Relative descendent GW/Pairs correspondence

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Let XX be a nonsingular 33-fold with a nonsingular divisor D⊂XD\subset X, let 0≠β∈H2(X,Z)0\ne\beta\in H_2(X,\mathbb Z)), and let μ\mu be a cohomology-weighted partition of ∫β[D]\int_\beta[D]. For γi∈H∗(X,Q)\gamma_i\in H^*(X,\mathbb Q)), let τα1−1(γ1)⋯ταℓ−1(γℓ)‾\overline{\tau_{\alpha_1-1}(\gamma_1)\cdots\tau_{\alpha_\ell-1}(\gamma_\ell)} be the relative descendent correspondence formed using the relative diagonal and the logarithmic tangent bundle TX[−D]T_X[-D].

Relative descendent GW/Pairs correspondence. Under the variable change −q=eiu-q=e^{iu},

(−q)−dβ/2ZP(X/D;q ∣τα1−1(γ1)⋯ταℓ−1(γℓ) ∣ μ)β=(−iu)dβ+ℓ(μ)−∣μ∣ZGW′(X/D;u ∣τα1−1(γ1)⋯ταℓ−1(γℓ)‾ ∣ μ)β.(-q)^{-d_\beta/2}\mathsf Z_{\mathsf P}\Big(X/D;q\ \Big|\tau_{\alpha_1-1}(\gamma_1)\cdots\tau_{\alpha_\ell-1}(\gamma_\ell)\ \Big|\ \mu\Big)_\beta =(-iu)^{d_\beta+\ell(\mu)-|\mu|}\mathsf Z'_{\mathsf{GW}}\Big(X/D;u\ \Big|\overline{\tau_{\alpha_1-1}(\gamma_1)\cdots\tau_{\alpha_\ell-1}(\gamma_\ell)}\ \Big|\ \mu\Big)_\beta.

The conjecture also includes rationality in qq of the stable-pairs descendent series, which makes the change of variables well-defined. No resolution status is supplied.

References

Primary source

R. Pandharipande and A. Pixton, “Gromov-Witten/Pairs correspondence for the quintic 3-fold”, arXiv:1206.5490 (2016).

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