Stable-pairs descendent correspondence for absolute 3-folds

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Let XX be a nonsingular projective 33-fold, let β∈H2(X,Z)\beta\in H_2(X,\mathbb Z), let dβ=∫βc1(X)d_\beta=\int_\beta c_1(X), and let γi∈H∗(X)\gamma_i\in H^*(X). The barred descendent insertion τα1−1(γ1)⋯ταℓ−1(γℓ)‾\overline{\tau_{\alpha_1-1}(\gamma_1)\cdots\tau_{\alpha_\ell-1}(\gamma_\ell)} is defined using the correspondence matrix described in the source. Stable-pairs descendent correspondence conjecture. Under the variable change −q=eiu-q=e^{iu},

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

This is the stable-pairs/Gromov–Witten descendent correspondence proposed in the cited work; the supplied text gives no resolution of the general statement.

References

Primary source

Rahul Pandharipande, “Descendents for stable pairs on 3-folds”, arXiv:1703.01747 (2017).

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