−∞ dx f ( x ) e πipx . (2.4)Harmonic analysis is essentially a Fourier transform on the group manifold. As discussedabove, a normalizable function on SO ( d +1 ,
1) for even d can be decomposed into the principalseries representations. Let R = [ d + ic, (cid:96) ] be a principal series representation. It plays therole of the Fourier basis. The above completeness relation is generalized to, f (1) = (cid:90) dR Tr[ ˆ f ( R )] , where ˆ f ( R ) ≡ (cid:90) dg f ( g ) R ( g ) , (cid:90) dR ≡ (cid:88) (cid:96) (cid:90) d + i ∞ d − i ∞ ρ (cid:96) (∆) d ∆ (2.5)Here dg is the Haar measure on the group. The sum (cid:80) (cid:96) is over all the spins. The function ρ (cid:96) (∆) is known as the Plancherel weight and is known to be equal to the normalizationconstant in the integration kernel relating representation [∆ , (cid:96) ] to its shadow [7, 8]. Forscalars, it is equal to N ∆ in equation (2.2). For odd dimensions the integral over the principalseries is supplemented with a sum over the discrete series. In odd dimensions, the Plancherelweight has poles, the discrete series of representations lives at those points and essentiallyserves to cancel the “fake” poles coming from the measure. Conformal theories are usually formulated in terms of the operator product expansion. Theproduct of local operators at two distinct points is presented in terms of a single local operatorinsertion. O ( x ) O ( x ) = (cid:88) O K O ,O ,O O ( x ) + desc . (2.6)The operators are normalized using their two point function. The tower of descendantsdenoted as desc. is completely fixed by symmetries. The constant K is a dynamical coefficientknown as the three point function coefficient or the structure constant. Apart from thespectrum, K is the only dynamical data defining the conformal theory. It is constrainedby the crossing equation (1.2) which follows from the associativity of the operator productexpansion. Each local operator is a representation of the conformal group. It is then naturalto look at this expansion from the point of view of representation theory i.e. to ask how is theoperator product expansion consistent with the representation ring of the conformal group.This question can be answered for the principal series representations.The direct product of two principal series representations can be decomposed into irre-ducible representations. As it turns out only the principal series representations appear inthis decomposition [7]. In other words, the principal series representation are closed underClebsch-Gordan decomposition. A principal series representation is a function on R d , f R =[∆ ,(cid:96) ] – 6 –abeled by the conformal dimension ∆ and spin (cid:96) . The vector in the direct product of twosuch representations is a bilocal function f ( x , x ) ≡ f R ( x ) f R ( x ). On general grounds,the Clebsch-Gordan takes the form, f ( x , x ) = (cid:90) dR (cid:90) dx C R ,R , ˜ R ( x , x , x ) f R ( x ) . (2.7)The integration measure dR is defined in (2.5). The kernel C is the Clebsch-Gordan coefficient.Let us emphasis that so far this is a purely group theoretic statement and does not includethe dynamical structure constant K . Comparing with the Clebsch-Gordan decomposition fora compact group, say SU (2): | j , m (cid:105) ⊗ | j , m (cid:105) = (cid:88) j,m C j ,j ,jm ,m ,m | j, m (cid:105) , (2.8)we see that the coordinates x i are analogous to the magnetic quantum number m i . Theexplicit form of the integration kernel is determined by matching conformal transformationproperties on both side of equation (2.7). Surprisingly, this exercise turns out to be easierthan determining the Clebsch-Gordan coefficients for the compact group. As an example, forscalar operators, C R ,R ,R ( x , x , x ) = C| x | ∆ +∆ − ∆ | x | ∆ +∆ − ∆ | x | ∆ +∆ − ∆ , x ij ≡ x i − x j . (2.9)Normalization constant C is fixed from the normalization of the 2-point function (2.3). It isclear from equation (2.9) that the Clebsch-Gordan coefficient for the conformal group has thesame form as the familiar 3-point function. Indeed the operator product expansion (2.6) canbe expressed as an integral over space with a Kernel that is the three point function instead ofas sum over descendents. This is described in [9]. The integral form of the operator productexpansion is termed the OPE block.The Clebsch-Gordan coefficient for decomposing the direct product of representations R and R into R is nonzero if the 3-point function of R , R and ˜ R is nonzero. A configurationof three points on the sphere breaks the conformal symmetry to SO ( d − (cid:96) , (cid:96) and (cid:96) admit an SO ( d −
