Conformal-helicity duality & the Hilbert space of free CFTs
IIPMU19-0015
Conformal-helicity duality & the Hilbert space of free CFTs
Brian Henning
1, 2 and Tom Melia Department of Physics, Yale University, New Haven, Connecticut 06511, USA D´epartment de Physique Th´eorique, Universit´e de Gen`eve,24 quai Ernest-Ansermet, 1211 Gen`eve 4, Switzerland Kavli Institute for the Physics and Mathematics of the Universe (WPI),The University of Tokyo Institutes for Advanced Study,The University of Tokyo, Kashiwa, Chiba 277-8583, Japan (Dated: February 20, 2019)We identify a means to explicitly construct primary operators of free conformal field theories(CFTs) in spacetime dimensions d = 2, 3, and 4. Working in momentum space with spinors, wefind that the N -distinguishable-particle Hilbert space H N exhibits a U ( N ) action in d = 4 ( O ( N )in d = 2 ,
3) which dually describes the decomposition of H N into irreducible representations of theconformal group. This U ( N ) is a natural N -particle generalization of the single-particle U (1) littlegroup. The spectrum of primary operators is identified with the harmonics of N -particle phasespace which, specifically, is shown to be the Stiefel manifold V ( C N ) = U ( N ) /U ( N −
2) (respec-tively, V ( R N ), V ( R N ) in d = 3 , G ( C N ) ⊂ V ( C N ). We provide a recipe to construct these harmonic polynomials using standard U ( N ) ( O ( N )) representation theory. We touch upon applications to effective field theory and nu-merical methods in quantum field theory. INTRODUCTION AND SUMMARY
The study of free systems is important in physics, asmany interacting systems of interest are deformations ofa free theory. Examples include the Standard Model,the Ising model, or essentially any theory defined by aLagrangian. Two lines of modern research that attemptto exploit this line of thinking are effective field theory(EFT) and Hamiltonian truncation. EFT parameter-izes all possible interactions of particles that enjoy theproperty of being non-interacting when asymptoticallyseparated. In spirit, it is the field-theoretic version ofan S -matrix program; so-called positivity [1–3] and re-lated [4, 5] results are encouraging hints that it might alsoadmit a bootstrap formulation. In another direction, thedecades old idea of Hamiltonian truncation is the fieldtheoretic version of familiar techniques from QM: take adeformation δH of a known, e.g. free, Hamiltonian H ,compute the matrix elements (cid:104) i | δH | j (cid:105) with | i (cid:105) the Hib-ert space states of H , and diagonalize the matrix. Inprinciple, this non-perturbative approach approximatesthe spectrum with increasing accuracy as one includesmore states. This idea has recently been revived in thehigh-energy community [6, 7] with encouraging results, e.g. [6–12]; a central idea in these works is the organiza-tion of the states of H according to conformal symmetry.In both the above examples, crucial ingredients arethe operators of the non-interacting theory. In EFT,one focuses on the so-called operator basis which con-sists of the set of independent operators one can add tothe Lagrangian; equivalently, one can think of this as theset of Lorentz preserving deformations of the free the-ory. In a series of papers, enumeration of the operatorbasis was solved and steps towards systematic construc- tion taken [13–15] (see also [16, 17]). While these re-sults follow from Poincar´e symmetry, it was shown thatthe conformal group can technically aid in addressingthese questions. In Hamiltonian truncation, one needsthe basis states, i.e. one needs to explicitly constructthe primary operators of the unperturbed theory. Thisis a technically difficult problem, even for free theories.While no systematic construction exists to date, subsetsof operators have been constructed and utilized for vari-ous problems, e.g. [18–21].In this letter we take a significant step towards solv-ing the operator construction problem for free, conformaltheories in low spacetime dimensions. Similar to [7, 15],we work in momentum space, except here we choose tomake use of spinor-helicity variables. The physical ideahere is that spinors capture both momentum and polar-ization information; therefore—since the spectrum of afree theory is entirely kinematic—one only needs spinors, i.e. there is no need to introduce polarization tensors inaddition to momentum vectors. Moreover, as we willsee, spinor-helicity variables reveal geometric structuresin the phase-space which govern the operator content.In the rest of this section we will explain the basicidea and state our central result, focusing on the casewith spacetime dimension d = 4 (the analogous resultsfor d = 2 , Phase space and the Stiefel manifold
The basic ideais simple: due to the Fock space structure of the Hilbertspace, in free theories operators simply consist of polyno-mials in the fields and their derivatives. Moreover, when a r X i v : . [ h e p - t h ] F e b FIG. 1.
Left:
Interpretation of kinematic data with referenceto both an SL (2 , C ) (Lorentz) and a U ( N ) action. Right:
Geometry of the Stiefel manifold. the theory is conformal this spans the Hilbert space,due to the operator-state correspondence. In momen-tum space, the operators translate into polynomials ofthe momentum and polarization tensors—or, in d = 4,simply polynomials in spinor variables λ and (cid:101) λ . Now consider N distinguishable particles carrying totalmomentum P a ˙ a = N (cid:88) i =1 λ ia (cid:101) λ ˙ ai . (1)In addition to the usual Lorentz group SL (2 , C ) actionon the spinors, there is generically a GL ( N, C ) actionrotating the spinors amongst themselves. Importantly,the U ( N ) ⊂ GL ( N, C ) subgroup—under which λ trans-forms in the fundamental (defining) representation and (cid:101) λ in the conjugate representation—leaves P invariant. Inthis sense the U ( N ) action naturally generalizes the var-ious U (1) little group scalings we can perform on eachof the spinors; in particular, the N little group scalingsform the torus, U (1) N ⊂ U ( N ).For a single particle, N = 1, the helicity is a represen-tation of the U (1) little group that leaves the momentuminvariant. This generalizes to N particles: besides carry-ing some net momentum P , the possible other propertiesof the system of particles is encoded by the U ( N ) in aparticular way.More precisely, P = (cid:80) λ (cid:101) λ carves out some manifold in C N . To elucidate this, write λ i = (cid:18) u i v i (cid:19) and go to thecenter of mass frame P a ˙ a = M δ a ˙ a , (cid:18) M M (cid:19) = (cid:18) | u | v † uu † v | v | (cid:19) . (2)Evidently, in this frame, the