Exact results for \( \mathcal{N} =2\) supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants

DOI10.1007/JHEP07(2016)023zbMATH Open1390.81302arXiv1509.00267WikidataQ64038481 ScholiaQ64038481MaRDI QIDQ1638892

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Publication date: 12 June 2018

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Abstract: We provide a contour integral formula for the exact partition function of $calN=2$ 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 $U(2)$ $calN=2$ theory on $mathbb{P}^2$ for all instanton numbers. In the zero mass case, corresponding to the $calN=4$ 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.

Full work available at URL: https://arxiv.org/abs/1509.00267

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