JHEP04(2018)052 ial and Springer April 11, 2018 March 26, 2018 : mal four-point : January 30, 2018 : Published Accepted allows us to reinterpret Received action of left/right invariant ar, we construct the potential l seed conformal blocks. Our heoretic objects. In particular, inning blocks in any dimension roigrequations arehr¨odinger mapped rnal fields of arbitrary and chnology, herland quantum mechanics with ] is extended to arbitrary dimensions 1 Published for SISSA by https://doi.org/10.1007/JHEP04(2018)052 [email protected] b , . 3 and Evgeny Sobko a 1711.02022 The Authors. Conformal and W Symmetry, Conformal Field Theory, Different c

In this paper we develop further the relation between confor , [email protected] Nordita, Stockholm University andRoslagstullsbacken KTH 23, Royal SE-106 Institute 91 of Stockholm, Te Sweden E-mail: DESY Hamburg, Theory Group, Notkestraße 85, 22607 Hamburg, Germany b a ArXiv ePrint: Algebraic Geometry, Field Theories in Higher Dimensions vector fields on the conformal group. Keywords: approach furnishes solutions ofdimension Casimir equations in for terms exte ofstandard operations functions on conformal on blocks thewe in shall terms conformal discuss of group. group-t the relationthrough between This differential the operators construction acting of sp on seed blocks and the and to the case offor any boundary set two-point of functions. external tensor In fields.explicitly particul to Some of the the resulting known Sc Casimir equations for 4-dimensiona Open Access Article funded by SCOAP blocks involving external spinning fields andmatrix-valued potentials. Calogero-Sut To this end, the analysis of [ Abstract: Volker Schomerus From spinning conformal blocksCalogero-Sutherland to models matrix JHEP04(2018)052 7 9 1 3 7 12 21 23 23 14 16 19 theory eight ltrami ] which uncovered 9 relevant mathemati- – 7 G relators in which all for s rn [ l conformal blocks seemed as a long history that goes y. It was realized from the n of conformal field theory ions of correlation functions he conception of the modern aints. Even though the numerical lving tensor fields, and in particu- G A – 1 – ]. This insight paved the way for the conformal boot- ], the demands of the scientific community gradually in- 3 , 10 2 ]. Unfortunately, there was no comprehensive mathematical 6 – 4 3.1 The metric3.2 on the conformal Formula for group 3.3 Laplacian on the Example: torus Casimir equation for 4-dimensional seed block shifting operator many highly non-trivial factsnumerical about bootstrap these program functions. [ creased. With t While Dolanexternal and fields Osborn are had scalar, focused itlar is the on clear stress blocks tensor, that for provide correlators important cor invo additional constr strap programme [ of conformal blocks atcal the background did time not anddormant yet we until exist. know the today widely The recognized that entire papers even subject of the of Dolan globa and Osbo provide a very cleanfrom way the to dynamical separate content the [ kinematical skeleto 1 Introduction The theory of conformalback partial almost waves and 50 conformal yearsbeginning blocks to of h early the studies subject in that conformal conformal field theor partial wave expans C Compact picture for principal series representations of 6 Discussion A Comparison with Casimir equations forB 4d seed Comparison blocks with Casimir equations for boundary two-point function 4 Boundary two-point functions 5 Harmonic analysis view on the diagonalization, seed blocks and w 3 Calogero-Sutherland Hamiltonian as a radial part of Laplace-Be Contents 1 Introduction 2 The space of conformal blocks JHEP04(2018)052 ], ], ti- 13 40 – – 11 38 3. In the utherland 6 d . Similarly, we ] and then later e matrix-valued A 33 – 31 .