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Enumerative mirror symmetry for zero-degree quasimap invariants

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Let r≥2{\mathsf r}\geq 2, and let QMˇd,0a\check{\mathsf{QM}}^{\mathsf a}_{{\mathsf d},0} and QM^d,0a\hat{\mathsf{QM}}^{\mathsf a}_{{\mathsf d},0} denote the zero-degree quasimap invariants indexed by d,a∈Zr{\mathsf d},{\mathsf a}\in\mathbb Z_{\mathsf r}. Let ω=e2π−1/r\omega=e^{2\pi\sqrt{-1}/{\mathsf r}}. Enumerative mirror symmetry. One has

rQMˇd,0a=∑d′∑a′ωd′⋅a+d⋅a′QM^d′,0a′,{\mathsf r}\check{\mathsf{QM}}^{\mathsf a}_{{\mathsf d},0}=\sum_{{\mathsf d}'}\sum_{{\mathsf a'}}\omega^{{\mathsf d'}\cdot{\mathsf a}+{\mathsf d}\cdot{\mathsf a'}}\hat{\mathsf{QM}}^{{\mathsf a'}}_{{\mathsf d'},0},

where the sums range over a′,d′∈Zr{\mathsf a'},{\mathsf d'}\in\mathbb Z_{\mathsf r}. This is the zero-degree SS-duality relation, proposed for arbitrary rank at least two.

References

Primary source

Denis Nesterov, “Enumerative mirror symmetry for moduli spaces of Higgs bundles and S-duality”, arXiv:2302.08379 (2023).

enumerative-geometryquasimapss-duality
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Progress summary

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The proposed symmetry remains a conjecture, with a genus-one special case reported as proved and no public proof or counterexample found.

A June 2023 preprint identifies the stated zero-degree relation for rank at least 22 as Conjecture D, or “Enumerative mirror symmetry, w=0w=0.” It is motivated by Vafa–Witten invariants and physical calculations, but is not presented as proved.

Known results

  • In genus 11, the source says the relevant conjectures are proved; for nonzero degree the moduli space is a point, so only the w=0w=0 statement is meaningful.

Current status (as of August 2026): the arbitrary-rank zero-degree relation remains Conjecture D; only the genus-one case is reported as proved, and no proof, counterexample, or verification of the general claim is recorded.

Sources

Solutions 0

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