Hostname: page-component-54dcc4c588-mz6gc Total loading time: 0 Render date: 2025-10-09T17:34:12.445Z Has data issue: false hasContentIssue false
  • English
  • Français

A NOTE ON GUNNINGHAM’S FORMULA

Part of: Quantum field theory; related classical field theories Projective and enumerative geometry

Published online by Cambridge University Press:  01 August 2018

JUNHO LEE Open the ORCID record for JUNHO LEE [Opens in a new window]
JUNHO LEE*
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA email junho.lee@ucf.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Gunningham [‘Spin Hurwitz numbers and topological quantum field theory’, Geom. Topol.20(4) (2016), 1859–1907]constructed an extended topological quantum field theory (TQFT) to obtain aclosed formula for all spin Hurwitz numbers. In this note, we use a gluingtheorem for spin Hurwitz numbers to re-prove Gunningham’s formula. We alsodescribe a TQFT formalism naturally induced by the gluing theorem.

Keywords

topological quantum field theoryspin Hurwitz numbersgluing theorem

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Secondary: 81T45: Topological field theories

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 3 , December 2018 , pp. 389 - 401
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The author was partially supported by NSF grant DMS-1206192.

References

Abrams, L., ‘Two-dimensional topological quantum field theories and Frobenius algebras’, J. Knot Theory Ramifications 5(5) (1996), 569–587.Google Scholar
Bryan, J. and Pandharipande, R., ‘Curves in Calabi–Yau threefolds and topological quantum field theory’, Duke Math. J. 126(2) (2005), 369–396.Google Scholar
Bryan, J. and Pandharipande, R., ‘The local Gromov–Witten theory of curves. With an appendix by Bryan, C. Faber, A. Okounkov and Pandharipande’, J. Amer. Math. Soc. 21(1) (2008), 101–136.Google Scholar
Cheng, S.-J. and Wang, W., Dualities and Representations of Lie Superalgebras, Graduate Studies in Mathematics, 144 (American Mathematical Society, Providence, RI, 2012).Google Scholar
Eskin, A., Okounkov, A. and Pandharipande, R., ‘The theta characteristic of a branched covering’, Adv. Math. 217(3) (2008), 873–888.Google Scholar
Gunningham, S., ‘Spin Hurwitz numbers and topological quantum field theory’, Geom. Topol. 20(4) (2016), 1859–1907.Google Scholar
Józefiak, T., ‘Semisimple superalgebras’, in: Algebra. Some Current Trends (Varna 1986), Lecture Notes in Mathematics, 1352 (Springer, Berlin, 1988), 96–113.Google Scholar
Józefiak, T., A Class of Projective Representations of Hyperoctahedral Groups and Schur Q-Functions, Topics in Algebra, Banach Center Publications, 26, Part 2 (PWN-Polish Scientific Publishers, Warsaw, 1990).Google Scholar
Joachim, K., Frobenius Algebras and 2D Topological Quantum Field Theories, London Mathematical Society Student Texts, 59 (Cambridge University Press, Cambridge, 2004).Google Scholar
Kiem, Y.-H. and Li, J., ‘Low degree GW invariants of spin surfaces’, Pure Appl. Math. Q. 7(4) (2011), 1449–1476.Google Scholar
Lee, J. and Parker, T. H., ‘A structure theorem for the Gromov–Witten invariants of Kähler Surfaces’, J. Differential Geom. 77(3) (2007), 483–513.Google Scholar
Lee, J. and Parker, T. H., ‘Spin Hurwitz numbers and the Gromov–Witten invariants of Kahler surfaces’, Comm. Anal. Geom. 21(5) (2013), 1015–1060.Google Scholar
Maulik, D. and Pandharipande, R., ‘New calculations in Gromov–Witten theory’, Pure Appl. Math. Q. 4(2, part 1) (2008), 469–500.Google Scholar
Sergeev, A. N., ‘Tensor algebra of the identity representation as a module over the Lie superalgebras Gl (n, m) and Q (n)’, Math. USSR Sbornik. 51(1985) 419–425.Google Scholar
Serre, J.-P., Topics in Galois Theory (Jones and Bartlett, Boston, 1992).Google Scholar