Download this article
 Download this article For screen
For printing
Recent Issues

Volume 28
Issue 7, 3001–3510
Issue 6, 2483–2999
Issue 5, 1995–2482
Issue 4, 1501–1993
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1364-0380 (online)
ISSN 1465-3060 (print)
Author Index
To Appear
 
Other MSP Journals
Stable maps to Looijenga pairs

Pierrick Bousseau, Andrea Brini and Michel van Garrel

Geometry & Topology 28 (2024) 393–496
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 = 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.

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

Open Access made possible by participating institutions via Subscribe to Open.