1) singlet.The Clebsch-Gordan coefficients also enjoy an orthogonality relation. It is most familiar inthe case of SU (2), (cid:88) m ,m C j,j ,j m,m ,m C j ,j ,j (cid:48) m ,m ,m (cid:48) = δ j,j (cid:48) δ m,m (cid:48) . (2.10)Similarly for the conformal group [7], (cid:90) dx dx C R,R ,R ( x, x , x ) C ˜ R , ˜ R ,R (cid:48) ( x , x , x (cid:48) ) = δ ( x − x (cid:48) ) δ R, ˜ R (cid:48) + G ( x − x (cid:48) ) δ R,R (cid:48) δ R,R (cid:48) ≡ δ (cid:96)(cid:96) (cid:48) δ (∆ − ∆ (cid:48) ) 2 πρ (cid:96) (∆) . (2.11)– 7 –ere ρ (cid:96) (∆) is the Plancherel weight defined in equation (2.5) and G ( x − x (cid:48) ) is the two pointfunction of two local operators with representation R . For example, if R = [∆ ,
0] then G = | x − x (cid:48) | − ∆ . This equation will be useful later in the paper.In a unitary euclidean conformal theory, the operator product expansion is best expressedin a way that makes the representation ring (2.7) manifest. The only additional piece requiredis the structure constant K . It is incorporated as follows, O R ( x ) O R ( x ) = (cid:90) dR (cid:90) dx K R ,R ,R C R ,R ,R ( x , x , x ) O ˜ R ( x ) . (2.12)Interestingly this suggests an extension of the notion of operator product expansion to com-pact groups as well as to any other non-compact groups. Taking the example of SU (2), asuitable “operator product expansion” would be, O j ,m O j ,m = (cid:88) j,m K j ,j ,j C j ,j ,jm ,m ,m O j,m . (2.13)The structure constants K are symmetric and the operators O j,m are normalized such that K j,j, = 1. Given that the group theory allows a nonzero 1-point only for the trivial repre-sentation, normalizing it to unity (cid:104) O , (cid:105) = 1, we get (cid:104) O j ,m O j ,m (cid:105) = δ j ,j δ m + m , C j,j, m, − m, .The constraint on K comes from the associativity of the operator product expansion. Foreuclidean conformal group, the associativity equation is (cid:90) dRK R ,R ,R K ˜ R,R ,R (cid:90) dx C R ,R ,R ( x , x , x ) C ˜ R,R ,R ( x, x , x ) = (cid:90) dR (cid:48) K R ,R ,R (cid:48) K R , ˜ R (cid:48) ,R (cid:90) dx C R ,R ,R (cid:48) ( x , x , x (cid:48) ) C R , ˜ R (cid:48) ,R ( x , x (cid:48) , x ) . (2.14)It is known that the position space integral on each side of the integral is proportional to thesum of conformal blocks g R and g ˜ R for representation R and its shadow ˜ R [10]. For scalarexternal operators, (cid:90) dx C R ,R ,R ( x , x , x ) C ˜ R,R ,R ( x, x , x ) = (2.15) | x | ∆ +∆ − ∆ | x | ∆ +∆ − ∆ ( g R ( u, v ) + g ˜ R ( u, v )) , u = | x x || x x | , v = | x x || x x | . It is clear that the equation (2.14) is very similar to the crossing equation (1.2), the maindifference is that the physical representations have been replaced by principal series repre-sentations and correspondingly the sum is replaced by an integral. In fact they are relatedeven more intimately. If the structure constants have appropriately slow asymptotic growththen due to the exponentially decaying behavior of the conformal block g R towards positivereal infinity, the contour for the first term can be deformed towards the positive real axis asshown schematically in figure 1. The shadow block g ˜ R decays exponentially towards negativereal infinity so its contour can be deformed towards negative real axis . Thus the integral We thank Balt van Rees for discussion on this issue. – 8 – d ic A B
Figure 1: The integral over the principal series representations is represented by the contourA. It can be expressed as a sum over physical representations corresponding to the poles onthe real axis given that the contour can be deformed to B without picking any additionalcomplex poles.over principal series is converted to a sum over physical representations. Here we have as-sumed that the poles of the integral lie only on the real line. The upshot is that if one findsan analytic solution to equation (2.14) with appropriate properties then one can build thesolution of the physical crossing equation (1.2). One subtle obstruction to this argument isthat, say on the left hand side of the equation, the product of structure constants that appearare K R ,R ,R K ˜ R,R ,R and not K R ,R ,R K R,R ,R ; similarly on the right. It is the later combi-nation that would produce the desired equation. So in addition to their symmetry under allpermutations of their labels, we also require the structure constants to be shadow symmetrici.e. K R ,R ,R = K ˜ R ,R ,R . (2.16)With this condition, the two terms in the integral consisting of g R and g ˜ R give the samecontribution.The above