Modularity of supersymmetric partition functions

We discover a modular property of supersymmetric partition functions theories with R-symmetry in four dimensions. This is, sense, the generalization invariance function two-dimensional on torus i.e. elliptic genus. The question are manifolds homeomorphic to ones obtained by gluing solid tori. Such involves choice large diffeomorphism boundary torus, along gauge transformation for background flavor symmetry connections, if present. Our is manifestation consistency procedure. used rederive Cardy formula four-dimensional that has played key role computing entropy black holes. To be concrete, we work N=1 but expect versions our result apply more widely other