manifold consists of twocomplex spheres of radius √ M that are tangential to one Although true in d = 4, in general masslessness does not guar-antee conformality; see [23] for the classification. Our conventions can be deduced from the following equations: p a ˙ a = p µ σ µa ˙ a = (cid:16) p + p p − ip p + ip p − p (cid:17) a ˙ a with magnitude p = g µν p µ p ν = det p a ˙ a = (cid:15) ab (cid:15) ˙ a ˙ b p a ˙ a p b ˙ b = p a ˙ a p a ˙ a . another ( v † u = 0) (see Fig. 1). It is easiest to thinkof this as a homogeneous space: the two U ( N ) vectors u and v take “vevs” and “break” U ( N ) → U ( N − U ( N ) /U ( N − V ( C N ).Physical observables of the N particles only havesupport on the Stiefel manifold. That is, observablesare functions f ( λ ia , (cid:101) λ ˙ ai ) on V ( C N ) ⊂ C N . Certainobservables—in particular, scattering amplitudes—areLorentz invariant: f ( gλ, g ∗ (cid:101) λ ) = f ( λ, (cid:101) λ ) for g ∈ SL (2 , C ).In this case, we can mod out by an additional U (2) cor-responding to the little group of the massive momentum P (and a complex phase), and consider functions on theGrassmann manifold G ( C N ) = U ( N ) /U ( N − × U (2).This is nothing but the “kinematic Grassmannian” [24]that proliferates the modern study of four-dimensionalscattering amplitudes. Intuitively, the “more general”correlation functions live on the “more general” Stiefelmanifold, while the Lorentz singlet data lives on a sub-manifold. Mathematically, the Stiefel manifold is a U (2)fiber bundle over the Grassmannian U (2) → V ( C N ) → G ( C N ); physically, this reflects familiar manipulationslike decomposing correlation functions or form factorsinto Lorentz spin structures times Lorentz invariant func-tions (see, e.g. , [25] for a systematic discussion in thecontext of conformal correlation functions).In consideration of observables, the harmonics on theStiefel/Grassmann manifold provide a natural basis forsuch functions. It becomes obvious how to ascertain theseharmonics upon reviewing the case of spherical harmon-ics, which we do in the supplemental material (this alsoprovides the analogous story for d = 2 dimensions). Theupshot is that (i) these originate from polynomials in thespinors—called harmonic polynomials—that are annihi-lated by the generalized Laplacian K a ˙ a = − (cid:88) i ∂ ai (cid:101) ∂ ˙ ai (3)and (ii) the polynomials furnish certain (finite-dimensional) representations of U ( N ).The physical content of these statements becomestransparent upon recognizing that this generalized Lapla-cian is the generator of special conformal transforma-tions: P and K —together with the Lorentz and dilata-tion generators M , (cid:102) M , and D —furnish the conformal Respectively (with “ · ” denoting a sum over Lorentz indices), M ba = − i (cid:88) i (cid:16) λ ia ∂ bi − δ ba λ i · ∂ i (cid:17) , (cid:102) M ˙ b ˙ a = − i (cid:88) i (cid:16)(cid:101) λ ˙ ai (cid:101) ∂ ˙ bi − δ ˙ b ˙ a (cid:101) λ i · (cid:101) ∂ i (cid:17) ,D = − i (cid:88) i (cid:16) λ i · ∂ i + (cid:101) λ i · (cid:101) ∂ i + 2 (cid:17) . (4) algebra su (2 , (cid:39) so (4 , K —implies thatthe corresponding operator is a primary operator.The linking of the harmonic and primary condition im-plies that the conformal representation theory is deter-mined by the U ( N ) representation theory and vice versa .This can be shown in numerous ways. One simple wayis by showing that the Casimir operators of su (2 ,
2) canbe written in terms of the Casimir operators of u ( N ), and vice-versa. A more constructive method fleshes outthe ideas sketched in the previous paragraphs, and willbe taken up in the next section.As the U ( N ) generalizes helicity, we call this link withthe conformal group conformal-helicity duality . Many ofthe ideas presented here have appeared in the mathe-matics literature, where the SU (2 , × U ( N ) action onthe space of polynomials in the spinors is an exampleof a reductive dual pair within the oscillator representa-tion [26, 27] (see the supplemental material for a briefintroduction). We note that the single particle N = 1case essentially coincides with the analysis of [28]. In ad-dition to the general theory developed by Howe [26, 27],the papers [29, 30] contain relevant results.Our main result can be summarized as: Main Result
For free, massless particles of arbitrary helicity in fourdimensions, denote by H N the Hilbert space of N dis-tinguishable particles ( H N = H ⊗ N ). By the operator-state correspondence, we equivalently think of H N asthe space of local operators characterizing the parti-cles. H N furnishes a (reducible) unitary representa-tion of SU (2 , × U ( N ) whose irreducible decomposi-tion consists of an infinite number of positive-energyirreducble representations (irreps) of SU (2 ,
2) [31] car-rying an irrep of U ( N ). The decomposition is suchthat the U ( N ) irrep specifies the SU (2 ,
2) irrep, andvice-versa.Using the usual partitions L = ( L , . . . , L N ) with L i ∈ Z and L ≥ L ≥ · · · ≥ L N to label U ( N ) irreps,the decomposition is H N = (cid:77) L ∈ Λ ( N ) V L , V L = R [∆ ,j ,j ]( L ) ⊗ W L , (6) The representation of the u ( N ) algebra is given by E ij = − i (cid:0) λ i · ∂ j − (cid:101) λ j · (cid:101) ∂ i (cid:1) , [ E ij , E kl ] = − i (cid:0) δ kj E il − δ il E kj (cid:1) . (5)As a rank-3 algebra, su (2 ,
2) nominally can have three indepen-dent Casimirs. For N ≥ su (2 ,
2) Casimirs are inde-pendent and can be written in terms of the u ( N ) Casimirs. For N = 1, resp. 2, there are only one, resp. two, independent con-formal Casimirs; a related fact is that these contain “short” rep-resentations (the shortening conditions coming from equations ofmotion, resp. current conservation). where W L is an irrep of U ( N ) labeled by partition L , R [∆ ,j ,j ]( L ) is a positive-energy irrep of SU (2 , L (see eq. (8) below), and Λ ( N ) is the set of L s appearingin the decomposition.For N ≥ ( N ≥ = (cid:110) ( l ,l , , . . . , , − (cid:101) l , − (cid:101) l ) suchthat l i , (cid:101) l i ∈ Z ≥ , l ≥ l , (cid:101) l ≥ (cid:101) l (cid:111) , (7)where the negative numbers are a standard notation for U ( N ) conjugate representations, e.g. sec. 15.5 of [32]. As we will see in the next section, l , ( (cid:101) l , ) are naturallabels for polynomials in λ ia ( (cid:101) λ ˙ ai ) (see Fig. 2).The corresponding SU (2 ,
