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Counting curves on Hirzebruch surfaces: tropical geometry and the Fock space

Part of: Projective and enumerative geometry

Published online by Cambridge University Press:  22 February 2021

RENZO CAVALIERI ,
PAUL JOHNSON ,
HANNAH MARKWIG  and
DHRUV RANGANATHAN
RENZO CAVALIERI
Affiliation:
Weber Building, Department of Mathematics, Colorado State University, Fort CollinsCO80523, U.S.A e-mail: [email protected]
PAUL JOHNSON
Affiliation:
Hicks Buliding, Mathematics and Statistics, University of Sheffield, SheffieldS10 2TN. e-mail: [email protected]
HANNAH MARKWIG
Affiliation:
Auf der Morgenstelle 10, Mathematisch-Naturwissenschaftliche Fakultät, Universität Tübingen, 72076Tübingen, Germany. e-mail: [email protected]
DHRUV RANGANATHAN
Affiliation:
Center for Mathematical Sciences, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, CambridgeCB3 0WA. e-mail: [email protected]
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Abstract

We study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams are then used to connect tropical curve counting to the algebra of operators on the bosonic Fock space, and are showno coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block–Göttsche, Cooper–Pandharipande and Block–Gathmann–Markwig.

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 1 , July 2021 , pp. 165 - 205
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

Supported by NSF grant FRG-1159964 and Simons collaboration grant 420720.

‡

Supported by DFG-grant MA 4797/6-1.

§

Supported by NSF grants CAREER DMS-1149054 (PI: Sam Payne) and DMS 1128155 (Institute for Advanced study).

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