Polology of Superconformal Blocks

TL;DR

This paper classifies all poles of superconformal blocks across various dimensions and supersymmetries, revealing infinite short multiplets and subtle algebraic features, with implications for understanding superconformal correlators.

Contribution

It provides a comprehensive classification of superconformal block poles using irreducibility criteria for Lie superalgebras, extending the understanding of superconformal multiplets and their algebraic structure.

Findings

Identified all possible poles of superconformal blocks in dimensions three and higher.

Discovered infinitely many short multiplets, mostly non-unitary.

Found poles shifted linearly with respect to the number of supersymmetries, $ $.

Abstract

We systematically classify all possible poles of superconformal blocks as a function of the scaling dimension of intermediate operators, for all superconformal algebras in dimensions three and higher. This is done by working out the recently-proven irreducibility criterion for parabolic Verma modules for classical basic Lie superalgebras. The result applies to correlators for external operators of arbitrary spin, and indicates presence of infinitely many short multiplets of superconformal algebras, most of which are non-unitary. We find a set of poles whose positions are shifted by linear in $\mathcal{N}$ for $\mathcal{N}$-extended supersymmetry. We find an interesting subtlety for 3d $\mathcal{N}$-extended superconformal algebra with $\mathcal{N}$ odd associated with odd non-isotropic roots. We also comment on further applications to superconformal blocks.