2) quantum numbers are∆ = 12 (cid:0) l + l + (cid:101) l + (cid:101) l (cid:1) + N,j = 12 (cid:0) l − l (cid:1) , j = 12 (cid:0)(cid:101) l − (cid:101) l (cid:1) , (8)together with a net helicity quantum number h , h = 12 (cid:0) l + l − (cid:101) l − (cid:101) l (cid:1) . (9)The N < l , and (cid:101) l , parameters they areΛ (1) = (cid:8) ( l ) and ( − (cid:101) l ) (cid:9) , (10a)Λ (2) = (cid:8) ( l , l ) , ( l , − (cid:101) l ) , and ( − (cid:101) l , − (cid:101) l ) (cid:9) , (10b)Λ (3) = (cid:8) ( l , l , − (cid:101) l ) and ( l , − (cid:101) l , − (cid:101) l ) (cid:9) . (10c)For each subspace V L , we focus on the primary oper-ator in the conformal representation, i.e. the state in R [∆ ,j ,j ] annihilated by K a ˙ a . These are the states whichare harmonics on the Stiefel manifold, and we refer tothem as Stiefel harmonics. It is important to recognizethat a harmonic in the W L representation of U ( N ) con-tains dim( W L ) primary operators, and that the differentprimaries may be composed of particles of different spin(see eq. (21) for an explicit example). The physical ba-sis for the states in a given harmonic are characterizedby the U (1) N little group scalings; a prescription to con-struct these states is supplied by semi-standard Youngtableau (SSYT). In the physics literature it is common to use an overline to denoteconjugate representations: ( L , . . . , L N ) = ( − L N , . . . , − L ). One can think of this either as the charge of the diagonal U (1) in U ( N ) (cid:39) SU ( N ) × U (1) or as that of U (2 , (cid:39) SU (2 , × U (1). FIG. 2. Young diagram for the U ( N ) representation Λ ( N ) given in eq. (7). CONSTRUCTING HARMONICS
We proceed algebraically by organizing the space ofpolynomials in the N spinors into finite-dimensional ir-reps of the Lorentz group and U ( N ) (we focus for nowon the Lorentz subgroup SL (2 , C ) ⊂ SU (2 , λ s are in an irrep of SL (2 , C ) ⊗ U ( N ), λ ia = V ⊗ W = ⊗ , (11)where V denotes the spinor rep of SL (2 , C ) and W = W (1 , ,..., the fundamental of U ( N ). We introducedYoung diagram notation (see e.g. [33]) to depict both SL (2 , C ) and U ( N ) reps (diagrams with at most two rowsand N rows, respectively). For the (cid:101) λ s, (cid:101) λ ˙ ai = V ∗ ⊗ W = ⊗ = ⊗ ·· , (12)where V ∗ denotes the complex conjugate spinor rep of SL (2 , C ), and W = W (0 ,..., , − the anti-fundamental of U ( N ). Young diagrams for conjugate U ( N ) represen-tations are barred; by using the epsilon tensor to raiseindices, these can be expressed by exchanging each col-umn of x boxes by a column of N − x boxes (and reversingcolumn order to make a valid diagram). We shade such“raised” boxes blue to keep track of their (cid:101) λ origin.The simple observation is that for a polynomial builtfrom an object with two indices, e.g. λ ia , symmetrizingon one set of indices according to some Young diagramautomatically symmetrizes the other set of indices ac-cording to the same Young diagram. That is, the or-ganisation of λ i a · · · λ i n a n (= Sym n ( λ ) = Sym n ( V ⊗ W ),with Sym denoting the symmetric tensor product) into SL (2 , C ) × U ( N ) multiplets takes the following form,Sym n ( V ⊗ W ) = (cid:77) ρ (cid:96) n Length( ρ ) ≤ min(dim V, dim W ) S ρ ( V ) ⊗ S ρ ( W ) , (13)where S ρ symmetrizes the indices according to the par-tition ρ and the cutoff on the number of rows in ρ is because one can only antisymmetrize so much before anobject vanishes. Because dim V = 2, for N ≥ ρ ) ≤
2, and so the sum is over partitions ( l , l )with l + l = n .Polynomials in the λ and (cid:101) λ live in a polynomial ring R N = C [ λ, (cid:101) λ ] that can be decomposed as R N = ∞ (cid:77) k =0 Sym k (cid:0) λ ⊕ (cid:101) λ (cid:1) = ∞ (cid:77) k =0 (cid:77) n + (cid:101) n = k Sym n ( λ ) ⊗ Sym (cid:101) n ( (cid:101) λ ) , = ∞ (cid:77) k =0 (cid:77) n + (cid:101) n = k (cid:40)(cid:32) (cid:77) l + l = n V ( l ,l ) ⊗ W ( l ,l , ,.., (cid:33) ⊗ (cid:77) (cid:101) l + (cid:101) l = (cid:101) n V ∗ ( (cid:101) l , (cid:101) l ) ⊗ W (0 ,.., , − (cid:101) l , − (cid:101) l ) , (14)where we have used eq. (13) and assummed N ≥ W ( l ,l , ,.., ⊗ W (0 ,.., , − (cid:101) l , − (cid:101) l ) are reducibleunder U ( N ). Representing W (0 ,.., , − (cid:101) l , − (cid:101) l ) by the conju-gate (blue shaded) diagram, the familiar “box-placing”(Littlewood-Richardson) rules for decomposing U ( N )tensor products can be applied. We take N ≥ N < l , l , (cid:101) l , (cid:101) l ,only one irrep in this decomposition is primary: the oneobtained by simply “gluing” the conjugated Young di-agram for W (0 ,.., , − (cid:101) l , − (cid:101) l ) to the left of the diagram for W ( l ,l , ,.., . We show the resulting Young diagram inFig. 2; it corresponds to the irrep W ( l ,l , ,.., , − (cid:101) l , − (cid:101) l ) ofthe main result, eq. (7).That the other terms in the decomposition of W ( l ,l , ,.., ⊗ W (0 ,.., , − (cid:101) l , − (cid:101) l ) are not primary is readilyverified by appealing to the “box-placing” rules, fromwhich two situations arise. A white box placed below acolumn of N − (cid:80) k (cid:101) λ k λ k = P in the operator; a white box placed below a columnof N − (cid:80) N − k =1 (cid:101) λ N ˙ a (cid:101) λ ˙ ak λ k a = (cid:101) λ N ˙ a P ˙ aa . The presence of a factorof P indicates the operator is a descendant and ensuresthe non-annihilation by K .Because the primary operators are in the U ( N ) irrep( l , l , , . . . , , − (cid:101) l , − (cid:101) l ), which is simply the partitionobtained by adding ( l , l , , . . . ) and ( . . . , , − (cid:101) l , − (cid:101) l ),for N < P ; these get factoredout of the polynomial W ( l ,l , ,... ) ⊗ W ( ..., , − (cid:101) l , − (cid:101) l ) whenan upper λ i index contracts a lower (cid:101) λ i index. For N < e.g. for N = 3, if both l , (cid:101) l (cid:54) = 0 in ( l , l − (cid:101) l , − (cid:101) l ).This rule readily gives eq. (10). An important conse-quence is that scattering operators—Lorentz invariantoperators with derivatives acting on fields ( l = l (cid:54) = 0and (cid:101) l = (cid:101) l (cid:54) = 0)—only occur for N ≥
4; this is thefamiliar statement that Mandelstam invariants are onlynon-trivial for N ≥ U ( N ) rep can be constructed usingSSYT: fillings of the Young diagram boxes with numbersfrom the set 1 , . . . , N , such that numbers weakly increaseacross rows and strongly increase down columns. Thenumber of valid SSYT is equal to the dimension of the U ( N ) rep. The highest weight state fills the k th row withthe number k . One can verify that this state is triviallyannihilated by K ; that all other states in the rep are toofollows because K is a U ( N ) singlet. EXAMPLE: N = 2 & HIGHER SPIN CURRENTS Consider the N = 2 sector H . The holomorphic—built only from λ s—primary operators belong to the U (2)representations ( l , l ) (likewise, the anti-holomorphicprimaries fall into the ( − (cid:101) l , − (cid:101) l ) representations). When l = l ≡ l the harmonics are Lorentz scalars; moreover,they contain only a single operator (they are singlets un-der the SU (2) ⊂ U (2)). For l ≥ l/ . . . (cid:124) (cid:123)(cid:122) (cid:125) l ⇔ ( λ a λ a ) l ∼ ( F ,L ) a ...a l ( F ,L ) a ...a l . Note that to describe indistinguishable particles 1 and 2,we must symmetrize over the indices (1 ,
