Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-26T12:04:03.869Z Has data issue: false hasContentIssue false
  • English
  • Français

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: renzo@math.colostate.edu
PAUL JOHNSON
Affiliation:
Hicks Buliding, Mathematics and Statistics, University of Sheffield, SheffieldS10 2TN. e-mail: paul.johnson@shef.ac.uk
HANNAH MARKWIG
Affiliation:
Auf der Morgenstelle 10, Mathematisch-Naturwissenschaftliche Fakultät, Universität Tübingen, 72076Tübingen, 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, CambridgeCB3 0WA. e-mail: dr508@cam.ac.uk
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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).

References

Abramovich, D. and Chen, Q.. Stable logarithmic maps to Deligne–Faltings pairs. II, Asian J. Math. 18 (2014), pp. 465–488.10.4310/AJM.2014.v18.n3.a5CrossRefGoogle Scholar
Abramovich, D., Chen, Q., Gross, M. and Siebert, B.. Decomposition of degenerate Gromov–Witten invariants. Available at http://www.math.brown.edu/˜abrmovic/PAPERS/LOG/decomposition-formula.pdf.Google Scholar
Ardila, F. and Block, F.. Universal polynomials for Severi degrees of toric surfaces. Adv. Math. 237 (2013), pp. 165–193.10.1016/j.aim.2013.01.002CrossRefGoogle Scholar
Ardila, F. and Brugalle, E.. The double Gromov–Witten invariants of Hirzebruch surfaces are piecewise polynomial. arXiv:1412.4563, (2014).Google Scholar
Block, F., Gathmann, A. and Markwig, H.. Psi-floor diagrams and a Caporaso–Harris type recursion. Israel J. Math. 191 (2012), pp. 405–449.10.1007/s11856-011-0216-0CrossRefGoogle Scholar
Block, F. and Göttsche, L.. Fock spaces and refined Severi degrees. Int. Math. Res. Not. 2016 (2016), p. 6553.Google Scholar
Brugallé, E.. Floor diagrams relative to a conic, and GW-W invariants of del Pezzo surfaces. Adv. Math. 279 (2015), pp. 438–500.10.1016/j.aim.2015.04.006CrossRefGoogle Scholar
Brugallé, E. and Mikhalkin, G.. Enumeration of curves via floor diagrams. C. R. Math. Acad. Sci. Paris. 345 (2007), pp. 329–334.10.1016/j.crma.2007.07.026CrossRefGoogle Scholar
Brugallé, E. and Mikhalkin, G.. Floor decompositions of tropical curves: the planar case. Proceedings of the 15th Gökova Geometry-Topology Conference. (2008), pp. 64–90. arXiv:0812.3354.Google Scholar
Cavalieri, R., Johnson, P. and Markwig, H.. Tropical Hurwitz numbers. J. Alg. Comb. 32 (2010), pp. 241–265.10.1007/s10801-009-0213-0CrossRefGoogle Scholar
Cavalieri, R., Johnson, P. and Markwig, H.. Wall crossings for double Hurwitz numbers. Adv. Math. 228 (2011), pp. 1894–1937.10.1016/j.aim.2011.06.021CrossRefGoogle Scholar
Cavalieri, R., Johnson, P., Markwig, H. and Ranganathan, D.. A graphical interface for the Gromov–Witten theory of curves. Proceedings of the 2015 Algebraic Geometry Summer Institute (arXiv:1604.07250).Google Scholar
Cavalieri, R., Markwig, H. and Ranganathan, D.. Tropicalizing the space of admissible covers. Math. Ann. 364 (2016), pp. 1275–1313.10.1007/s00208-015-1250-8CrossRefGoogle Scholar
Cavalieri, R., Markwig, H. and Ranganathan, D.. Tropical compactification and the Gromov–Witten theory of ${\mathbb {P}^1}$ . Sel. Math., New Ser. 23 (2017), pp. 1027–1060.10.1007/s00029-016-0265-7CrossRefGoogle Scholar
Chen, Q.. Stable logarithmic maps to Deligne–Faltings pairs I. Ann. of Math. 180 (2014), pp. 455–521.10.4007/annals.2014.180.2.2CrossRefGoogle Scholar
Chen, Q.. The degeneration formula for logarithmic expanded degenerations. J. Algebr. Geom. 23 (2014), pp. 341–392.10.1090/S1056-3911-2013-00614-1CrossRefGoogle Scholar
Cooper, Y. and Pandharipande, R.. A Fock space approach to Severi degrees Proc. London Math. Soc. 114 (2017), pp. 476–494.10.1112/plms.12017CrossRefGoogle Scholar