argument is for when the external representations R , . . . , R are kept fixed.As the external representations are also principal series representations, it is not sufficientto just convert the intermediate representations to physical ones. We propose the followingmethod to change all the representations from principal series to physical set together. Take R = R and R = R . The poles of the integral, say on the left hand side of equation (2.14),are then determined in terms of principal series representations R and R . This gives theequation, R = f i ( R , R ) . (2.17)Here i labels the poles. As the structure constant is symmetric in all three representationlabels, we symmetrize the above equation to obtain two additional equations. The phys-ical representations then live at the intersection of three divisors f i ( R , R ) , f j ( R, R ) and f k ( R , R ). This procedure is summarized in figure 2. We have denoted the Clebsch-Gordan– 9 – dR Z dR Z dR X r X r X r Figure 2: Converting the principal series representations to physical representations.coefficient C R ,R ,R by a trivalent vertex. The accompanying factor of K is denoted as a blackdot. Before embarking on finding the solutions to equation (2.14), it is instructive to considerthe more familiar case of SU (2). The associativity constraint coming from the product (2.13). (cid:88) j,m K j ,j ,j K j,j ,j C j ,j ,jm ,m ,m C j,j ,j m,m ,m = (cid:88) j (cid:48) ,m (cid:48) K j ,j ,j (cid:48) K j ,j (cid:48) ,j C j ,j ,j (cid:48) m ,m ,m (cid:48) C j ,j (cid:48) ,j m ,m (cid:48) ,m . (2.18)This equation is simplified further using what is known as the Racah coefficient. We havedefined the Racah coefficient in appendix A. It is the recoupling coefficient for three angularmomenta. Even though it is defined in an algebraic fashion, for our current purposes it canbe thought of as the coefficient that expresses the product of two Clebsch-Gordan coefficientsin one channel in terms of linear combination of the product of Clebsch-Gordan coefficientsin the other channel, see equation (A.3). C j ,j ,jm ,m ,m C j,j ,j m,m ,m = (cid:88) j (cid:48) (cid:112) (2 j + 1)(2 j (cid:48) + 1) W ( j , j , j , j ; j, j (cid:48) ) C j ,j ,j (cid:48) m ,m ,m (cid:48) C j ,j (cid:48) ,j m ,m (cid:48) ,m . (2.19)The quantity W is the Racah coefficient . Substituting this expression in equation (2.18)and comparing each term in the j sum we get an equation that doesn’t depend on themagnetic quantum number m i . In comparing individual terms in the j sum we make use ofthe orthogonality condition (2.10). K j ,j ,j (cid:48) K j ,j (cid:48) ,j = (cid:88) j (cid:48) (cid:112) (2 j + 1)(2 j (cid:48) + 1) W ( j , j , j , j ; j, j (cid:48) ) K j ,j ,j K j,j ,j . (2.20)This form of the associativity constraint is more invariant than (2.18) as it does not refer to theindividual vectors in the representations. It is easy to generalize to other Lie groups especiallyto non-compact Lie groups. Due to its abstract origin, the Racah coefficient can defined forany group. The representations that would appear are the ones arising in the harmonicanalysis. In the case of compact groups this set consists of all irreducible representationswhile for non-compact groups it consists of certain types of induced representations described A more symmetric object is known as the 6 j symbol, it is defined as ( − j + j j + j W . It enjoys a groupof symmetries consisting of the symmetries of a tetrahedron. – 10 –n section 2. We refer to the generalization of the associativity equation (2.20) or equivalently(2.18) to Lie group G as G -associativity equation or G -crossing equation. For even d , the SO ( d + 1 , K R ,R ,R K R ,R ,R = (cid:90) dR (cid:48) W ( R , R , R , R ; R, R (cid:48) ) K R ,R ,R (cid:48) K R ,R ,R (cid:48) . (2.21)In this equation the function W is the Racah coefficient for the conformal group and theintegral is over the principal series. This equation is equivalent to (2.14) and the commentsbelow the latter about contour deformation and changing the principal series integral to asum over physical representations are equally applicable to this equation i.e. a solution to(2.21) with appropriate analytic and asymptotic behavior leads to a physical solution to theconformal crossing equation. From the relationship between equations (2.21) and (2.14) wesee that the ambient space does not play a profound role in defining the conformal theory butcan be thought of as merely an artifact of the group theory. In the remainder of the paperwe find explicit solutions to G -associativity equation for any G .
