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 $(T^{2}\times {\mathbb R})^N$ and the vacuum geometry turns
out to be a solution to a generalization of monopole equations in $3N$
dimensions where the relevant topological ring is that of line operators. We
compute the generalization of the 2d cigar amplitudes, which lead to $S^2\times
S^1$ or $S^3$ 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 $(T^3\times {\mathbb R})^M\times T^{3N}$, 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.