Fomin, S. and Mikhalkin, G.. Labelled floor diagrams for plane curves. J. Eur. Math. Soc. 12 (2010), pp. 1453–1496.10.4171/JEMS/238CrossRefGoogle Scholar
Gathmann, A. and Markwig, H.. The Caporaso–Harris formula and plane relative Gromov-Witten invariants in tropical geometry. Math. Ann. 338 (2007), pp. 845–868.10.1007/s00208-007-0092-4CrossRefGoogle Scholar
Gathmann, A. and Markwig, H.. The numbers of tropical plane curves through points in general position. J. Reine Angew. Math. (Crelle’s Journal), 602 (2007), pp. 155–177.Google Scholar
Gathmann, A. and Markwig, H.. Kontsevich’s formula and the WDVV equations in tropical geometry. Adv. Math. 217 (2008), pp. 537–560.10.1016/j.aim.2007.08.004CrossRefGoogle Scholar
Graber, T. and Vakil, R.. Relative virtual localisation and vanishing of tautological classes on moduli spaces of curves. Duke Math. J. 130 (2005), pp. 1–37.10.1215/S0012-7094-05-13011-3CrossRefGoogle Scholar
Gross, A.. Intersection theory on tropicalisations of toroidal embeddings. arXiv preprint arXiv:1510.04604, (2015).Google Scholar
Gross, M.. Mirror Symmetry for ${\mathbb{P}^2}$ and tropical geometry. Adv. Math. 224 (2010), pp. 169–245.10.1016/j.aim.2009.11.007CrossRefGoogle Scholar
Gross, M., Pandharipande, R. and Siebert, B.. The tropical vertex. Duke Math. J. 153 (2010), pp. 297–362.CrossRefGoogle Scholar
Gross, M. and Siebert, B.. Logarithmic Gromov-Witten invariants. J. Amer. Math. Soc. 26 (2013), pp. 451–510.10.1090/S0894-0347-2012-00757-7CrossRefGoogle Scholar
Kim, B.. Logarithmic stable maps. arXiv:0807.3611, (2008).Google Scholar
Kim, B., Lho, H. and Ruddat, H.. The degeneration formula for stable log maps. arXiv:1803.04210, (2018).Google Scholar
Li, J.. A degeneration formula of GW-invariants. J. Diff. Geom. 60 (2002), pp. 199–293.Google Scholar
Mandel, T. and Ruddat, H.. Descendant log Gromov-Witten invariants for toric varieties and tropical curves. arXiv:1612.02402, (2016).Google Scholar
Mandel, T. and Ruddat, H.. Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves. arXiv:1902.07183, (2019).Google Scholar
Markwig, H., The enumeration of plane tropical curves, 2006. PhD thesis, TU Kaiserslautern.Google Scholar
Markwig, H. and Rau, J., Tropical descendant Gromov-Witten invariants. Manuscripta Math. 129 (2009), pp. 293–335.CrossRefGoogle Scholar
Mikhalkin, G.. Enumerative tropical geometry in ${\mathbb{R}^2}$ . J. Amer. Math. Soc. 18 (2005), pp. 313–377.CrossRefGoogle Scholar
Mikhalkin, G.. Moduli spaces of rational tropical curves. In Proceedings of the 13th Gökova geometry–topology conference, Gökova, Turkey. (May 28–June 2, 2006). (Cambridge, MA: International Press, 2007) pp. 39–51.Google Scholar
Nishinou, T. and Siebert, B.. Toric degenerations of toric varieties and tropical curves. Duke Math. J. 135 (2006), pp. 1–51.10.1215/S0012-7094-06-13511-1CrossRefGoogle Scholar
Okounkov, A. and Pandharipande, R.. Gromov–Witten theory, Hurwitz theory, and completed cycles. Ann. of Math. 163 (2006), pp. 517–560.10.4007/annals.2006.163.517CrossRefGoogle Scholar
Overholser, P. D.. Descendant tropical mirror symmetry for ${\mathbb{P}^2}$ . Commun. Number Theory Phys. 10 (2016), pp. 739–803.CrossRefGoogle Scholar
Ranganathan, D.. Skeletons of stable maps I: rational curves in toric varieties. J. Lond. Math. Soc. (to appear) arXiv:1506.03754, (2017).CrossRefGoogle Scholar
Ranganathan, D.. Logarithmic gromov-witten theory with expansions. arXiv:1903.09006, (2019).Google Scholar
Richter–Gebert, J., Sturmfels, B. and Theobald, T.. First steps in tropical geometry. Idempotent Mathematics and Mathematical Physics, Proceedings Vienna. (2003). arXiv:math/0306366.Google Scholar
Wick, G. C.. The evaluation of the collision matrix. Physical Rev. (2). 80 (1950), pp. 268–272.10.1103/PhysRev.80.268CrossRefGoogle Scholar