Superconformal Blocks for SCFTs with Eight Supercharges
Nikolay Bobev, Edoardo Lauria, Dalimil Mazac
TL;DR
The paper develops a dimensionally unified framework for SCFTs with eight Poincaré supercharges in 2 < d ≤ 6, using the quadratic superconformal Casimir to derive closed differential equations for four-point functions of moment map operators. By formal dimensional reduction from d=6 and a careful SU(2)_R selection of external indices, the authors obtain decoupled Casimir equations and compute explicit superconformal blocks as finite sums of ordinary conformal blocks across dimensions. They classify admissible unitary multiplets and fix the block coefficients through the Casimir constraints, providing complete formulas for L[Δ,ℓ,0], B[ℓ,R], and D[ℓ,R] multiplets, with consistency checks including agreement with known d=4 results and a free-hypermultiplet check. The results enable systematic numerical bootstrap studies of eight-supercharge SCFTs in diverse dimensions and offer insights into connections with non-supersymmetric blocks and potential integrability structures.
Abstract
We show how to treat the superconformal algebras with eight Poincaré supercharges in a unified manner for spacetime dimension $2 < d\leq 6$. This formalism is ideally suited for analyzing the quadratic Casimir operator of the superconformal algebra and its use in deriving superconformal blocks. We illustrate this by an explicit construction of the superconformal blocks, for any value of the spacetime dimension, for external protected scalar operators which are the lowest component of flavor current multiplets.