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Systematic understanding of automorphism enhancement (USp(2)^{⊗g} → USp(2g))

Determine a systematic mechanism by which multiple USp(2) automorphisms acting on vertex operator algebras obtained via relative semi-infinite cohomology combine and enhance to larger symplectic automorphism groups, including establishing the expected enhancement USp(2)^{⊗ g} → USp(2g) for genus‑g class S theories and characterizing the underlying algebraic and physical structures that drive this enhancement.

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Background

The authors observe examples where several USp(2) actions merge into a larger group, such as USp(4) for a genus‑2 class S theory of type A1. They expect broader enhancements (USp(2){⊗ g} to USp(2g)) across class S constructions, often realized via relative semi-infinite cohomology of certain VOAs.

A general, uniform explanation for this enhancement is not provided. The problem seeks a coherent framework that derives and predicts the enhanced automorphism groups from the construction data.

References

The results reported here leave open several interesting avenues to pursue, some of which we summarize here. In some examples, multiple of the ${\rm USp}(2)$ automorphisms found in this note combine and enhance to a larger automorphism group. For example, the VOAs coming from a genus-$g$ class $\mathcal{S}$ theory, e.g. when realized as a suitable relative semi-infinite cohomology of the VOAs appearing in , should witness an enhancement of ${\rm USp}(2){\otimes g}$ to ${\rm USp}(2g)$; see for the genus-2 class $\mathcal{S}$ theory of type $A_1$, where two copies of ${\rm USp}(2)$ combine into a ${\rm USp}(4)$ automorphism group. How can this enhancement be understood systematically?

— On the semi-infinite cohomology of graded-unitary vertex algebras (2509.10364 - Beem et al., 12 Sep 2025) in Section 1 (Introduction), end, Item 2