The main result of the paper under review establishes a Gromov-Witten/Pairs (GW/P) correspondence for Fano or Calabi-Yau 3-folds that are complete intersections in products of projective spaces.
Let \(X \subset \mathbb P^{n_1}\times \cdots \mathbb P^{n_m}\) be a Fano or Calabi-Yau complete intersection and \(\gamma_i\in H^{2*}(X,\mathbb Q)\) be even classes, and \(\beta\in H_2(X,\mathbb Z)\) with \(d_\beta=\int_\beta c_1(X)\), then GW/P correspondence can be expressed as follows:
Firstly, the generating series of class \(\beta\) stable pair invariants with descendents is a rational function, i.e., \[Z_P \left(X;q \mid \tau_{\alpha_1-1}(\gamma_1)\cdots \tau_{\alpha_\ell-1}(\gamma_\ell)\right)_\beta\in\mathbb Q(q)\] and secondly there is a correspondence \[\begin{aligned} (-q)^{-d_\beta/2}Z_P \bigg(X;q \mid \tau_{\alpha_1-1}(\gamma_1)&\cdots \tau_{\alpha_\ell-1}(\gamma_\ell)\bigg)_\beta \\&=(-iu)^{d_\beta}Z'_{GW} \bigg(X;q \mid \overline{\tau_{\alpha_1-1}(\gamma_1)\cdots \tau_{\alpha_\ell-1}(\gamma_\ell)}\bigg)_\beta \end{aligned}\] under the change of variable \(-q=e^{iu}\). Here, the right hand side is the generating series of class \(\beta\) Gromov-Witten invariants with descendants (of possibly disconnected domain curves). The over line in the right hand side is a correspondence rule defined by means of a universal correspondence matrix indexed by partitions of positive size and constructed from the capped descendent vertex in an earlier work of the authors of this paper. In the case where all descendents are primary or stationary, the over line correspondence is the identity and one recovers the standard GW/P correspondence conjectured earlier. The proof of the main theorem is by means of the degeneration scheme established by Maulik-Pandharipande. To run this scheme the authors of this paper prove GW/P correspondences for relative and descendent insertions in several simpler geometries. Moreover, the degeneration scheme requires the study of relative theories of \(\mathbb P^1\)-bundles over surfaces \(S\) where \(S\) is either(i) a toric surface, (ii) a \(K3\) surface,(iii) or a \(\mathbb P^1\)-bundle over a higher genus curve \(C\).The authors prove the descendent correspondences for the relative surface geometries (i)-(iii) among which the most difficult ones to establish are the surface geometries (iii).
Pandharipande-Thomas’ recent proof of the full Katz-Klemm-Vafa conjecture for the Gromov-Witten theory of \(K3\) surfaces uses the GW/P correspondences for nontoric hypersurface Calabi-Yau 3-folds established in this paper.

Reviewer: Amin Gholampour (College Park)

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