Instanton on toric singularities and black hole countings

We compute the instanton partition function for ${\cal N}=4$ U(N) gaugetheories living on toric varieties, mainly of type $\R^4/\Gamma_{p,q}$including $A_{p-1}$ or $O_{\PP_1}(-p)$ surfaces. The results providemicroscopic formulas for the partition functions of black holes made out ofD4-D2-D0 bound states wrapping four-dimensional toric varieties inside aCalabi-Yau. The partition function gets contributions from regular andfractional instantons. Regular instantons are described in terms of symmetricproducts of the four-dimensional variety. Fractional instantons are built outof elementary self-dual connections with no moduli carrying non-trivial fluxesalong the exceptional cycles of the variety. The fractional instantoncontribution agrees with recent results based on 2d SYM analysis. The partitionfunction, in the large charge limit, reproduces the supergravity macroscopicformulae for the D4-D2-D0 black hole entropy.Comment: 29 pages, 3 fig Section 5 is improved by the inclusion of a detailed comparison between the instanton partition function and the D4-D2-D0 black hole entropy formula coming from supergravit