We consider $\mathcal{N}=2$ supersymmetric gauge theories on four manifolds admitting an isometry. Generalized Killing spinor equations are derived from the consistency of supersymmetry algebrae and solved in the case of four manifolds admitting a $U(1)$ isometry. This is used to explicitly compute the supersymmetric path integral on $S^2 \times S^2$ via equivariant localization. The building blocks of the resulting partition function are shown to contain the three point functions and the conformal blocks of Liouville Gravity.

1 aBawane, Aditya1 aBonelli, Giulio1 aRonzani, Massimiliano1 aTanzini, Alessandro uhttps://doi.org/10.1007/JHEP07(2015)05401436nas a2200145 4500008004100000245008000041210006900121260001000190520096800200100002001168700002401188700002401212700001801236856003601254 2013 en d00aN=2 gauge theories on toric singularities, blow-up formulae and W-algebrae0 aN2 gauge theories on toric singularities blowup formulae and Wal bSISSA3 aWe compute the Nekrasov partition function of gauge theories on the\r\n(resolved) toric singularities C^2/\\Gamma in terms of blow-up formulae. We\r\ndiscuss the expansion of the partition function in the \\epsilon_1,\\epsilon_2\r\n\\to 0 limit along with its modular properties and how to derive them from the\r\nM-theory perspective. On the two-dimensional conformal field theory side, our\r\nresults can be interpreted in terms of representations of the direct sum of\r\nHeisenberg plus W_N-algebrae with suitable central charges, which can be\r\ncomputed from the fan of the resolved toric variety.We provide a check of this\r\ncorrespondence by computing the central charge of the two-dimensional theory\r\nfrom the anomaly polynomial of M5-brane theory. Upon using the AGT\r\ncorrespondence our results provide a candidate for the conformal blocks and\r\nthree-point functions of a class of the two-dimensional CFTs which includes\r\nparafermionic theories.1 aBonelli, Giulio1 aMaruyoshi, Kazunobu1 aTanzini, Alessandro1 aYagi, Futoshi uhttp://hdl.handle.net/1963/6577