Effective lagrangian for 3d N=4 SYM theories for any gauge group and monopole moduli spaces -: art. no. 027

We construct low energy effective lagrangians for 3d N = 4 supersymmetric Yang-Mills theory with any gauge group. They represent supersymmetric sigma models at hyper-Kahler manifolds of dimension 4r (r is the rang of the group). In the asymptotic region, perturbatively exact explicit expression for the metric are written. We establish the relationship of this metric with the TAUB-NUT metric describing the perturbatively exact effective lagrangians for unitary groups and monopole moduli spaces: the former is obtained out of the latter by a proper hyper-Kahler reduction. We describe in details the reduction procedure for SO/Sp/G2 gauge groups, where it can also be given a natural interpretation in D-brane language. We conjecture that the exact nonperturbative metrics can be obtained by a similar hyper-Kahler reduction from the corresponding multidimensional Atiyah-Hitchin metrics.