Descendent GW/Pairs correspondence for nonsingular projective 3-folds

Let XX be a nonsingular projective 33-fold, let 0βH2(X,Z)0\ne\beta\in H_2(X,\mathbb{Z}), and let γiH(X,Q)\gamma_i\in H^*(X,\mathbb{Q}). Let dβ=βc1(X)d_\beta=\int_\beta c_1(X)), and let τα11(γ1)τα1(γ)\overline{\tau_{\alpha_1-1}(\gamma_1)\cdots\tau_{\alpha_\ell-1}(\gamma_\ell)} denote the descendent correspondence defined using the matrix K~\widetilde{\mathsf K}, with the prescribed signs for odd cohomology classes.

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

(q)dβ/2ZP(X;q τα11(γ1)τα1(γ))β=(iu)dβZGW(X;u τα11(γ1)τα1(γ))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 descendent form of the GW/Pairs correspondence for all nonsingular projective 33-folds. The source gives no evidence of a resolution, so the conjecture is recorded as open.

Sources & referencesView supporting material

Primary source

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

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