On the Witten index of 3d $\mathcal{N}=2$ unitary SQCD with general CS levels

Abstract

We consider unitary SQCD, a three-dimensional \mathcal{N}=2𝒩=2 supersymmetric Chern-Simons-matter theory consisting of one U(N_c)_{k, k+l N_c}U(Nc)k,k+lNc vector multiplet coupled to n_fnf fundamental and n_ana antifundamental chiral multiplets, where kk and ll parameterise generic CS levels for U(N_c)=(SU(N_c)\times U(1))/\mathbb{Z}_{N_c}U(Nc)=(SU(Nc)×U(1))/ℤNc. We study the moduli space of vacua of this theory with n_a=0na=0, for generic values of the parameters N_c, k, l, n_fNc,k,l,nf and with a non-zero Fayet-Ilopoulos parameter turned on. We uncover a rich pattern of vacua including Higgs, topological and hybrid phases. This allows us to derive a closed-form formula for the flavoured Witten index of unitary SQCD for any n_f\neq n_anf≠na, generalising previously known results for either l=0 or n_f=n_anf=na. Finally, we analyse the vacuum structure of recently proposed infrared-dual gauge theories and we match vacua across the dualities, thus providing intricate new checks of those dualities. Incidentally, we also discuss a seemingly new level/rank duality for pure CS theories with U(N) x U(N’) gauge group.