2) for the bosonic(even l ) case, or else anti-symmetrize for the fermionic(odd l ) case. In both cases, the operator does not vanish.The non-holomorphic harmonics are particularly inter-esting for N = 2, as they correspond to conserved cur-rents like the stress tensor and higher spin currents. Herewe describe them and provide a generating function; fur-ther details can be found in the supplemental material.The non-holomorphic harmonics in H are the U (2)representations ( n, − m ) ⇔ · · · (cid:124) (cid:123)(cid:122) (cid:125) m · · · (cid:124) (cid:123)(cid:122) (cid:125) n (15)with n, m > j , j ) = ( n , m ). Recallthat the conjugated diagrams involve raising the U (2)index on the (cid:101) λ s, → ⇔ (cid:101) λ i ˙ a → (cid:101) λ i ˙ a = (cid:15) ij (cid:101) λ j ˙ a , (16)so that the polynomial encoded by eq. (15) simply sym-metrizes over all of the (raised) flavor indices. We definethis operator J ( n,m ) = (cid:0) J ( n,m ) (cid:1) ( i ...i n + m )( a ...a n )( ˙ a ... ˙ a m ) = λ ( i a · · · λ i n a n (cid:101) λ i n +1 ˙ a · · · (cid:101) λ i n + m )˙ a m . (17) There are n + m + 1 primaries inside this U (2) rep-resentation, corresponding to setting ( i , . . . , i n + m ) =(1 , . . . , , (1 , . . . , , , . . . , (1 , , . . . , , or (2 , . . . , J ( n,m )1 n + m − k k ≡ (cid:0) J ( n,m ) (cid:1) ( n + m − k (cid:122) (cid:125)(cid:124) (cid:123) . . . k (cid:122) (cid:125)(cid:124) (cid:123) . . . )( a ...a n )( ˙ a ... ˙ a m ) . (18)We can readily come up with a generating function forthese states. Let λ a ≡ λ a and λ a ≡ η a and define f ( n,m ) ≡ n (cid:89) i =1 (cid:0) cλ + η (cid:1) a i m (cid:89) j =1 (cid:0)(cid:101) η − c (cid:101) λ (cid:1) ˙ a j , (19)with c ∈ C some arbitrary parameter. The chain rulereadily verifies that K a ˙ a = − ( ∂ λ (cid:101) ∂ λ + ∂ η (cid:101) ∂ η ) a ˙ a annihilates f ( n,m ) . Since c is arbitrary, this further implies that eachterm in the expansion f ( n,m ) = (cid:80) n + mk =0 c n − k f ( n,m ) k is alsoannihilated by K . With a little more effort, one deduces f ( n,m ) = n + m (cid:88) k =0 c n − k (cid:18) n + mk (cid:19) J ( n,m )1 n + m − k k , (20)thereby providing an efficient means to obtain the J ( n,m )1 n + m − k k . Finally, note that P · f ( n,m ) ≡ P a ˙ a f ( n,m ) aa ...a n ˙ a ˙ a ... ˙ a m = 0; this implies P · J ( n,m )1 n + m − k k = 0, i.e. these are conserved currents.As an example, consider the stress-tensor harmonic n = m = 2 → ( j , j ) = (1 , J (2 , are J (2 , = ( λλ (cid:101) η (cid:101) η ) ab ˙ a ˙ b ( − λλ (cid:101) η (cid:101) λ − λλ (cid:101) λ (cid:101) η + λη (cid:101) η (cid:101) η + ηλ (cid:101) η (cid:101) η ) ab ˙ a ˙ b ( λλ (cid:101) λ (cid:101) λ − λη (cid:101) η (cid:101) λ − ηλ (cid:101) η (cid:101) λ − λη (cid:101) λ (cid:101) η − ηλ (cid:101) λ (cid:101) η + ηη (cid:101) η (cid:101) η ) ab ˙ a ˙ b ( λη (cid:101) λ (cid:101) λ + ηλ (cid:101) λ (cid:101) λ − ηη (cid:101) η (cid:101) λ − ηη (cid:101) λ (cid:101) η ) ab ˙ a ˙ b ( ηη (cid:101) λ (cid:101) λ ) ab ˙ a ˙ b (21)corresponding to operators of the schematic form F L F R , ψ L ∂ψ R , φ ∂∂φ , ψ R ∂ψ L , and F R F L .There is something remarkable about this result.These operators carry spin ( j , j ) = (1 , P · J (2 , − k k = 0, reducing thenumber of independent components to five. This is pre-cisely the dimension of the U (2) representation. In fact,as a result of conservation, this happens for all ( n, m ):the number of independent components of J ( n,m )1 n + m − k k isequal to n + m + 1. Mathematically, this follows fromthe reductive dual pair structure and is a consequence ofFrobenius reciprocity, on which we elaborate further inthe supplementary material. DISCUSSION
A very useful extension of the results we present abovewould be a similar systematic understanding of identi-cal particles: an operator containing identical (fermions)bosons lies within the appropriately (anti-)symmetrizedFock space ⊂ H N . In practice, such an operator is se-lected out by applying the permutation group to the( N -distinguishable) operators whose construction we de-tailed. This operation preserves the primary conditionbut mixes states within a U ( N ) representation (in d = 4the permutation belongs to the Weyl group S N ⊂ U ( N )).In this vein, the works [21, 34–36] could be useful.We have focused on the spectrum of free CFTs, andseen how a generalized notion of helicity encodes thisinformation. The other data in CFTs are the OPE co-efficients. What, if anything, does the U ( N ) (or O ( N )in d = 2 ,
3) say about these? Could it be that the OPEcoefficients are related in some way to Clebsch-Gordancoefficients of U ( N )? In this spirit—and discussed brieflyin the supplementary material—we note that the use ofspinors opens up potentially more efficient methods forevaluating OPE coefficients (as compared to traditionalmomentum variables).A supersymmetric version of the oscillator represen-tation is obtained with the inclusion of anticommutingcounterparts to the spinors (see e.g. [37] for such a rep-resentation for N = 4 SYM); it would be interesting toflesh out the reductive dual pair part of this story.Underlying the conformal-helicity duality/reductivedual pair structure described in this work is the sym-plectic action on spaces of polynomials, i.e. on theoscillator representation. We have utilized this for fi-nite N , in particular to decompose H N into irreps ofthe conformal group. Important in this regard wherehow the SU (2 ,
2) and U ( N ) generators were quadraticin the spinors and their derivatives. The question natu-rally arises about an infinite dimensional generalizationwhere the oscillator representation arises as automor-phisms of the field theory canonical commutation rela-tions, [ φ ( x ) , π ( y )] = iδ ( x − y ). (In fact, this infinite di-mensional case is where the oscillator representation wasformally introduced by Segal and Shale [38, 39].) We ex-pect the higher spin currents (which are quadratic in thefields) to play an important role; it would be interestingto see if this allows a more efficient means to constructingthe spectrum and the OPE coefficients of free theories.What more of the U ( N )? Certainly its represen-tation theory is relevant to the discussion of non-renormalisation theories in EFT [40–42], where thegrouping of operators [41] corresponds to those that be-long to the same U ( N ) representation. And we have seenabove how the U (2) encodes properties of higher spin cur-rents at a deep mathematical level; it would be nice togain a physical understanding of this, if indeed such anunderstanding exists. ACKNOWLEDGEMENTS