[ we have sketched above. l review how spinning con- pins. The general expres- lies to spinning four-point riate hypergeometric func- = 4. This will allow us to ughly around the same time gero-Sutherland Hamiltoni- ral case and work out a few ocks and the bootstrap was cends from the Laplacian on logero-Sutherland potentials ed to the study of these new d ls and group theory had been nts in the theory of spinning of 4-dimensional seed blocks. e associated eigenvalue prob- uite recently, see e.g. [ s. On the other hand, in the erefore also hope that some of ] in appendix therland Hamiltonians. These os and take values in the space on the conformal group. Then ed to dimension tions, derive the corresponding amiltonian for a single particle 20 operator, the potential takes the intriguing dualities and deforma- g blocks. For somewhat technical ]. 26 – ] and developed further e.g. in [ ]. As we will see below, the application 14 ] for certain quotients of compact groups. 37 42 44 – ] as an interacting multi-particle (or multi- , 35 29 – 43 – 2 – 27 ] on spinning seed blocks in 4-dimensional conformal 20 ]. The investigation of these models uncovered an extremely 30 ] and references therein, as well as many others, Calogero-S ]. This observation makes an enormous body of recent mathema 34 41 ) we derive takes the form of a operator Schr¨odinger with som 3.19 ] we described a general algorithm to construct certain Calo The plan of this paper is as follows. In the next section we wil The present work lies at the intersection of the two subjects There is another area of theoretical physics that was born ro 1 we explain how the Casimirthe equation conformal for group conformal and blocksblocks des work out in an any explicit dimensionsion formula and that ( with app any assignment of external s formal blocks can be obtained from vector-valued functions of tensor structures.After The insertion latter into the is general constructedform expression for for of the the a Casimir case matrix-valuedlem Calogero-Sutherland is potential. shown to Th bealso equivalent discuss to the the case Casimir of equation (spinning) of boundary [ two-point func potential that is similar to theThis one Hamiltonian found acts in on [ functions that depend on cross rati field theory. of such techniques goescontext well of beyond spinning scalar blocks, four-pointappear that new block have classes received little ofthe attention matrix-valued methods so far. form Ca the Onequantum conformal may th mechanical bootstrap models. could be transferr tions. Even though a linkobserved between by Calogero-Sutherland mode Olshanetzki andit Perelomov was in not [ untilfirst last pointed year out that [ thecal relevance results for available for conformal conformal bl field theory [ dimensional) generalization of thein famous potential a 1-dimensional [ P¨oschl-Teller H rich structure including spectrum generatingtions. symmetries, Through the seminalof work Cherednik, of see Heckman [ andmodels Opdam, eventually see gave e.g birth to the modern theory of multiva the general challenge hasconformal boosted blocks significant over new the developme last few years, see e.g. [ bootstrap for spinning correlators has only been explored q In [ ans with matrix-valued potentials for thereasons theory of our spinnin general analysis and the examples were restrict as the conformal bootstrap, namelyHamiltonians the study were of first Calogero-Su written down in [ examples of relevant matrix Calogero-Sutherland models in present work we want to overcome this restriction in the gene make contact to the recent work [ JHEP04(2018)052 en are . It et of (2.1) (2.3) (2.2) G G ⊂ -dimensional d DR admits Bruhat o ) = G h ( ∈ . K f h ) ,µ 1 tages over conventional s and special conformal ∆ s entirely universal, i.e. − π r ( V R, y relations between blocks µ group al group in a equations descend from the ∈ λ ∈ fields as in this work, or is of the conformal group f conformal four-point blocks ion, also known as conformal al spinning blocks from scalar ∆ tions on the conformal group element