From Free Fields to Interacting SCFTs via Representation Theory

TL;DR

This paper investigates the conditions under which unitary multiplets of superconformal algebras can be constructed from free field theories across various dimensions, revealing new constraints and connections to deformations in theory space.

Contribution

It provides a systematic analysis of constructing superconformal multiplets from free theories and introduces conjectures that imply new non-perturbative constraints in 4d $ ext{N}=2$ SCFTs.

Findings

In 2, 3, and 5 dimensions, criteria for constructing multiplets are established.

Conjectures in 4d suggest new non-perturbative constraints on short multiplets.

Connections made between deformations in theory space and superconformal representations.

Abstract

We ask when it is possible to construct arbitrary unitary multiplets of the superconformal algebra with eight Poincar\'e supercharges that are compatible with locality from (continuous deformations of) representations in free field theories. We answer this question in two, three, and five dimensions. In four dimensions, we resort to an intricate but self-consistent web of conjectures. If correct, these conjectures imply various new non-perturbative constraints on short multiplets in any local unitary 4d $\mathcal{N}=2$ superconformal field theory and on an unusual set of related vertex algebras. Throughout, we connect our results with properties of deformations in the space of theories.