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DONALDSON–THOMAS INVARIANTS OF LOCAL ELLIPTIC SURFACES VIA THE TOPOLOGICAL VERTEX

Part of: Cycles and subschemes Projective and enumerative geometry

Published online by Cambridge University Press:  21 March 2019

JIM BRYAN  and
MARTIJN KOOL Open the ORCID record for MARTIJN KOOL [Opens in a new window]
JIM BRYAN
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2
MARTIJN KOOL
Affiliation:
Mathematical Institute, Utrecht University, Room 502, Budapestlaan 6, 3584 CD Utrecht, The Netherlands

Abstract

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We compute the Donaldson–Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the invariants in terms of the topological vertex. Utilizing identities for the topological vertex proved in Bryan et al. [‘Trace identities for the topological vertex’, Selecta Math. (N.S.)24 (2) (2018), 1527–1548, arXiv:math/1603.05271], we derive product formulas for the partition functions. The connected version of the partition function is written in terms of Jacobi forms. In the special case where the elliptic surface is a K3 surface, we get a derivation of the Katz–Klemm–Vafa formula for primitive curve classes which is independent of the computation of Kawai–Yoshioka.

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 14C05: Parametrization (Chow and Hilbert schemes)
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e7
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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