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Chiral quantization of the twistor circle and relation to hyperkähler metrics

Investigate whether the graded‑unitary structure on vertex operator algebras arising from the SCFT/VOA correspondence can be constructed as a chiral quantization of the twistor S^1 of the hyperkähler Higgs branch, and derive whether such a construction yields vertex‑algebraic data that determine or relate to the hyperkähler metric of the parent four‑dimensional N=2 SCFT.

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Background

VOAs associated to four-dimensional N=2 SCFTs are often regarded as chiral quantizations of the Higgs branch, a hyperkähler cone equipped with an SU(2)_R action that also underlies the VOA’s graded‑unitary filtration. The authors speculate that a twistor S1 construction may be directly linked to the graded‑unitary structure.

A precise mathematical procedure connecting the VOA’s graded‑unitary structure to twistor geometry — and in particular to the hyperkähler metric — is currently lacking, motivating this open problem.

References

The results reported here leave open several interesting avenues to pursue, some of which we summarize here. The VOAs appearing in the SCFT/VOA correspondence are rather geometric in flavor and are often thought of as chiral quantizations of their parent SCFT's Higgs branch of vacua. These Higgs branches are naturally hyperkähler cones and their twistor $S1$ of complex structure structures are rotated by the same ${\rm SU}(2)_R$ symmetry used to define the filtration in their graded-unitary structures. Can the graded-unitary structure on such a VOA be understood as arising from a kind of chiral quantization of the twistor $S1$ of the Higgs branch and can this be used to relate the hyperkähler metric on the parent SCFT's Higgs branch to vertex algebraic data?

— On the semi-infinite cohomology of graded-unitary vertex algebras (2509.10364 - Beem et al., 12 Sep 2025) in Section 1 (Introduction), end, Item 3