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1) (whichis the Stiefel manifold V ( R N )). Functions on the sphereare the induction of the trivial representation of O ( N − O ( N ), Ind O ( N ) O ( N − , whose decomposition into irrepsof O ( N ) is Ind O ( N ) O ( N − = ∞ (cid:77) l =0 V ( l, ,..., . (22)The irreps V ( l, ,..., are the “spin l ” reps of O ( N ), i.e. the traceless symmetric rank- l tensors, i.e. the polyno-mials x { i . . . x i l } . These representations have eigenvalue l ( l + N −
2) under the O ( N ) quadratic Casimir. Notethat these show up with unit multiplicity in the decom-position.On the space of polynomials in the x i there is aless obvious SL (2 , R ) action which commutes with the O ( N ). In particular, P = x , K = −∇ , and D = − i (cid:0) x · ∂ + N (cid:1) (which essentially measures the degree ofhomogeneous polynomials), close under commutation togive the sl (2 , R ) algebra. The SL (2 , R ) is the analog ofthe SU (2 ,
2) in the main text, and we have named theabove generators accordingly to highlight this fact. Theharmonic polynomial condition—annihilation by K —determines the lowest weight state of an SL (2 , R ) repre-sentation, with the rest of the states in the representationobtained by repeated applicaiton of P .In complete analogy with the main text, specifying an O ( N ) harmonic polynomial specifies the SL (2 , R ) repre-sentation and vice-versa. This SL (2 , R ) × O ( N ) action isthe prototypical example of a reductive dual pair [26]. This SL (2 , R ) × O ( N ) duality is more than a simplifiedexample of the SU (2 , × U ( N ) duality in d = 4 dimen-sions covered the main text. It is precisely the story for d = 2 dimensions! To see this, recall that massless par-ticles split into left- and right-movers in d = 1 + 1, andthat the (global) conformal group factorizes accordingly, SO (2 , (cid:39) SL (2 , R ) × SL (2 , R ). Working with, say, theleft movers, particles carry non-zero lightcone momentum p − i = p i − p i , where i = 1 , . . . , N is a particle numberindex. In analogy to spinors, the “square root of momen-tum” in d = 1 + 1 is just a simple change of variables, p − i = λ i . The construction of N -particle primaries forfree theories in d = 2 thus boils down to finding the har-monic polynomials in the λ i , the solution to which wegave above. The oscillator representation and conformalrepresentations
Here we provide some basic information about the os-cillator representation, as well as some technical detailsabout the conformal representations encountered in themain text. Among others, we have found the refer-ences [26, 27, 29, 31, 43–49] useful in helping us stitchtogether the following story.Consider a set of n harmonic oscillators of unit massand unit frequency. The Hamiltonian is H = n (cid:88) I =1 (cid:0) q I + π I (cid:1) , (23)with the coordinates obeying the usual canonical com-mutation relations (CCR)[ q I , π J ] = iδ IJ . (24)Considered as a whole, the symplectic group Sp (2 n, R )acts on phase space; it is an automorphism preservingthe CCR, (cid:104) (cid:18) q π (cid:19) , (cid:18) q π (cid:19) (cid:105) = iJ, J = (cid:18) − (cid:19) , (25)where J is the usual symplectic bilinear form. As isreadily checked, the generators of the symplectic alge-bra sp (2 n, R ) are the symmetric, quadratic parts of the The following appendix gives a more general introduction to re-ductive dual pairs. We use λ for the variable name here just to emphasize the con-nection to the main text, but note that λ i ∈ R is not a spinor inthe present context. phase space coordinates,12 (cid:110) (cid:18) q π (cid:19) , (cid:18) q π (cid:19) (cid:111) = q I q J { q I , π J } π I π J . (26)As is standard material, the Hilbert space is spannedby the | n (cid:105) = | n . . . n n (cid:105) states. We can readily see thisby passing to creation/annihilation operators, (cid:18) aa † (cid:19) = 1 √ (cid:18) − i (cid:19) (cid:18) q π (cid:19) , (27)whereupon [ a I , a † J ] = δ IJ and H = (cid:80) nI =1 { a † I , a I } = (cid:80) nI =1 (cid:0) a † I a I + (cid:1) . Analogous to eq. (26), (cid:8)(cid:0) aa † (cid:1) , (cid:0) aa † (cid:1)(cid:9) provide a (unitarily equivalent) realization of the sym-plectic algebra.The Hamilitonian makes apparent that the energyeigenstates are degenerate and fall into representations of K = Sp (2 n, R ) ∩ O (2 n ) (cid:39) U ( n ), which is the maximalcompact subgroup of G = Sp (2 n, R ). The transforma-tions in coset G/K mix states of different energies. Inthis way, the Hilbert space of the harmonic oscillatorsfurnishes a representation of Sp (2 n, R ), known as the os-cillator representation. The abstract action of the symplectic group on the q s and π s (or a s and a † s) is unitarily realized on aspace of square integrable functions. The coordinatespace wavefunctions (cid:104) q | n (cid:105) = h n ( q ) e − q / , with h n ( q ) = h n ( q ) · · · h n n ( q n ) the appropriate product of Hermitepolynomials, are a basis for this space with the stan-dard inner product on R n , (cid:104) ψ | ψ (cid:48) (cid:105) = (cid:82) d n q ψ ( q ) ψ (cid:48) ( q )with the bar denoting complex conjugation. Note that π I → − i∂/∂q I is Hermitian with respect to this innerproduct. On the other hand, the creation/annihilationoperators are realized on a complex space with basis (cid:104) ξ | n (cid:105) = ξ n = ξ n · · · ξ n n n and inner product (cid:104) ψ | ψ (cid:48) (cid:105) = (cid:82) d n ξ d n ξ e −| ξ | ψ ( ξ ) ψ (cid:48) ( ξ ). Note that with this measure a I → ∂/∂ξ I is the adjoint to a † I → ξ I .We call the representation in terms of q s and π s theSchr¨odinger representation, and that in terms of the a s and a † s the Bargmann-Fock representation. Theintertwining operator which maps us between theserepresentations— i.e. implements the transform ineq. (27)—is given by the Bargmann-Fock transform (in As a result of the 1 / Sp (2 n, R )—and hence is often called the metaplectic rep-resentation. This representation is also reducible, splitting intoan even and odd component (schematically, this is because thegenerator a † a † causes jumps of two in the occupation number). Historically in physics, symmetries where the Hamiltonian wasa generator in a larger symmetry group were called spectrumgenerating symmetries or dynamical symmetries.
