General geometric-engineering correspondence

Prove that the equivariant Gromov–Witten series of X_{N,m}×C² equals the Nekrasov partition function Z_{N,m} for pure supersymmetric SU(N) gauge theory with Chern–Simons level m whenever |m|<N, for arbitrary Calabi–Yau torus actions.

Background

Geometric engineering identifies toric Calabi–Yau threefolds X_{N,m} with five-dimensional pure SU(N) gauge theory. The paper proves the corresponding Pandharipande–Thomas/instanton identity, while the Gromov–Witten identity is established only in the anti-diagonal specialization and checked numerically beyond it.

The conjecture would extend refined topological-string/gauge-theory matching to general Ω-background parameters.

References

For $|m|<N$ we have

— Experiments with membranes, maps and sheaves  (2609.03152 - Holmes et al., 2 Sep 2026) in Conjecture C (label conj: gauge GW correspondence), Introduction and Section 3

Suppose $X$ is a strip geometry and $(X\timesC{2},T)$ is equivariantly Calabi--Yau. Then if we denote the tangent $T$-weights of the affine plane by $\epsilon_4,\epsilon_5$ we have

— Experiments with membranes, maps and sheaves  (2609.03152 - Holmes et al., 2 Sep 2026) in Conjecture strip GW formula (label conj: strip GW formula), Section 2.4