Logarithmic primary PT/GW correspondence

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Let (X,D)(X,D) be the logarithmic pair, let β\beta be a curve class, and set

dβ=c1(TX)⋅β∈Z.d_\beta=c_1(T_X)\cdot\beta\in\mathbb{Z}.

Let ζ1,…,ζk\zeta_1,\ldots,\zeta_k be cohomology insertions and let μ(δ)\bm{\mu}(\delta) be relative conditions. Under the identification of variables −q=eiu-q=e^{iu}, primary PT/GW correspondence. One has

(−q)−dβ/2ZPT(X,D;q∣∏i=1kτ0(ζi)∣μ(δ))β=(−iu)dβ+∑ℓ(μj)−∣μj∣ZGW(X,D;u∣∏i=1kτ0(ζi)∣μ(δ))β.(-q)^{-d_\beta/2}\mathsf{Z}_{\mathsf{PT}}\left(X,D;q\mid\prod_{i=1}^k\tau_0(\zeta_i)\mid\bm{\mu}(\delta)\right)_\beta=(-iu)^{d_\beta+\sum\ell(\mu_j)-|\mu_j|}\mathsf{Z}_{\mathsf{GW}}\left(X,D;u\mid\prod_{i=1}^k\tau_0(\zeta_i)\mid\bm{\mu}(\delta)\right)_\beta.

This is the simplest form of the curve/sheaf correspondence, relating primary PT and GW generating series after the stated change of variables. The supplied text gives no resolution status.

References

Primary source

Davesh Maulik and Dhruv Ranganathan, “Logarithmic enumerative geometry for curves and sheaves”, arXiv:2311.14150 (2025).

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