On the moduli spaces of 4d $\mathcal{N} = 3$ SCFTs I: triple special K\"ahler structure

TL;DR

This paper explores the geometric structure of moduli spaces in 4d $ abla=3$ superconformal field theories, introducing a new 'triple special K"ahler' structure characterized by singularities and stratifications.

Contribution

It introduces the triple special K"ahler (TSK) structure as a novel geometric framework for understanding $ abla=3$ SCFT moduli spaces, based on chiral rings and supersymmetry constraints.

Findings

TSK moduli spaces have complex dimension 3n for rank n.

Singularities occur at co-dimension 3 subspaces with massless charged states.

Moduli space stratification is described by lower-dimensional TSK manifolds.

Abstract

We initiate a systematic analysis of moduli spaces of vacua of four dimensional $\mathcal{N}=3$ SCFTs. Our analysis is based on the one hand on the properties of $\mathcal{N}=3$ chiral rings --- which we review in detail and contrast with chiral rings of theories with less supersymmetry --- and on the other hand on constraints coming from low-energy supersymmetry. This leads us to introduce a new type of geometric structure, which characterizes $\mathcal{N}=3$ SCFT moduli spaces, and that we call $triple\ special\ K\"ahler$ (TSK). A rank-$n$ TSK moduli space has complex dimension $3n$, and is singular at complex co-dimension 3 subspaces where charged states become massless. The structure of singularities defines a stratification of the TSK space in terms of lower-dimensional TSK manifolds.