3. Solutions to generalized crossing equation
Let us first consider the SU (2)-associativity equation (2.20). The advantage of working witha compact group is that one has to deal with a discrete set of representations rather than acontinuous one. Interestingly one solution to the SU (2) associativity is already known in theliterature [11, 12]: K j ,j ,j = 1[(2 j + 1)(2 j + 1)(2 j + 1)] (cid:104) ( j + j + j + 1)!( j + j − j )!( j − j + j )!( − j + j + j )! (cid:105) (3.1)It is obtained using the associativity of multiplication of the so called “symplecton polyno-mials” [13, 14]. From the modern perspective, the idea is to consider the quantization of CP and study the associated ring of operators. The phase space enjoys an action of SU (2) andso do the functions on the phase space. After quantization these functions become operatorsacting on the Hilbert space. They form a vector space. If the quantization is performedcovariantly, the vector space of quantum mechanical operators also inherits the SU (2) action.We can organize the operators into irreducible representations of this action. Let P jm to bean operator belonging the representation j of SU (2) with charge m under the Cartan gen-erator. Multiplying together two vectors from different representations and decomposing theresulting operator into irreducible representations we expect P j m · P j m = (cid:88) j,m K j ,j ,j C j ,j ,jm ,m ,m P jm . (3.2)Here · stands for operator multiplication. The m -dependence of the right hand side is uniquelyfixed by group theory while K is an undetermined constant allowed by the symmetry. Lo– 11 –nd behold, the equation (3.2) is exactly the same as equation (2.13). We expect the oper-ator multiplication to be associative. This associativity constraint is the same as equation(2.18). In this way the CP quantum mechanical system has generated a solution to the SU (2)-associativity equation. The authors of [13, 14] find the solution (3.1) by explicitly con-structing operators P jm . For similarly constructed solutions to the associativity equation forthe quantum group SU (2) q see [15].From this discussion it is clear that quantization of the classical phase space admitting G -symplectomorphism can be use to generate a solution to the G -associativity equation. Ingeneral the G -action will not be carried over to the Hilbert space but only to the space ofoperators but if the coordinate space itself enjoys a G -action then the group will also acton the Hilbert space. In this case we can find the solution to G -associativity explicitly. Aswe will describe momentarily, the idea is in fact more basic and can be formulated in analgebraic way without relying on quantum mechanics. Identifying the underlying algebraicstructure will get us infinitely many solutions in closed form. They will be in one to onecorrespondence with representations of G . We will describe the construction for SU (2), itadmits straightforward generalization to any Lie group including the conformal group.Let the Hilbert space form a representation α of SU (2). The linear operators acting onthis space belong to the representation α ⊗ α ∗ where α ∗ is the dual to α . In the case of SU (2), α ∗ = α . We organize these operators according to their irreducible representations j i.e. α ⊗ α ∗ = ⊕ αj =0 j . In figure 3a we have denoted the operator O j in representation j graphically. The two directed lines represent its matrix indices valued in representations α .Multiplication of two operators is obtained by contracting the matrix indices appropriately.Graphically, the index contraction is represented by joining one of the “internal” α lines ofthe two operators. The component of the operator O j in irreducible representation j inthe multiplication of operators O j and O j is denoted in figure 3b. More specifically, ifwe pick vectors m i belonging to representation j i then the figure 3b refers to the coefficient K j ,j ,j C j ,j ,j m ,m ,m that appears on the right hand side of equation (3.2). If the “internal”representation α is irreducible