eats all variables on the same and on an auxiliary space that ce with the Casimir equations n the other hand, our approach e ons ,µ 1 ) is a subgroup of rotations and s a lower dimensional subspace of zero ∆ nformal blocks which involve two G, f D, r d π ) = ∈ ∈ ′ ) is given by the left regular action of ,µ ] and the weight shifting operators that hndr = SO( ∆ ( N, d π 46 f R , | ∈ ,V , h, h µ ) 20 ′ ,µ V , h ∆ – 3 – 1 π → ˜ 19 + 1). Almost any − V N, n , h G ( ∈ , d : 13 f . ˜ n f B Hom( n ) = ′ → = h ]. All these possess a simple origin in group theory which = SO(1 G ]( 23 of an irreducible representation of the rotation group. Giv : G f nndr, ) ) µ ,µ ,µ h = ˜ ( ∆ (∆ G/NDR h along with some consequences. In particular, we describe a s π ,µ Γ ∆ 5 and appendix π ∼ = [ 4 1) is a dilatation subgroup, ,µ , ∆ π V = SO(1 ], the concept of seed blocks [ D are two abelian subgroups that are generated by translation 20 ], see section Finite dimensional irreducible representations of the sub Our approach to spinning blocks has several important advan N The set of elements that do not have such representation form 45 1 , ˜ the conformal group N transformations, respectively. and the representation where Euclidean space is given by 2 The space of conformalIn this blocks section we willas review a the space construction of equivariant of functions the on space the o group. The conform footing. This throws a newones light [ on the constructionwere of introduced speci recently inwe [ outline in section seed blocks for conformal field theories in any dimension. Laplacian(s) on the conformalmanifest, (super)group. see further comments This in the makesrealizes concluding man solutions section. of O theitself. Casimir In equations comparison in withsets terms conventional of realizations of coordinates, of namely func co coordinatesserves on to embedding space encode spin degrees of freedom, our realization tr ones. On theregardless one hand, of as the anextended setup, approach to to whether supersymmetric Casimir models it equations and involves it defects, local i the spinning Casimir can be realised on the following space of equivariant functi measure. decomposition: in [ Calogero-Sutherland model, and demonstrate the equivalen these data we can induce an irreducible representation parametrized by the eigenvalue ∆weight, of and the by the generator weight of dilatat JHEP04(2018)052 . ) ) n µ 4 G d d . π 4 of ctor of (2.5) (2.4) ) ⊗ 2 3 G π ]. Let us ,..., ⊂ 1 ]. It states ⊗ of external G D 2 = 1 47 µ π ) and Spin( ⊂ ∈ of the rotation d , i ) ⊗ i R on the conformal µ λ π 1 ( ∈ π ) that is twisted by d within the maximal r . As a vector space ) ion } -orbits. In this sense, D ,µ rotations group SO( blocks from [ ) can be realised on the wrw , w (∆ G/NDR ( e applicable to reflection 1 Γ nclude external fermions. µ ) for 2.2 { ocks is given by the space polynomial in ∆, they can NDR h heorem 9.4 of [ ∈ ( rmations laws ld theories on a space with esentations f f n between SO( ) = ) to fermionic fields and spinorial r ) for on of the Calogero-Sutherland 1 ions ( ( h ′ . − ( µ r f ! ( ) ′ j j λ λ µ owever they will not appear in our discussion. ∆ -valued functions on the d-dimensional − ⊗ µ i ) V sinh cosh 1 (∆ ). Correspondingly, the values our label ) rather than spinorial representations, i.e. λ − λ d λ d e r ( i of restricted Weyl group. The latter is given by µ – 4 – sinh cosh . ′ )) = w µ λ

( ) = as hd hr ) = ( ( ) as a space of λ f f D (

