Logarithmic descendent PT/GW correspondence

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

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

Let α=(α1,,αk)\alpha=(\alpha_1,\ldots,\alpha_k) be a partition, let ζ1,,ζk\zeta_1,\ldots,\zeta_k be cohomology classes, and let τα11(ζ1)ταk1(ζk)\overline{\tau_{\alpha_1-1}(\zeta_1)\cdots\tau_{\alpha_k-1}(\zeta_k)} denote the corrected descendent insertion defined in the source. Let μ(δ)\bm{\mu}(\delta) be relative conditions. Under the identification q=eiu-q=e^{iu}, descendent PT/GW correspondence. One has

(q)dβ/2ZPT(X,D;qτα11(ζ1)ταk1(ζk)μ(δ))β(-q)^{-d_\beta/2}\mathsf{Z}_{\mathsf{PT}}\left(X,D;q\mid\tau_{\alpha_1-1}(\zeta_1)\cdots\tau_{\alpha_k-1}(\zeta_k)\mid\bm{\mu}(\delta)\right)_\beta =(iu)dβ+(μj)μjZGW(X,D;uτα11(ζ1)ταk1(ζk)μ(δ))β.=(-iu)^{d_\beta+\sum\ell(\mu_j)-|\mu_j|}\mathsf{Z}_{\mathsf{GW}}\left(X,D;u\mid\overline{\tau_{\alpha_1-1}(\zeta_1)\cdots\tau_{\alpha_k-1}(\zeta_k)}\mid\bm{\mu}(\delta)\right)_\beta.

This is the full descendent form of the curve/sheaf correspondence for PT theory, extending the primary correspondence through corrected descendents. The supplied text gives no resolution status.

Sources & referencesView supporting material

Primary source

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

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.