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

Jacques Distler, Monica Jinwoo Kang, Craig Lawrie

TL;DR

The work identifies a fundamental ambiguity in the geometric construction of 6d $(1,0)$ SCFTs: theories can share tensor-branch data and conventional invariants yet have distinct Higgs branches. By augmenting the tensor-branch description with Higgs-branch chiral-ring generators and analyzing very-even nilpotent Higgsings of $(D,D)$ conformal matter, the authors predict and verify differences in Higgs spectra that surface at specific conformal dimensions, confirmed independently via torus compactifications to 4d class $\mathcal{S}$ and Hall–Littlewood indices. They derive explicit dimensional thresholds $\Delta$ for several families, showing that HL indices match the predicted distinctions, thereby establishing a reliable 6d-to-4d consistency check. The results generalize beyond minimal $(D,D)$ matter, highlight the role of E-string flavor algebras and $E_8$-homomorphisms, and emphasize that a complete classification of 6d $(1,0)$ SCFTs requires incorporating Higgs-branch data into the geometric construction. This provides a robust framework for resolving ambiguities in tensor-branch decompositions and clarifies the landscape of high-dimensional SCFTs with identical conventional invariants.

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.

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

TL;DR

The work identifies a fundamental ambiguity in the geometric construction of 6d SCFTs: theories can share tensor-branch data and conventional invariants yet have distinct Higgs branches. By augmenting the tensor-branch description with Higgs-branch chiral-ring generators and analyzing very-even nilpotent Higgsings of conformal matter, the authors predict and verify differences in Higgs spectra that surface at specific conformal dimensions, confirmed independently via torus compactifications to 4d class and Hall–Littlewood indices. They derive explicit dimensional thresholds for several families, showing that HL indices match the predicted distinctions, thereby establishing a reliable 6d-to-4d consistency check. The results generalize beyond minimal matter, highlight the role of E-string flavor algebras and -homomorphisms, and emphasize that a complete classification of 6d SCFTs requires incorporating Higgs-branch data into the geometric construction. This provides a robust framework for resolving ambiguities in tensor-branch decompositions and clarifies the landscape of high-dimensional SCFTs with identical conventional invariants.

Abstract

We provide a new extension to the geometric construction of 6d SCFTs that encapsulates Higgs branch structures with identical global symmetry but different spectra. In particular, we find that there exist distinct 6d 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 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 SCFTs give rise to 4d SCFTs of class 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 conformal matter, and hence should be understood for more general classes of 6d SCFTs.
Paper Structure (12 sections, 87 equations, 1 figure)

This paper contains 12 sections, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. (D,D) conformal matter and nilpotent Higgsing
  3. Very even Higgsing in 6d via examples
  4. (so₈, so₈) with 2×[2⁴]
  5. (so₄ₖ, so₄ₖ) with 2×[(2k-2)², 2²]
  6. (so₄ₖ, so₄ₖ) with 2×[(2k-4)², 2⁴]
  7. (so₄ₖ, so₄ₖ) with 2×[(2k-2l)², 2²ˡ]
  8. A non-example: the uniqueness of conformal matter
  9. 6d (1,0) on T² and class S
  10. Conclusion
  11. Flavor algebras from E-strings
  12. From nilpotent orbits to E₈-homomorphisms

Figures (1)

  • Figure 4.1: The $(N+2)$-punctured spheres associated to the class $\mathcal{S}$ theories where the Higgs branch operator spectrum differs as described in equations \ref{['eqn:2to2kredred']} and \ref{['eqn:2to2kredblue']}. The former contains a $\hat{B}_{(k-1)(N-1)-1}$ operator that the latter does not.