0a more physics-oriented language, this is the operator (cid:82) d n ξ d n ξ (cid:104) q | ξ (cid:105) (cid:104) ξ | and is commonly obtained via the in-troduction of coherent states). That such a map exists isguaranteed by the Stone-von Neumann theorem, whichessentially states that there is only one unitary represen-tation of the CCR, so that any two realizations must beunitarily equivalent.The symplectic group contains many subgroups. A re-ductive dual pair is a pair of groups G × G (cid:48) ⊂ Sp (2 n, R )such that G and G (cid:48) are maximal commutants with re-spect to one another, i.e. G (cid:48) is the largest subgroup in Sp (2 n, R ) that commutes with G . When restricted to G × G (cid:48) , the reduction of the oscillator representation of Sp (2 n, R ) is such that the irreps of G appearing deter-mine those of G (cid:48) and vice-versa.For our applications of decomposing the N -distinguishable-particle Hilbert space of free CFTs— i.e. decomposing H N = H ⊗ N into irreps of the conformalgroup—the relevant dual pairs are d = 2 : SL (2 , R ) × O ( N ) ⊂ Sp (2 N, R ) , (28a) d = 3 : Sp (4 , R ) × O ( N ) ⊂ Sp (4 N, R ) , (28b) d = 4 : SU (2 , × U ( N ) ⊂ Sp (8 N, R ) . (28c)In each case, the conformal primaries of H N are the har-monics of the relevant Stiefel manifold, d = 2 : V ( R N ) (cid:39) O ( N ) /O ( N − , (29a) d = 3 : V ( R N ) (cid:39) O ( N ) /O ( N − , (29b) d = 4 : V ( C N ) (cid:39) U ( N ) /U ( N − . (29c)Analogous to eqs. (7) and (22), for completeness we in-clude the decomposition of functions on the Stiefel man-ifolds into irreps of O ( N ) or U ( N ), i.e. the spectrum ofprimary operators in H N :Ind O ( N ) O ( N − = ∞ (cid:77) l =0 V ( l, ,..., , (30a)Ind O ( N ) O ( N − = (cid:77) l ≥ l l − l +1 (cid:77) A =1 V A ( l ,l , ,..., , (30b)Ind U ( N ) U ( N − = (cid:77) l ≥ l (cid:77) (cid:101) l ≥ (cid:101) l ( l − l +1)( (cid:101) l − (cid:101) l +1) (cid:77) A =1 V A ( l ,l , ,..., , − (cid:101) l , − (cid:101) l ) . (30c)A few comments. As in the main text, we have labeledthe finite-dimensional representations by their associatedpartitions/Young diagrams; recall that, in contrast to U ( N ), for O ( N ) these realize tensors which are traceless.For d = 3 , A accounts for the multiplicity It is possibly true that a similar mechanism works in d = 6. V L shows up with in the decomposition; in accordancewith the dual pair structure, it is simply the dimensionof the Lorentz representation ( SL (2 , R ) or SL (2 , C ) in d = 3 ,
4, respectively). (The d = 3 spinorial realizationof the conformal group is discussed below.) For d = 3 , N ≥
4, with exceptionalcases for
N < d = 4; the d = 3 ana-logue is straightforward to work out. For d = 2 , d = 4, we had toidentify the primary in a decomposition of a product of U ( N ) representations (see discussion following eq. (14));the analogue in d = 2 , GL ( N, R ) representation labeled by par-tition L to O ( N ). Just as the d = 4 primary was the“leading term” in the decomposition, so it is in d = 2 , O ( N ) irrep labeled by the same par-tition L (but now, as an O ( N ) irrep, it corresponds to atraceless polynomial). ∗ ∗ ∗ With the above said, we now wish to make a few com-ments on the conformal representations encountered inthe main text. We give this discussion in the context ofthe d = 3 case in order to provide some details for thatstory; the analogous statements for d = 4 (main text)and d = 2 (supplemental material on spherical harmon-ics) are easily generalized.In d = 3 a massless momentum can be written as p ab = λ a λ b with λ a a two-dimensional real spinor. For N particles, the representation of the conformal algebra sp (4 , R ) (cid:39) so (3 ,
2) is given by P ab = (cid:88) i λ ia λ ib , (31a) K ab = − (cid:88) i ∂ ai ∂ bi , (31b) D = (cid:88) i { λ ia , − i∂ ai } = − i (cid:88) i (cid:0) λ i · ∂ i + 1 (cid:1) , (31c) M ba = (cid:88) i (cid:18) { λ ia , − i∂ bi } − δ ba { λ ic , − i∂ ci } (cid:19) = − i (cid:88) i (cid:18) λ ia ∂ bi − δ ba λ i · ∂ i (cid:19) . (31d)This is the Schr¨odinger representation where the “canon-ical momentum” to the λ ia is π ai = − i∂ ai .For free QFT in d = 3 the scalar and fermion are theonly massless particles which are conformal; respectively, See, e.g. , [50] for a discussion of restricting GL ( N ) reps to O ( N ). O ( N = 1) (cid:39) Z .As the theory is free, we can introduce fields containingboth the positive- and negative-energy components, e.g. for the scalar φ ( x ) = (cid:90) d λ π ) (cid:16) e − i λ a λ b x ab A λ + e i λ a λ b x ab A † λ (cid:17) , (32)and for the fermion ψ a ( x ) = (cid:82) d λ π ) (cid:0) λ a e − i λλx A λ +h.c. (cid:1) . An operator is just some polynomial in fieldsand their derivatives; acting on the vacuum it creates astate O ( N ) ( x ) | (cid:105) = (cid:90) (cid:32) N (cid:89) i =1 d λ i π ) (cid:33) e i (cid:80) i λ ia λ ib x ab × f O ( λ ia ) | λ · · · λ N (cid:105) . (33)Here, we took O ( N ) to contain N fields. f O ( λ ) is somepolynomial in the spinors; as discussed in the main text,if O ( N ) is a conformal primary, then the polynomial isharmonic: Kf O = 0. From here one can readily calculate2-point functions ( n -point functions follow with obviousgeneralization), (cid:104)O ( N ) ( x ) O (cid:48) ( N (cid:48) ) ( y ) (cid:105) = δ NN (cid:48) (cid:90) (cid:32) N (cid:89) i =1 d λ i π ) (cid:33) (34) × e − i (cid:80) i λ ia λ ib ( x − y ) ab f O ( λ ) f (cid:48)O (cid:48) ( λ ) . In order to discuss unitarity of the representation (andtherefore the hermiticity of the generators in eq. (31)), weneed to define the inner-product. A natural candidate isto define the in states at t = i where x ab = iδ ab (and outstates at the complex conjugate t = − i ). At this point,the states have the appropriate Gaussian wave functionfactor for harmonic oscillators, (cid:104) λ · · · λ N |O ( t = i ) (cid:105) = f O ( λ ) e − (cid:80) a,i λ ia λ ia . (35)This is the scheme in [31, 46] and in modern literaturethis goes by the name “NS