3d N=1 Chern-Simons-matter theory and localization

We consider the most general, classically-conformal, three-dimensional N=1 Chern-Simons-matter theory with global symmetry Sp(2) and gauge group U(N)×U(N). We show that the Lagrangian in the on-shell formulation of the theory admits one more free parameter as compared to the theory formulated in off-shell N=1 superspace. The theory on T3 can be formally localized. We partially carry out the localization procedure for the theory on T3 with periodic boundary conditions. In particular we show that restricting to the saddle points with vanishing gauge connection gives a trivial contribution to the partition function, i.e. the bosonic and fermionic contributions exactly cancel each other.