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K-theoretic Donaldson invariants via instanton counting

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arxiv math/0611945 v1 pith:ENN6TY55 submitted 2006-11-30 math.AG hep-thmath.DG

Lothar Göttsche , Hiraku Nakajima , Kota Yoshioka This is my paper · ORCID
classification math.AGhep-thmath.DG
keywords invariantsdonaldsonfunctionk-theoreticsurfacetermswallcrossingalgebraic
verification ladder T0 review T1 audit T2 compute T3 formal
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abstract

In this paper we study the holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 semistable sheaves on an algebraic surface X, which can be viewed as $K$-theoretic versions of the Donaldson invariants. In particular, if X is a smooth projective toric surface, we determine these invariants and their wallcrossing in terms of the K-theoretic version of the Nekrasov partition function (called 5-dimensional supersymmetric Yang-Mills theory compactified on a circle in the physics literature). Using the results of math.AG/0606180 we give an explicit generating function for the wallcrossing of these invariants in terms of elliptic functions and modular forms.

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Cited by 2 Pith papers

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  1. Generalised global symmetries in 5d $\mathcal{N}=1$ theories from the blow-up equations

    hep-th 2026-07 conditional novelty 7.0 of 10

    Fractional exponents of the blow-up prefactor exp(-V_n) on 1-form backgrounds encode cubic and mixed anomalies of 5d N=1 SCFTs, deciding 2-groups versus mixed anomalies once the faithful UV symmetry is known from the index.

  2. Refined Vafa-Witten invariants for toric surfaces from supersymmetric localization in 5D gauge theory

    hep-th 2026-07 conditional novelty 6.0 of 10

    5D N=1* supersymmetric localization on toric surfaces equals refined Vafa–Witten invariants for odd first Chern class; the even-c1 case fails without a hand-added constant.

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