Prepared for submission to JHEP tt ∗ Geometry in 3 and 4 Dimensions

Abstract: We consider the vacuum geometry of supersymmetric theories with 4 supercharges, on a flat toroidal geometry. The 2 dimensional vacuum geometry is known to be captured by the tt ∗ geometry. In the case of 3 dimensions, the parameter space is (T 2 × R)N and the vacuum geometry turns out to be a solution to a gen-eralization of monopole equations in 3N dimensions where the relevant topological ring is that of line operators. We compute the generalization of the 2d cigar ampli-tudes, which lead to S2 × S1 or S3 partition functions which are distinct from the supersymmetric partition functions on these spaces, but reduce to them in a certain limit. We show the sense in which these amplitudes generalize the structure of 3d Chern-Simons theories and 2d RCFT’s. In the case of 4 dimensions the parameter space is of the form (T 3 × R)M × T 3N, and the vacuum geometry is a solution to a mixture of generalized monopole equations and generalized instanton equations (known as hyper-holomorphic connections). In this case the topological rings are associated to surface operators. We discuss the physical meaning of the generalized Nahm transforms which act on all of these geometries. ar X i