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Vafa-Witten invariants for projective surfaces I: stable case

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arxiv 1702.08487 v4 pith:HNABC5ZP submitted 2017-02-27 math.AG hep-thmath.DG

classification math.AGhep-thmath.DG
keywords vafa-witteninvariantsmodulispacesurfaceswhenactionadmits
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abstract

On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a $\mathbb C^*$ action with compact fixed locus. Applying virtual localisation we define invariants constant under deformations. When the vanishing theorem of Vafa-Witten holds, the result is the (signed) Euler characteristic of the moduli space of instantons. In general there are other, rational, contributions. Calculations of these on surfaces with positive canonical bundle recover the first terms of modular forms predicted by Vafa and Witten.

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Cited by 2 Pith papers

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  1. Revisiting the Quantum Geometry of Torus-fibered Calabi-Yau Threefolds

    hep-th 2025-10 conditional novelty 7.0 of 10

    The conjectured Jacobi modularity of topological string amplitudes on torus-fibered Calabi-Yau threefolds is derived conditionally from the wave-function property under the relative conifold monodromy, which also maps...

  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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