Compactifying 5d superconformal field theories to 3d

TL;DR

This paper explores the systematic reduction of 5d superconformal field theories on Riemann surfaces to 3d theories, revealing new elements like Chern-Simons terms and monopole superpotentials, and matching symmetries and spectra.

Contribution

It introduces a framework for compactifying 5d SCFTs to 3d, including novel elements specific to three dimensions, and verifies the resulting theories through various consistency checks.

Findings

Matching IR symmetries with expected global symmetry commutants

Identification of operator spectra and conformal manifolds

Discovery of Chern-Simons terms and monopole superpotentials in 3d theories

Abstract

Building on recent progress in the study of compactifications of $6d$ $(1,0)$ superconformal field theories (SCFTs) on Riemann surfaces to $4d$ $\mathcal{N}=1$ theories, we initiate a systematic study of compactifications of $5d$ $\mathcal{N}=1$ SCFTs on Riemann surfaces to $3d$ $\mathcal{N}=2$ theories. Specifically, we consider the compactification of the so-called rank 1 Seiberg $E_{N_f+1}$ SCFTs on tori and tubes with flux in their global symmetry, and put the resulting $3d$ theories to various consistency checks. These include matching the (usually enhanced) IR symmetry of the $3d$ theories with the one expected from the compactification, given by the commutant of the flux in the global symmetry of the corresponding $5d$ SCFT, and identifying the spectrum of operators and conformal manifolds predicted by the $5d$ picture. As the models we examine are in three dimensions, we encounter novel elements that are not present in compactifications to four dimensions, notably Chern-Simons terms and monopole superpotentials, that play an important role in our construction. The methods used in this paper can also be used for the compactification of any other $5d$ SCFT that has a deformation leading to a $5d$ gauge theory.