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Exact results for {cal N}=2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants

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arxiv 1509.00267 v3 pith:SDQBSOZQ submitted 2015-09-01 hep-th math-phmath.AGmath.MPmath.RT

Exact results for {cal N}=2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants

Mikhail Bershtein , Giulio Bonelli , Massimiliano Ronzani , Alessandro Tanzini This is my paper
classification hep-th math-phmath.AGmath.MPmath.RT
keywords supersymmetriccasedonaldsonequivariantfunctiongaugecompactcontour
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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We provide a contour integral formula for the exact partition function of ${\cal N}=2$ supersymmetric $U(N)$ gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for $U(2)$ ${\cal N}=2$ theory on $\mathbb{P}^2$ for all instanton numbers. In the zero mass case, corresponding to the ${\cal N}=4$ supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a long-standing conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new.

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