one can compute the structure constants K j ,j ,j explicitly.Let us use the notation α ( i ) to distinguish the three irreducible isomorphic representationsappearing in figure 3b and let n ( i ) index their vectors. Then, j ∈ α (1) ⊗ α (2) , j ∈ α (2) ⊗ α (3) , j ∈ α (3) ⊗ α (1) . (3.3)Correspondingly, their vectors are related by Clebsch-Gordan coefficients, | j , m (cid:105) = (cid:88) n (1) ,n (2) C j ,α (1) ,α (2) m ,n (1) ,n (2) | α (1) , n (1) (cid:105)| α (2) , n (2) (cid:105) , . . . . The quantity denoted in figure 3b is a combination of product of three Clebsch-Gordancoefficients. (cid:88) n (1) ,n (2) ,n (3) C α (2) ,j ,α (1) n (2) ,m ,n (1) C α (3) ,j ,α (1) n (3) ,m ,n (1) C α (3) ,j ,α (2) n (3) ,m ,n (2) = K j ,j ,j C j ,j ,j m ,m ,m . (3.4)– 12 – ↵↵ (a) The operator in representation j is representedgraphically. The two lines denote the vector spaces α . When the operator is represented as a matrix,they also represent its two indices. j j j ↵ (1) ↵ (2) ↵ (3) (b) The operator product algebra. This figure de-notes the j component in the multiplication of rep-resentation j and j respectively. The subscript on α distinguishes the three isomorphic vector spaces. Figure 3: Graphical representation of the operators and their product algebra.Thanks to the definition of the Racah coefficient (2.19) and orthogonality (2.10), the lefthand side can be evaluated in terms of the Racah coefficient, see equation (A.4). It is indeedproportional to the Clebsch-Gordan coefficient as on the right hand side. The proportionalityconstant, appropriately normalized is: K ( α ) j ,j ,j = ( − α + j + j W ( α, j , α, j ; α, j ) (cid:112) N j N j N j /N , N j = ( − α + j ) W ( α, j, α, j ; α, . (3.5) K ( α ) j ,j ,j = ( − j − j − j (2 α + 1) [(2 j + 1)(2 j + 1)(2 j + 1)] W ( α, j , α, j ; α, j ) . We claim that the proposed structure constant K ( α ) j ,j ,j is symmetric under the permutationsof its labels and satisfies the SU (2)-associativity equation (2.18) or equivalently (2.20). Thesymmetries of K follow straightforwardly from the tetrahedral symmetry group of the Racahcoefficient (more accurately, the 6 j -symbol) (A.5). The proof of associativity follows just asstraightforwardly from the Pentagon identity or the Biedenharn-Elliot identity of the Racahcoefficient: (cid:88) j (cid:48) (2 j (cid:48) + 1) W ( j , j , j , j ; j, j (cid:48) ) W ( k , j , k , j ; k , j (cid:48) ) W ( k , j , k , j (cid:48) ; k , j )= W ( k , j , k , j ; k , j ) W ( k , j, k , j ; k , j ) . (3.6)The proof of this identity is discussed in Appendix A. Replacing all the k type labels by α and using the normalizations in (3.5), the above equation immediately reduces to the SU (2)-associativity equation. In this way, we can construct infinitely many solutions to theassociativity equation, each labeled by a representation α .– 13 –n fact a more general class of solutions can be constructed in this way. Essentially bynot requiring α to be a representation of G but only requiring the operators acting on α be arepresentation of G . The associativity of the operator product still guarantees that the ringcoefficients are a solution to the required G -associativity equation. A closed form expressionfor the solutions of this general type would be difficult to obtain.The pentagon identity holds for any Lie group including the conformal group becauseit follows from a simple argument involving tensor product of four representations, as sum-marized in figure 6. Hence the 6 j -symbol for the conformal group, specialized to the aboveform gives a solution to the conformal crossing equation valued in the principal series. Wehave already outlined the procedure to go from the principal series representations to thephysical representations by way of residue integral. So in effect, we have produced infinitelymany solutions to the usual conformal crossing equations. In doing so, at no point, havewe imposed the unitarity from the Lorentzian point of view. It would be interesting to seewhat constraints the Lorentzian unitarity imposes on