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

Published online by Cambridge University Press:  22 February 2021

RENZO CAVALIERI
Affiliation:
Weber Building, Department of Mathematics, Colorado State University, Fort Collins CO80523, U.S.A e-mail: renzo@math.colostate.edu
PAUL JOHNSON
Affiliation:
Hicks Buliding, Mathematics and Statistics, University of Sheffield, Sheffield S10 2TN. e-mail: paul.johnson@shef.ac.uk
HANNAH MARKWIG
Affiliation:
Auf der Morgenstelle 10, Mathematisch-Naturwissenschaftliche Fakultät, Universität Tübingen, 72076 Tübingen, Germany. e-mail: hannah@math.uni-tuebingen.de
DHRUV RANGANATHAN
Affiliation:
Center for Mathematical Sciences, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WA. e-mail: dr508@cam.ac.uk

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.

Type
Research Article
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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