We show that equivariant Donaldson polynomials of compact toric surfaces can be calculated as residues of suitable combinations of Virasoro conformal blocks, by building on AGT correspondence between $\mathcal{N}=2$ supersymmetric gauge theories and two-dimensional conformal field theory. Talk presented by A.T. at the conference Interactions between Geometry and Physics — in honor of Ugo Bruzzo’s 60th birthday 17–22 August 2015, Guarujá, São Paulo, Brazil, mostly based on Bawane et al. (0000) and Bershtein et al. (0000).

%B Journal of Geometry and Physics %V 118 %P 40 - 50 %G eng %U http://www.sciencedirect.com/science/article/pii/S0393044017300165 %R https://doi.org/10.1016/j.geomphys.2017.01.012 %0 Journal Article %J Journal of High Energy Physics %D 2016 %T Exact results for N=2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants %A Mikhail Bershtein %A Giulio Bonelli %A Massimiliano Ronzani %A Alessandro Tanzini %XWe provide a contour integral formula for the exact partition function of $\mathcal{N}=2$ 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 $U(2)\; \mathcal{N}=2^\star$ theory on $\mathbb{P}^2$ for all instanton numbers. In the zero mass case, corresponding to the $\mathcal{N}=4$ 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 longstanding conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new.

%B Journal of High Energy Physics %V 2016 %P 23 %8 Jul %G eng %U https://doi.org/10.1007/JHEP07(2016)023 %R 10.1007/JHEP07(2016)023