d ∈ 2.2 ′ j µ d V is a normalizer of the dilation subgroup ) belongs to the unitary principal series representations . In what follows we will assume that ∆ is of this form. Since -invariants on the four-fold tensor product ′ ⊗ R + 1) and it consists of two elements R G i 2.3 µ d ∈ V of with and we add the prime in order to stress that we consider this ve c µ G → ) V where ) 4 . Let us add a few comments on the representation of the rotatio j G π with : ,π i /R ′ ⊗ π f G/K ( ic 3 R Γ ( π ∼ = = denotes the finite dimensional carrier space of representat 2 + G/NDR from the values it assumes on representatives of the ⊗ ) j . We wrote elements µ j 2 d/ π G R V π ,π i ⊗ π The representation ( We are now prepared to review the construction of four-point coincides with ⊗ G/K ( There are also unitary discrete and supplementary series, h i 2 ′ Γ 1 π µ π does not matter, except for the choice of admissible transfo fields. The latter can simply assume more values if we want to i rather than its universal covering group Spin( assumes correspond to representations of SO( we focus on blocks for bosonicrepresentations external fields. is The extension straightHamiltonian forward. is local Since on our the constructi conformal group, the distinctio the conformal group.( By definition, the space of conformal bl In order to construct the space of such invariants we employ t Lorentz signature. all equations for conformalbe blocks continued we to shall arbitrarypositive derive ∆. Euclidean below theories In are as this well way, all as our unitary equations conformal ar fie group we can also think of the space ( group. Throughout the bulk of this work, we will consider the that the tensor product offollowing two space principal of series equivariant representat functions pick four external fields which are associated with four repr quotient if ∆ = group Here, Using the equivariance law we can reconstruct any function where we used the prime symbol for the representation space as a carrier of the representation conjugation with the nontrivial element the quotient compact subgroup SO( V JHEP04(2018)052 . ) 3 B to ) K ′ 4 2.5 . acts diag f (2.7) (2.8) (2.6) V on of ) are ∈ ) of the = on the G ⊗ , k 2.7 2 g G 3 π o V ⊂ K ⊗ on any such ⊗ G 1 . D resentation is through right ′ ∈ θξθ, θ 2 π ) f ⊂ r ). r V ∈ 4 ( π ) ′ 4 R , k ⊗ ) of the subgroup that is isomorphic l 2.2 λ ) = µ ′ 4 ⊗ ( k 1 ∈ ξ ). e following we shall not µ d by elements V K | d 3 ⊗ ph in a bit more de- r educe a function π ) in the space ( ⊗ × h r SO( ( 3 , f or Cartan decomposition K 3 × ) uct µ ) as Θ( h , respectively, according to b, µ 1) ( the form ( bλ ′ 4 , ) for . In this case, the underlying G , the conformal group 2 ) for − f µ ion of h ) h − V ( its orthogonal complement. To r ( KAK e . f k -invariants ( f = ( ⊗ p ) ( ) G/K 3 r 1 B ( µ ) = = Lie( R . Given the value of ) we can also think of the space ( ∆ ′ 2 r V − ⊗ R G ) µ g 2 ) and SO(1 2.2 λ ) = d ) and l ( ⊗ (∆ ∈ k d ) λ R O( K ( ( ) and r ], e ξ V ′ 2 L ( 1 R 1 µ µ ) = orbits of ⊗ , ) = and – 5 – h = Lie( 1 1 ) 2 ) may not be independent. In fact, it turns out that ′ − ) = r k K ) to possess a unique extension to the entire orbit, λ µ r ( a ( V G on the entire orbit. But this might meet an obstacle . Note that the elements ( rh d ′ 2 hk a, µ × l 4 ( ( ⊗ ), we arrive at the following realization of the space f µ f k 1 f f on ∆ ( K µ = ( ⊗

2.5 f where V ) ′ 2 | − K -invariants. L µ r p = ), we can realise the two-fold tensor product † 3 ( R V G 1 L V ⊕ V µ 2.5 of the ⊗ k ⊗ does no longer admit an action of the conformal group as one 1 aλ = ∆ G µ 2 -dimensional. Once again we only used the right regular acti L = e d V b with V g ) -valued function on the quotient ⊂ . In order for G/K ′ j of four-point invariants [ → , 2 \ µ possesses a nontrivial stabilizer subgroup in → G is 2 ) = A LR 1 as ) ) act on V G K ( 1) act on states by multiplication with a scalar factor. In th r 2 a G ) d , G ⊂ ) ∆ ∈ 4 ,π G ) through left regular transformations. The associated rep G : ⊗ : λ 1 \ π ( i a − π 2) f f 1). The map Θ determines a decomposition of the G/K K ( µ d SO( 2.5 2 ⊗ ( V Γ ( − n × 3 L d π ∼ = = = 1) ,..., ) 2 , = ∆ ⊗ 1 2 π , 2 ,π G ⊗ ) a 1 1 \ 1 G/K π , we would like to reconstruct π = SO( \ π − K ( Let us explain the statements we made in the previous paragra As in the previous cases, we can use the equivariance