Distinguishing 6d (1,0) SCFTs: an extension to the geometric construction

TL;DR

This paper extends the geometric construction of 6d (1,0) SCFTs to distinguish theories with identical global symmetries but different spectra, especially in their Higgs branches, and explores implications for 4d theories via compactification.

Contribution

It introduces a method to differentiate 6d (1,0) SCFTs with similar tensor branches by augmenting the description with Higgs branch operators, revealing subtleties beyond previous classifications.

Findings

Distinct 6d SCFTs can share tensor branch data but differ in Higgs branch spectra.

The proposed method predicts the conformal dimension where Higgs branch operators differ.

Differences in 4d class S theories are confirmed via Hall--Littlewood indices.

Abstract

We provide a new extension to the geometric construction of 6d $(1,0)$ SCFTs that encapsulates Higgs branch structures with identical global symmetry but different spectra. In particular, we find that there exist distinct 6d $(1,0)$ SCFTs that may appear to share their tensor branch description, flavor symmetry algebras, and central charges. For example, such subtleties arise for the very even nilpotent Higgsing of $(\mathfrak{so}_{4k}, \mathfrak{so}_{4k})$ conformal matter; we propose a method to predict at which conformal dimension the Higgs branch operators of the two theories differ via augmenting the tensor branch description with the Higgs branch chiral ring generators of the building block theories. Torus compactifications of these 6d $(1,0)$ SCFTs give rise to 4d $\mathcal{N}=2$ SCFTs of class $\mathcal{S}$ and the Higgs branch of such 4d theories are captured via the Hall--Littlewood index. We confirm that the resulting 4d theories indeed differ in their spectra in the predicted conformal dimension from their Hall--Littlewood indices. We highlight how this ambiguity in the tensor branch description arises beyond the very even nilpotent Higgsing of $(\mathfrak{so}_{4k}, \mathfrak{so}_{4k})$ conformal matter, and hence should be understood for more general classes of 6d $(1,0)$ SCFTs.