tt* geometry in 3 and 4 dimensions

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 (T2 × $$ \mathbb{R} $$)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 …