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Stable maps to Looijenga pairs

Pierrick Bousseau, Andrea Brini and Michel van Garrel

Geometry & Topology 28 (2024) 393–496

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 = D1 + + Dl |KY | 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):

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

  2. the Gromov–Witten theory of the total space of i𝒪Y (Di),

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

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

  5. 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.

Gromov–Witten invariants, mirror symmetry, log Calabi–Yau surfaces, Donaldson–Thomas invariants
Mathematical Subject Classification
Primary: 14J33, 14J81, 14N35, 16G20, 53D45
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
Pierrick Bousseau
Department of Mathematics
University of Georgia
Athens, GA
United States
Andrea Brini
School of Mathematics and Statistics
University of Sheffield
United Kingdom
Michel van Garrel
School of Mathematics
University of Birmingham
United Kingdom

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