On N $$ mathcal{N} $$ = 2 supersymmetric gauge theories on S 2 × S 2

We construct a supergeometry based on s 2×s 2 on which four dimensional $$ mathcal{n} $$ = 2 gauge theories can be placed supersymmetrically while preserving all supersymmetries. by embedding the supergeometry in four dimensional $$ mathcal{n} $$ = 2 supergravity we are able to construct an arbitrary $$ mathcal{n} $$ = 2 gauge theory on s 2 × s 2. we show that $$ mathcal{n} $$ = 2 gauge theories are invariant under the exceptional superalgebra d(2, 1, α), where α is the ratio of the radii of the two s 2’s. we solve the supersymmetry fixed points equations for a choice of supercharge in d(2, 1, α). the solution of these bps equations, which we find, would serve as the exact saddle point configurations of a localization computation of the partition function of $$ mathcal{n} $$ = 2 gauge theories on s 2 × s 2.