Stable maps to Looijenga pairs

Bousseau, Pierrick¹ ; Brini, Andrea² ; van Garrel, Michel³

¹ Department of Mathematics, University of Georgia, Athens, GA, United States
² School of Mathematics and Statistics, University of Sheffield, Sheffield, United Kingdom
³ School of Mathematics, University of Birmingham, Birmingham, United Kingdom

Geometry & topology, Tome 28 (2024) no. 1, pp. 393-496.

Résumé

A log Calabi–Yau surface with maximal boundary, or Looijenga pair, is a pair $(Y,D)$ with $Y$ a smooth rational projective complex surface and $D=D_1+\cdots +D_l\in |-K_Y|$ an anticanonical singular nodal curve. Under some natural conditions on the pair, we propose a series of correspondences relating five different classes of enumerative invariants attached to $(Y,D)$:

the log Gromov–Witten theory of the pair $(Y,D)$,

the Gromov–Witten theory of the total space of ${\oplus }_i{\mathcal{𝒪}}_Y(-D_i)$,

the open Gromov–Witten theory of special Lagrangians in a Calabi–Yau $3$–fold determined by $(Y,D)$,

the Donaldson–Thomas theory of a symmetric quiver specified by $(Y,D)$, and

a class of BPS invariants considered in different contexts by Klemm and Pandharipande, Ionel and Parker, and Labastida, Mariño, Ooguri and Vafa.

We furthermore provide a complete closed-form solution to the calculation of all these invariants.

Détail

DOI : 10.2140/gt.2024.28.393

Keywords: Gromov–Witten invariants, mirror symmetry, log Calabi–Yau surfaces, Donaldson–Thomas invariants

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