laws to r In close analogy to eq. ( Just as in our previous discussion of the space ( LR A , V ⊗ K ( Elements in SO(1 Γ B 1 3 1 Γ ∈ π − which is actually the space of four-point tensor structures representatives would expect from a space of a any representative since the left and right action of we should restrict it to take values in the subspace of Hence, the space Γ and introduce an automorphism Θ acting on to tail. To this end we mimic the usual construction of the a subgroup to formulate the equivariance law. Consequently equivariant functions, as in eq. ( K = SO(1 where the two representations quotient space equivariant with respect to both left and right multiplicat ( as a space of highly reducible and may be decomposed into irreducibles of distinguish between carrier space of representations for S If we now combine this with the realization of the tensor prod conformal group ( Here 2 space of left-equivariant functions, on the space ( JHEP04(2018)052 . . ) ,µ 2) 2) µν R G 0 V 2.7 − M − (2.9) M tions leads takes ⊗ (2.12) (2.10) d d orbits. L f B V K . Indeed, = . A × ) is embedded = SO( B A = SO( a ) ( K biliser ∈ by conjugation to B-invariants G R B f B ) V of the conformal of 2 A )) P ⊂ . ⊗ b ji ) ( , τ L ∈ K is given through the 1 K V G × τ M a ( G 1. Once we know that K a ⊂ ⊗R − ponentiation we pass to + 1. Obviously, the Lie ) O b coincides with that of 2) identity matrix in the B = ( A ∈ d > in turn is spanned by , a L − a ij oices, we impose that p d acts on ) = dim( , . . . d ∪             ))( from its values on ( M 2 r B , KAK B 0 0 1 0 0 0 . These two generators commute = k with × G 1 ( 3 , , ∈ d is trivial and ( we can write the conformal group 1 2) on the group b R dim( ...... M B ⊗R = 0 − ) B − l KAK 0 0 = d ) = (2.11) ) k ∈ G/K 1 of dilations along with the elements ( , the subgroup . We shall select a 2-dimensional abelian − i, j A \ 2 − r b 2 L µ 1 a τ 2 A , 2 k τ ) may be considered as functions of the two K 2) matrices and such matrices do commute 0 1 = 3, the stabilizer subgroup K 0 0 0 0 π − commute with the subgroup sin M d 4 − ) = ( and . From the equivariance law in definition ( , – 6 – ) of a d − 2 A bab ( G ( , [0 l 0 ) + dim( f 1 2.9 1 2 k 2 τ ) τ ∈ ∈ ∈ × ( K b M f 2 r r 0 cos 0 a τ 2) k k run through = ( 1 + 1) that consists of matrices of the form so that an element ), ) = − − + 1 a 2 as a formal union 2 , d i, j d 2 2 K ∞ higher than − τ r τ ⊗R , bab ) 0 cosh 0 sinh l d × b ak contains the d-dimensional abelian subalgebras of transla l k l ) = 2dim( -invariants. We can construct the projection k k p SO(1 K −∞ ( = 3 the stabiliser subgroup ( = as the latter contain a ( B 1 1 ( 2 KAK 2 τ f L τ d ⊂ ∈ r ∈ A 0 0 0 0 0 0 0 0 0 1 0 cos 0 sin ) = KAK A ...... 1 1 ak ) = l τ ∈ − a sinh cosh + 1) where k ( a f dim(             b, b , d ) + 1 that generate rotations. Our subspace ( = 2 and r is spanned by the generator . As we have stressed before, the existence of nontrivial sta k d . In particular, the dimension of the space a ( ) = 7→ that is spanned by coincides with the double coset 2 K b = a A , τ ⊗R of 1 , . . . , d 1 = SO(1 . The subalgebra ) τ l − k KAK ( ,µ k G 1 a ( = 2 = bab L M µ and of special conformal transformations G 7→ µ for and the space P with each other since theythe have abelian no subgroup index in common. Through ex subalgebra Once we pass to dimensions lower-right corner. The action of such elements into the lower rightwith corner all of elements ( In order to ensure that the extension is independent of our ch group values in the subspace of becomes non-trivial. With our choice ( as embedding algebra where two variables anharmonic ratios one can build from four points in Indeed, we can think of conclude that let us take a point to ambiguity when we try to reconstruct a function on be more concrete let us introduce usual set of generators In dimensions a embedded into the lower right corner we obtain Taking into account