these solutions. As the solution to thecrossing symmetry is essentially the 6 j -symbol for the conformal group, it would be impor-tant to compute it explicitly [16]. The asymptotic property of the structure constant controlsthe convergence of the integral in equation (2.14) which will in turn decide whether non-normalizable states, such as the identity, appear in the physical conformal spectrum. Thisquestion is also under investigation.The easiest method for computing the conformal 6 j -symbol relies on version of the rela-tion (A.3) for the conformal group. Denoting the three point function by a trivalent vertex,the relation is succinctly expressed as R R ˜ R ˜ R R R ˜ R R R R R R = W ( R , R , R , R ; R, R ) ⇥ From this equation we also see that the 6 j -symbol for the conformal group is invariant underexchanging on the of representations with its shadow representation. This means that inaddition to the permutation symmetry of its labels, the proposed structure constant also hasthe symmetry K R ,R ,R = K ˜ R ,R ,R (3.7)as promised around equation (2.16).Let us comment about the physical relevance of the solutions we have obtained. Thesolution for the G -crossing equation in terms of the 6 j symbol appear as the boundary con-– 14 –ormal theory of the SU (2) k WZW model [17, 18]. Here the group G is taken to be thequantum group SU (2) q , q = exp 2 πi/ ( k + 2). This is expected to be true for other groupsas well. The solution label α is the one that labels the conformal boundary conditions. Theappearance of the quantum group in the context should not be surprising as there is a wellknown connection between affine symmetries and quantum groups, first observed in [19]. Infact the representation theory of the Virasoro algebra is also intimately linked to the repre-sentation theory of the quantum group SL (2) q . Indeed the structure constants given in termsof the 6 j -symbol of SL (2) q appear as the structure constants for the FZZT brane, the famousconformal boundary condition of the Liouville theory [20]. In both these examples, the rel-evant structure constants are of the boundary theory and not of the bulk. This is because,on the boundary only one copy of the affine symmetry acts while on the bulk two copies,holomorphic and anti-holomorphic, of the same act. As a result the bulk structure constantsobey the G × G crossing equation. The naive solution to this namely, the tensor productof the solutions to the G crossing equation, doesn’t work because the G -representations onholomorphic and anti-holomorphic side are typically entangled in a certain way as requiredby the modular invariance.
4. Outlook
In this paper we have looked at the conformal crossing equation from the viewpoint of con-formal representation theory. As the conformal group is non-compact, its harmonic analysisplays an important role in this approach. Nevertheless, thanks to its well-developed theory,we are able to think of harmonic analysis as an additional feature for non-compact groupsof what really is a problem for any Lie group. Using the crossing equation for SU (2) groupas a toy model, we have constructed infinitely many solutions to the crossing equation forany Lie group G in terms of its 6 j -symbol. In particular, we have obtained infinitely manysolutions to the conformal crossing equation. It would be interesting to investigate if theylead to physical unitary conformal field theories.What is the significance of the G -associativity equation for other groups G ? Inspired bythe kinematics of the AdS-CFT correspondence, we would like to postulate a correspondencebetween a “ G -Theory” i.e. solution of G -associativity equation, and string theory on a certain G -symmetric space. In the paper we have outlined how a topological quantum mechanicalsystem with G symmetry leads to the solution of G -associativity equation. The argument isequally valid if one replaces topological quantum mechanics with string theory. The relationbetween the space-time operator product expansion and world-sheet operator product expan-sion