Stable maps to Looijenga pairs

Bousseau, Pierrick; Brini, Andrea; van Garrel, Michel

doi:10.2140/gt.2024.28.393

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Stable maps to Looijenga pairs

Pierrick Bousseau, Andrea Brini and Michel van Garrel

Geometry & Topology 28 (2024) 393–496

DOI: 10.2140/gt.2024.28.393

Abstract

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 $D = D_{1} + \dots + D_{l} \in | - K_{Y} |$ 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 $\oplus_{i} 𝒪_{Y} (- D_{i})$ ,
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.

Keywords

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

Mathematical Subject Classification

Primary: 14J33, 14J81, 14N35, 16G20, 53D45

References

Publication

Received: 8 April 2022

Revised: 8 April 2022

Accepted: 6 May 2022

Published: 27 February 2024

Proposed: Richard P Thomas

Seconded: Mark Gross, Dan Abramovich

Authors

Pierrick Bousseau
Department of Mathematics University of Georgia Athens, GA United States

Andrea Brini
School of Mathematics and Statistics University of Sheffield Sheffield United Kingdom

Michel van Garrel
School of Mathematics University of Birmingham Birmingham United Kingdom

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Volume 28, issue 1 (2024)