K-theoretic Gromov–Witten/Pandharipande–Thomas correspondence

Establish the K-theoretic Gromov–Witten/Pandharipande–Thomas correspondence for every local equivariant Calabi–Yau fivefold Z=Tot_X(L_4⊕L_5), identifying the K-theoretic Pandharipande–Thomas series with the exponential of the equivariant Gromov–Witten series after the natural change of variables.

Background

The paper studies a proposed five-dimensional analogue of the classical Gromov–Witten/Pandharipande–Thomas correspondence. For a Calabi–Yau fivefold obtained as the total space of two line bundles over a threefold, the authors compare equivariant Gromov–Witten invariants of the fivefold with K-theoretic stable-pair invariants of the threefold, including a Nekrasov–Okounkov insertion.

The correspondence is proved only in selected strip limits and supported elsewhere by numerical calculations. A general proof is explicitly identified as beyond the available methods.

References

For all local equivariant Calabi--Yau fivefolds $(Z,T)$ of the form eq: intro CY5 PT we have

— Experiments with membranes, maps and sheaves  (2609.03152 - Holmes et al., 2 Sep 2026) in Conjecture A (label conj: GW PT), Introduction and Section 2.3

There exist rational functions $\Omega_\beta$ on $T$ with constrained poles and coefficients in $\bZ[\frac{1}{2}]$ so that the series eq: intro omega gen series equates to the GW series\ of $(Z,T)$.

— Experiments with membranes, maps and sheaves  (2609.03152 - Holmes et al., 2 Sep 2026) in Conjecture B (label conj: gen GV), Introduction and Section 2.3

Suppose $(X\times C{2},T)$ is equivariantly Calabi--Yau and $\beta$ is a curve class in $X$ for which $_g(X,\beta)$ is proper for all $g\geq 0$.

— Experiments with membranes, maps and sheaves  (2609.03152 - Holmes et al., 2 Sep 2026) in Conjecture rigidity (label conj: rigidity), Section 2.3.2

We expect that the PT and GW series\ of $X$, respectively $X \times C{2}$, admit a closed formula similar to the one we presented for strip geometries.

— Experiments with membranes, maps and sheaves  (2609.03152 - Holmes et al., 2 Sep 2026) in Conjecture closed vertex formula (label conj: closed vertex formula), Section 2.5