Abstract
We extend the Mellin space techniques of [1] for computing holographic four-point correlation functions in maximally superconformal theories to theories with only eight Poincaré supercharges. The one-half BPS operators in these correlators are taken to be the superconformal primary in the mathcal{D}left[kright] multiplet (with k = 2 corresponding to the flavor current multiplet), and transform in the adjoint representation of a flavor group G. Because of the smaller R-symmetry group SU(2), each individual superconformal Ward identity is less powerful. On the other hand, the constraining power is compensated in number by the different flavor channels in the four-point function. As concrete test cases, we study the Seiberg theories in five dimensions and E-string theory in six dimensions at the large N limit. We show that the flavor current multiplet four-point functions are fixed by superconformal symmetry up to two free parameters, which are proportional to the squared OPE coefficients for the flavor current multiplet and the stress tensor multiplet.
Highlights
Correlators have been computed in the past two decades using this recipe, to wit: a small number of examples in the AdS5 × S5 background [4,5,6,7,8,9,10]; only the four-point of the stress tensor multiplet in AdS7 × S4 [11]; and no results whatsoever for other backgrounds
We extend the Mellin space techniques of [1] for computing holographic fourpoint correlation functions in maximally superconformal theories to theories with only eight Poincare supercharges
We show that the flavor current multiplet four-point functions are fixed by superconformal symmetry up to two free parameters, which are proportional to the squared OPE coefficients for the flavor current multiplet and the stress tensor multiplet
Summary
Superconformal symmetry organizes operators into superconformal multiplets. In particular, the flavor conserved current of a d-dimensional SCFT with eight Poincare supercharges resides in the superconformal multiplet D[2] whose superconformal primary is the moment map operator Oαa1α2. The moment map operator is a one-half BPS operator with conformal dimension ∆ = d − 2 It has R-charge jR = 1 under the SU(2)R R-symmetry (captured by α1, α2 = 1, 2) and transforms in the adjoint representation under the flavor group G7 (captured by the flavor index a). There are more one-half BPS operators in D[k] multiplets These operators Oαa1...αk are the scalar superconformal primaries and have conformal dimension ∆ = k. They transform in the rank-k symmetric tensor of the fundamental representation of SU(2)R. To write down its crossing equations, it is convenient to project the four-point function Gabcd into the flavor irreducible representations that appear in the s-channel tensor product Adj1 ⊗ Adj with the projection matrix PIab|cd.
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