Bootstrapping SCFTs with Four Supercharges

TL;DR

This paper develops a formalism for superconformal blocks across dimensions 2 to 4, and uses numerical bootstrap to identify bounds and special features in the spectrum of theories with four supercharges.

Contribution

It introduces a universal approach to superconformal blocks for any dimension up to 4 and applies numerical bootstrap to reveal structure in operator dimensions.

Findings

Identified upper bounds on the conformal dimension of certain operators.

Discovered three distinct kinks in the bounds, indicating special theories.

Connected known theories across dimensions via a continuous interpolation.

Abstract

We study the constraints imposed by superconformal symmetry, crossing symmetry, and unitarity for theories with four supercharges in spacetime dimension $2\leq d\leq 4$. We show how superconformal algebras with four Poincar\'{e} supercharges can be treated in a formalism applicable to any, in principle continuous, value of $d$ and use this to construct the superconformal blocks for any $d\leq 4$. We then use numerical bootstrap techniques to derive upper bounds on the conformal dimension of the first unprotected operator appearing in the OPE of a chiral and an anti-chiral superconformal primary. We obtain an intriguing structure of three distinct kinks. We argue that one of the kinks smoothly interpolates between the $d=2$, $\mathcal N=(2,2)$ minimal model with central charge $c=1$ and the theory of a free chiral multiplet in $d=4$, passing through the critical Wess-Zumino model with cubic superpotential in intermediate dimensions.