quantization” [51]; in additionto [51] (the discussion of which is in Euclidean space), agood discussion in Lorentzian signature is the appendixof [52].Imaginary time is familiar from the i(cid:15) prescriptionto guarantee causality. However, it raises the ques-tion of where the conformal representation actually“lives”. Physically, we want quantum fields to live onMinkowski space, or its appropriate covering if we have The measure is simply the transform of d p (2 π ) δ ( p ) θ ( p ) intospinor variables; we note that the additional factor of 1 / / vol(little group) = 1 / vol( Z ). non-integer scaling dimensions [46]. This is manifest inthe standard construction of conformal representationsas finite-component field representations `a la Mack andSalam [47], which constructs them as induced represen-tations on G/ P = SO ( d, / [ ISO ( d − , (cid:110) R ] (cid:39) M d ,where M d is d -dimensional Minkowski space and theparabolic subgroup is generated by Lorentz transfor-mations, special conformal translations, and dilatations,which all preserve the origin of Minkowski space ( i.e. thecoset is generated by translations, which obviously giveall of Minkowski space).On the other hand, it is natural to consider (induced)representations which live on G/K , where K = SO ( d ) × SO (2) is the maximal compact subgroup of the conformalgroup. G/K is a Hermitian symmetric space; physi-cally it is a certain analytic continuation of Minkowskispace (dim(
G/K ) = 2 d ). In particular, it is an ana-logue of an “upper-half space”. For example, in d = 2 SL (2 , R ) /SO (2) (cid:39) Sp (2 , R ) /SO (2) (cid:39) H is the upper-half plane H , while in d = 3 it is the Siegel upper-halfspace H , Sp (4 , R ) (cid:46) Sp (4 , R ) ∩ O (4) (36)= (cid:110) z ∈ Mat × ( C ) : z = z T , Im z > (cid:111) . That is, z ab = x ab + iy ab consists of complex, symmetric2 × Importantly, Minkowski space lives at the bound-ary Im z →
0. These upper-half spaces are also knownas “tube domains”; they are the same tube domains ap-pearing in e.g. [53]; here we see that they naturally ariseas a homogeneous space of the conformal group. An important result, originally due to Harish-Chandra,is that when
G/K is a Hermitian symmetric space, then G possesses holomorphic discrete series representations,which are K -valued functions that have an analytic ex-pansion on all of G/K . In particular, the function can beexpanded in a Taylor series in z , O ( z ) ∼ (cid:80) k [ ∂ k O ](0) z k . For the spinorial form of the conformal groups in eq. (28), K = SO (2) (cid:39) U (1), Sp (4 , R ) ∩ O (4) (cid:39) U (2), and S ( U (2) × U (2)). For the spinorial conformal groups, G = SL (2 , R ), Sp (4 , R ), SU (2 , G acts on (real or complexified) Minkowski space viafractional-linear transformations. That is, for g = (cid:0) A BC D (cid:1) ∈ G (with A..D × d = 3 , g : z (cid:55)→ ( Az + B )( Cz + D ) − . One readily verifies that this gives thefamiliar conformal transformations on the d -vector z µ . (Veryschematically, Lorentz transformations and dilatations corre-spond to elements (cid:16) A A − (cid:17) , translations to (cid:0) B (cid:1) , and specialconformal transformations to (cid:0) C (cid:1) ; note that the groups may begenerated from Lorentz transformations + dilatations + trans-lations + the inversion element (cid:0) − (cid:1) which takes z → − z − .) For general d , SO ( d, / ( SO ( d ) × SO (2)) basically complexifiesMinkowski space via z µ = x µ + iy µ with y µ in the open forwardlight cone ( y > y > functions on G/K , and when wetake them to the boundary they are realized as distribu-tions on Minkowski space, e.g. [48, 49].Now, for the Schr¨odinger representation in eq. (31) thegenerator δ ab (cid:0) P + K (cid:1) ab = P + K = (cid:88) i,a (cid:0) λ ia λ ia + π ai π ai (cid:1) (37)is the Hamiltonian of the harmonic oscillators; thecombination P + K is called the conformal Hamilto-nian [31, 46, 51]. Per our earlier discussion, it is obvi-ously invariant under the maximal compact subgroup ofthe conformal group, K = Sp (4 , R ) ∩ O (4). The confor-mal group G = Sp (4 , R ) acts transitively on the upperhalf-space H in eq. (36); geometrically, K stabalizes thepoint i ( ) and the transformations in G/K act effec-tively, generating all of H from the point i , whenceeq. (35).The upper half spaces under consideration are un-bounded. They can be compactified by mapping themto generalized open unit disks. The map that imple-ments this transformation is none other than the Cayleytransform in eq. (27). For example, in d = 3 we have H → D = (cid:110) z ∈ Mat × ( C ) : z = z T , − z † z > (cid:111) . (38)Importantly, just as Minkowski space laid at the bound-ary of the upper-half space, at the boundary of the opendisk sits conformally compactified Minkowski space. The passage to the Bargmann-Fock representation viathe Cayley transform aligns closer to the typical wayconformal representations are discussed in the modernliterature with radial quantization in Euclidean space, e.g. [51, 54]. In particular, the point i in the upper-half space is mapped to the origin of the unit disk, sothe “in-states” lie at the origin. Moreover, the gener-ators P BF ∼ a † a † and K BF ∼ aa are now adjoints ofone another P † BF = K BF , while the dilatation operator D BF ∼ (cid:80) (cid:0) a † a + (cid:1) is the Hamiltonian. At the boundary z † z = 1, so z is a unitary, symmetric matrix.We can parameterize it as u (cid:16) t + it it it t − it (cid:17) with | u | = 1 a phaseand t i ∈ R with t + t + t = 1. Hence, the boundary istopologically S × S ; for interacting theories the timelike circleneeds to be unwrapped for obvious causality reasons, S → R —in this way higher (possibly infinite) sheeted coverings of theconformal group come into play. We emphasize that the special conformal generator is a general-ized Laplacian in both the Schr¨odinger and the Bargmann-Fockrepresentations, and so the harmonic polynomials take the exactsame form in either representation.
