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Verlinde formulae on complex surfaces: K-theoretic invariants

Part of: Families, fibrations Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  11 January 2021

L. Göttsche ,
M. Kool  and
R. A. Williams
L. Göttsche
Affiliation:
Mathematics Group, International Centre for Theoretical Physics, Strada Costiera 11, 34100, Trieste, Italy; E-mail: gottsche@ictp.it.
M. Kool
Affiliation:
Mathematical Institute, Utrecht University, P.O. Box 80010, 3508 TA, Utrecht, The Netherlands; E-mail: m.kool1@uu.nl.
R. A. Williams
Affiliation:
University of the Bahamas, Department of Mathematics, University Drive, Nassau, The Bahamas; E-mail: runako.williams@ub.edu.bs.

Abstract

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We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between K-theoretic Donaldson invariants studied by Göttsche and Nakajima-Yoshioka and K-theoretic Vafa-Witten invariants introduced by Thomas and also studied by Göttsche and Kool. We verify our conjectures in many examples (for example, on K3 surfaces).

Keywords

Moduli of (Higgs) sheaves on surfacesVerlinde formulaK-theoretic invariants

MSC classification

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles
Secondary: 14D21: Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14J80: Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) 14J81: Relationships with physics
Type
Algebraic and Complex Geometry
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e5
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Creative Commons
Creative Common License - CCCreative Common License - BY
Published by Cambridge University Press. 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), 2020. Published by Cambridge University Press

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