is explained in [21]. The example of the string theory of WZW model and its solution interms of 6 j -symbol [22,23], mentioned at the end of the last section, is indicative of such a cor-respondence. A connection between the de-Sitter gravity and a conformal theory of temperedrepresentations has been pointed out in [24, 25]. This raises a tantalizing possibility, couldthe CMB 3-point functions be 6 j -symbols? If such a generalized correspondence between a We thank David Simmons-Duffin for asking this question. – 15 – -theory and a string theory exists then we believe it would have most teeth in the case ofnon-compact groups because, in that case a G -theory can be expressed as a local quantumfield theory on the coset G/M AN where
M AN is the distinguished parabolic subgroup of G appearing in the harmonic analysis. It would be a natural question is to develop a notionof large- N expansion and match it with the string perturbation theory, `a la [26, 27]. Recentwork on p-adic AdS/CFT correspondence [28] could also fit in this framework by consideringgroups over p-adic numbers instead of over real/complex numbers.What can one say about solutions to conformal boundary conditions in higher dimen-sions? The local operators supported on the boundaries transform under a reduced symmetrygroup SO ( d, i.e. the crossing equation for the SO ( d,
1) group. Thesolutions are found in the same way in terms of the 6 j -symbol of SO ( d, ,
2) superconformal theory isbelieved to be completely fixed, modulo a discrete label, by symmetries. If a physical solutionis obtained for the (0 ,
2) superconformal group crossing equation using our method thenin all likelihood it is the (0 ,
2) superconformal theory. One can make a similar argumentfor the four dimensional N = 4 superconformal theory, in this case the conformal data isbelieved to depend on one additional continuous complex parameter. As our solutions areconstructed from group theory, it is our hope that the unwieldy problem of conformal fieldtheory classification gets reduced to relatively wieldy problem in representation theory, at leastpartially.Recently, a connection has been found between d -dimensional Euclidean conformal theorydata and scattering amplitudes in d + 2-dimensional Lorentzian quantum field theory [30].The conformal representations arising in this relation are not the physical ones but ratherprecisely the ones appearing in the harmonic analysis. These are also natural from ourviewpoint. Hence, our solutions could be useful in describing scattering in d + 2 dimensions.As a d + 2-dimensional theory is not just Lorentz invariant but rather Poincare invariant, onewould need to impose the translational symmetry by hand. It would be nice to see how sucha constraint can be naturally imposed on our solutions. Acknowledgements
We would like to thank Clay Cordova, Juan Maldacena, Pavel Putrov, Balt van Rees, DavidSimmons-Duffin and Douglas Stanford for illuminating discussions. We are also thankful toShiraz Minwalla whose comments about conformal representation theory inspired this work.– 16 –he author’s research is supported by the Roger Dashen Membership Fund and the NationalScience Foundation grant PHY-1314311.
A. The SU (2) Racah coefficient
The Racah coefficient is the recoupling coefficient for three angular momenta j , j , j into j . Such a coupling can be achieved via two schemes: by first coupling j and j into j andthen coupling j and j into j or by first coupling j and j into j (cid:48) and then coupling j and j (cid:48) into j . They respectively yield | j , m ; ( j ) (cid:105) = (cid:88) m,m C j,j ,j m,m ,m (cid:16) (cid:88) m ,m C j ,j ,jm ,m ,m | j , m (cid:105)| j , m (cid:105) (cid:17) | j , m (cid:105)| j , m ; ( j (cid:48) ) (cid:105) = (cid:88) m ,m (cid:48) C j ,j (cid:48) ,j m ,m (cid:48) ,m | j , m (cid:105) (cid:16) (cid:88) m ,m C j ,j ,j (cid:48) m ,m ,m (cid:48) | j , m (cid:105)| j , m (cid:105) (cid:17) . (A.1)Here the coefficients C are the Clebsch-Gordan coefficients. The bracketed label of | j , m (cid:105) denotes the scheme used to construct it. Both coupling schemes construct orthonormal basesin the vector space j ⊗ j ⊗ j . Hence there must be a unitary matrix relating them. | j , m ; ( j ) (cid:105) = (cid:88) j (cid:48) (cid:112) (2 j + 1)(2 j (cid:48) + 1) W ( j , j , j , j ; j, j (cid:48) ) | j , m ; ( j (cid:48) ) (cid:105) (A.2)The matrix coefficient W is known as the Racah coefficient. Substituting (A.1) into thisequation and using orthonormality of the basis vectors, we get a fundamental relation betweenthe Racah coefficients and the Clebsch-Gordan coefficients. C j ,j ,jm ,m ,m C j,j ,j m,m ,m = (cid:88) j (cid:48) (cid:112) (2 j + 1)(2 j (cid:48) + 1) W ( j , j , j , j ; j, j (cid:48) ) C j ,j ,j (cid:48) m ,m ,m (cid:48) C j ,j (cid:48) ,j m ,m (cid:48) ,m . (A.3)Using orthogonality and completeness of the Clebsch-Gordan coefficients (2.10) further rela-tions can be derived. (cid:88) m ,m C j ,j ,jm ,m ,m C j,j ,j m,m ,m C j (cid:48) ,j ,j m (cid:48) ,m ,m = (cid:112) (2 j + 1)(2 j (cid:48) + 1) W ( j , j , j , j ; j, j (cid:48) ) C j ,j ,j (cid:48) m ,m ,m (cid:48) . (A.4) (cid:88) m ,m C j ,j ,jm ,m ,m C j,j ,j m,m ,m C j (cid:48) ,j ,j m (cid:48) ,m ,m C j ,j ,j (cid:48) m ,m ,m (cid:48) = (cid:112) (2 j + 1)(2 j (cid:48) + 1) W ( j , j , j , j ; j, j (cid:48) ) . It is sometimes convenient to use a more symmetric form of the Racah coefficient knownas the 6 j -symbol. It is defined as, (cid:110) j j jj j j (cid:48) (cid:111) = ( − j + j + j + j W ( j , j , j , j ; j, j (cid:48) ) . (A.5)If we associate the six representations involved to six edges of a tetrahedron, as indicated infigure 4, then the 6 j -symbol is invariant under the tetrahedral symmetry group.– 17 – j j jj j j o = j j j j jj Figure 4: Geometric interpretation of the 6 j -symbol as a tetrahedron. j j j j j j j j j j j j k k k k k k k j jj j k k k k j k j Figure 5: Two triangulations of octahedron corresponding to the two sides of the pentagonidentity.The Racah-coefficients obey a powerful identity known as the Pentagon identity or theBiedenharn-Elliot identity: (cid:88) j (cid:48) (2 j (cid:48) + 1) W ( j , j , j , j ; j, j (cid:48) ) W ( k , j , k , j (cid:48) ; k , j ) W ( k , j , k , j ; k , j (cid:48) )= W ( k , j, k , j ; k , j ) W ( k , j , k , j ; k , j ) . (A.6)From a geometric point of view, where we associate a tetrahedron to the Racah coefficient,the Pentagon identity is the equivalence between two ways of obtaining an octahedron: eitherby gluing three tetrahedra along faces around a common edge or by gluing two tetrahedraalong a face. This is shown in figure 5. The pentagon identity follows when we considercoupling four angular momenta. Different ways of coupling them are sequentially related bymultiplying Racah coefficients. Then the equivalence of two sets of ordered moves yields thepentagon identity. Again it is best to explain it in the graphical language, figure 6. Due to– 18 – j j j j k k j j j j j k k k j k j j j k k k k j j j k j k k j j j k W ( j , j , j , j ; j,j ) W ( k ,j ,k ,j ; k ,j ) W ( k ,j ,k ,j ; k ,j ) W ( k ,j ,k ,j ; k ,j ) W ( k ,j,k ,j ; k ,j ) Figure 6: The pentagon identity follows from considering the associativity of four angularmomenta addition. The circular arcs denote the two sides of the pentagon identity.the simple and robust origin of the pentagon identity, it holds for any group including theconformal group SO ( d + 1 , eferences [1] S. Ferrara, A. F. Grillo, and R. Gatto, Tensor representations of conformal algebra andconformally covariant operator product expansion , Annals Phys. (1973) 161–188.[2] A. M. Polyakov, Nonhamiltonian approach to conformal quantum field theory , Zh. Eksp. Teor.Fiz. (1974) 23–42.[3] R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, Bounding scalar operator dimensions in 4DCFT , JHEP (2008) 031, [ arXiv:0807.0004 ].[4] D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT , arXiv:1612.0847 .[5] F. A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion , Nucl.Phys.
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