One can, of course, also construct the inner product inthe Bargmann-Fock representation and use this realiza-tion to compute n -point functions. This is straightfor-ward. We wish to mention that in both the Schr¨odingerrepresentation (eq. (34) and its generalization to n -pointfunctions) and the Bargmann-Fock representation, one isessentially dealing with Gaussian integrals. This is to becontrasted against the “usual” momentum variables— i.e. no spinors—where we would be dealing with Fourier in-tegrals e ip µ x µ . As physicists, Gaussian integrals are niceras the machinery of Wick contractions and such is avail-able. We also point out that the Bargmann-Fock repre-sentation may be even cleaner than the Schr¨odinger rep-resentation; although much remains to explore, at leastfor 2-point functions, many cross-terms in the expansionof the polynomials f O f (cid:48)O (cid:48) are trivially zero by homogene-ity conditions (in the Schr¨odinger representation thesecross-terms are generally non-zero, but the sum of allterms gives the appropriate cancellations).As a final point, we discuss finite conformal trans-formations, i.e. how the infinitesmal representation ofthe conformal algebra g integrates to a representationof the group, G = exp (cid:0) i g (cid:1) . The finite transformationsof translations, Lorentz transformations, and dilatationsare straightforward. The tricky one is special conformaltransformations.Again, we proceed with our discussion in d = 3 where G = Sp (4 , R ). First, note that all of G may be gener-ated from translations, Lorentz transformations, dilata-tions, and the element J = (cid:0) − (cid:1) . J acts like theinversion element, sending Minkoski space coordinates J : x → − x − . The action of J on the oscillator rep-resentation is easy to deduce. For N particles, J actsby (recall π ai = − i∂ ai ) (cid:0) − (cid:1) (cid:16) λ ··· λ N π ··· π N (cid:17) , so that J es-sentially exchanges λ and π . In other words, up to amultiplicative constant, J is realized as a Fourier trans-form! Recall that harmonic oscillator eigenstates—Hermite polynomials times a Gaussian factor—transforminto themselves under Fourier transforms. An importantfact, e.g. [29], is that for f ( λ ) e i (cid:80) i λ i λ i x with f ( λ ) a har-monic polynomial, the Fourier transform returns a har-monic polynomial. This implies the correct transforma-tion of a primary operator O ( x ) in Minkowski space [29].To the best of our knowledge, it is not known whatthe unitary transformation/operator is which implementsa finite special conformal transformation in momentumspace for general (positive energy) representations of theconformal group. Above we explained how this trans-formation is realized on half-integer scaling dimension This is a unitary transformation by Plancherel’s theorem. (Infact, logically speaking, we can use the oscillator representationto prove unitarity of the Fourier transform [43].) The infintesmal transformations are well-known and easy to ob-tain: just Fourier transform the position space generators in [47], i.e. when there are anomalous dimensions,so the scaling dimension is not necessarily half-integer).In this spirit, we note that the connection of finite spe-cial conformal transformations—and, in fact, the wholesetup of the oscillator representation—is reminiscent oftwistors, where one might hope to interpret e i λ a x ab λ b as e iλ a π a with π a = x ab λ b / Sp (4 , R ) has an easy to under-stand linear action on ( λπ ), so that our setup feels quiteclose to a spinorial form of “embedding space” e.g. [51] (aspinorial form of the projective null cone), and twistorsare probably the natural language to formalize this. Spin current
Here we elaborate on (i) a convenient form for ma-nipulating the currents J ( n,m )1 n + m − k k and (ii) a mathemati-cal reason—Frobenius reciprocity—for why the currentsmust be conserved.As all of the Lorentz indices are symmetrized, to sim-plify matters we introduce an index-free notation whereinwe contract the J ( n,m ) , f ( n,m ) , etc . with auxiliary spinors ξ a and (cid:101) ξ ˙ a (similar constructs are frequently used in theCFT bootstrap literature, e.g. [57]). We denote con-tracted objects with a hat, e.g. (cid:98) f ( n,m ) ≡ ξ a · · · ξ a n (cid:101) ξ ˙ a · · · (cid:101) ξ ˙ a m f ( n,m ) a ...a n ˙ a ... ˙ a m . (39)The uncontracted object is recovered simply by differen-tiating, f ( n,m ) a ...a n ˙ a ... ˙ a m = 1 n ! (cid:18) ∂∂ξ a (cid:19) n m ! (cid:18) ∂∂ (cid:101) ξ ˙ a (cid:19) m (cid:98) f ( n,m ) , (40)which ultimately is the same as instructing one to sym-metrize the object into which the auxiliarly spinors arecontracted, 1 n ! (cid:18) ∂∂ξ a (cid:19) n (cid:0) ξ b (cid:1) n = δ b ( a · · · δ b n a n ) . (41)For a simpler notation, we further define x ≡ (cid:98) λ ≡ ξ a λ a , y ≡ (cid:98) η ≡ ξ a η a ,x ≡ (cid:98)(cid:101) λ ≡ (cid:101) ξ ˙ a (cid:101) λ ˙ a , y ≡ (cid:98)(cid:101) η ≡ (cid:101) ξ ˙ a (cid:101) η ˙ a . (42) i.e. − i∂ x → p , x → i∂ p . The infinitesmal form and their impliedWard identities has been the subject of intensive study in recentyears, e.g. [52, 55, 56]. With these preliminaries, (cid:98) f ( n,m ) takes the form of asimple polynomial (cid:98) f ( n,m ) = ( cx + y ) n ( y − c x ) n (43a)= n + m (cid:88) k =0 c n − k (cid:18) n + mk (cid:19) (cid:98) J ( n,m )1 n + m − k k , (43b)where the second equality comes from eq. (20). By usingthe binomial expansion ( a + b ) n = (cid:80) i =0 (cid:0) ni (cid:1) a n − i b i , onecan expand (cid:98) f ( n,m ) and match powers of c to obtain anexpression for the currents (cid:98) J ( n,m )1 n + m − k k . The result is apiecewise expression, which can be written as a hyperge-ometric series. This is not particularly illuminating, sowe omit the formula. More to the point, a main purposeof this discussion was to introduce the index-free nota-tion, which often renders expressions more amenable forcomputation.Now we proceed to give a mathematical justificationfor why the non-holomorphic primaries in H must beconserved currents. This is a consequence of Frobeniusreciprocity, which roughly codifies the idea that induc-tion and restriction are inverse actions. In more detail,consider the induced representation Ind GH W of a repre-sentation W of H ⊂ G ( i.e. a H -valued function on thecoset G/H ). For compact groups, the finite-dimensionalirreps of G provide a basis for such functions. Frobe-nius reciprocity states that the multiplicity of a repre-sentation V of G appearing in Ind GH W is equal to themultiplicity for which W appears in the restriction of V to H : mult( V, Ind GH W ) = mult( W, Res GH V ). For theproblems we consider, this multiplicity is reflected thatthe harmonic polynomials carry both U ( N ) indices andLorentz indices, i.e. the multiplicity is equal to the di-mension of the Lorentz representation.For the currents in the main text, these are in U (2)representations labeled by Young diagrams with n + m boxes. Moreover, the Stiefel manifold is just U (2) in thiscase, U ( N ) /U ( N − → U (2) / U (2). Therefore, sincethe restriction of any representation to the trivial group H = 1 is simply the dimension of the representation,Frobenius reciprocity tells us that the multiplicity thatthis representation appears with is n + m + 1. At firstglance it seems there is a paradox since the dimensionof the SL (2 , C ) representation is ( n + 1)( m + 1). Thisis resolved by the fact that the currents are conserved: (cid:0) P · J ( n,m )1 n + m − k k (cid:1) a ...a n − ˙ a ... ˙ a m − transforms in the spin( n − , m − ) rep of SL (2 , C ). Hene P · J ( n,m )1 n + m − k k = 0 givesdim( n − , m − ) = nm constraints, leaving ( n + 1)( m +1) − nm = n + m + 1 independent components. A simple example are spherical harmonics. The traceless sym-metric tensors ( l, , . . . ,
0) of O ( N ) all contain the trivial repre-sentation of O ( N −
1) exactly once, e.g. an O ( N ) vector splitsinto a vector plus a singlet of O ( N −−