Superconformal blocks for mixed 1/2-BPS correlators with SU(2) R-symmetry

Abstract

For SCFTs with an SU (2) R-symmetry, we determine the superconformal blocks that contribute to the four-point correlation function of a priori distinct half-BPS superconformal primaries as an expansion in terms of the relevant bosonic conformal blocks. This is achieved by using the superconformal Casimir equation and the superconformal Ward identity to fix the coefficients of the bosonic blocks uniquely in a dimension-independent way. In addition we find that many of the resulting coefficients are related through a web of linear transformations of the conformal data. a very nice playground to study non-perturbative effects and relations to string theory, for instance their connection to the swampland program [38].As six is the largest dimension allowing for a superconformal algebra [39], N = (1, 0)representation theory provides an overarching language encompassing lower dimensions via dimensional reduction 3 . Let us review the possible multiplets allowed in theories with SU (2) R R-symmetry [19, 20]. There can exist states which are annihilated by a subset of the supercharges. These null states must be absent in unitary theories, and lead to what are referred as short multiplets, as opposed to long multiplets, which do not have null states. It is standard to write long multiplets as L[∆, , J R ], where ∆ is the conformal dimension of the superconformal primary, denotes how the superconformal primary transforms as a traceless-symmetric 4 representation of so(1, d − 1) rotations of the Poincaré algebra, and J R is the charge under SU (2) R . The different short multiplets are denoted as the A-, B-, C-, and D-type multiplets. Unitarity gives lower bounds on the allowed conformal dimensions, ∆, of the superconformal primaries of long multiplets, and is moreover strong enough to completely fix the conformal dimension of the short multiplets as a function of the other group theoretical data and the spacetime dimension. The superconformal multiplets can be summarised as follows : L[∆, , J R ] : ∆ > 2ε J R + + µ , A[ , J R ] : ∆ = 2ε J R + + 4ε − 2 , B[ , J R ] : ∆ = 2ε J R + + 2ε , (1.1) C[J R ] : ∆ = 2ε J R + 2 , D[J R ] : ∆ = 2ε J R ,with ε = (d − 2)/2, µ = 2ε for 2 < d ≤ 4 and µ = 4ε − 2 for 4 ≤ d ≤ 6. We stress again that this corresponds to the standard notation for d = 6. Indeed, A-type multiplets correspond to the unitarity bound of long multiplets, which for d ≤ 4 coincides with the bound of type B. Type C is unique to six dimensions and can be traced back to the presence of self-dual two-forms. Moreover, type C and D will appear only with = 0, since in this work we 3 We note that for d ≤ 4, the nomenclature we are using here might not match the one the reader is familiar with. For instance, in four dimensions, the classification of half-BPS multiplets is refined into Higgs and Coulomb type, commonly denoted E r andB R respectively [40]. We refer to [20] for a dictionary between the 6D notation and lower dimensions.4 In this paper we will consider only multiplets that have a superconformal primary in a traceles...

Gluon scattering in AdS from CFT

“…The superconformal blocks of this multiplet with external s I p satisfy the same superconformal Ward identities in N = 3 and N = 4. Since the N = 4 case admits a unique solution [59,66,67], it would require the conformal blocks associated with t I p scalars alone to form another solution to the Ward identities, if they were to appear in the exchange. Such solutions do not exist (see appendix B for details and further comments).…”

3d N=4 Bootstrap and Mirror Symmetry

“…In particular, the most minimal known theories [89] have no chiral operators, and are inaccessible by the study of such four-point functions. Some superconformal blocks that are necessary in the bootstrap of these more general four-point functions have been computed in [45,90].…”

“…1 We study the single-and mixed-branch four-point functions of the moment map op-erators (scalar primaries in the flavor current multiplets for G C and G H ), exploiting constraints from superconformal symmetry and crossing using the numerical bootstrap methods. 2 This is made possible by first determining the mixed-branch 3d N = 4 superconformal blocks (the single-branch blocks were determined in [25,44,45]). By comparison with certain OPE data computable by supersymmetric localization, we observe that all known candidate theories are consistent with the numerical bounds.…”

Bootstrapping (D, D) Conformal Matter

“…This allows one to write each coefficient as a rational function depending solely on the quantum numbers of the superconformal primary. For theories with eight supercharges and 2 < d ≤ 6, this analysis was performed in detail for the moment map in [40,58] and generalized to arbitrary D-type half-BPS multiplets in [60], to which we refer for the exact expressions.…”

“…We refer to [40,58,60] for additional details on the derivations of the sum rules and the form of the conformal blocks.…”