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

Abstract

We consider unitary SQCD, a three-dimensional $\mathcal{N}=2$ supersymmetric Chern-Simons-matter theory consisting of one $U(N_c{)}_{k,k+l{N}_c}$ vector multiplet coupled to $n_f$ fundamental and $n_a$ antifundamental chiral multiplets, where $k$ and $l$ parameterise generic CS levels for $U(N_c)=(SU(N_c)\times U(1))/{\mathbb{Z}}_{N_c}$. We study the moduli space of vacua of this theory with $n_a=0$, for generic values of the parameters $N_c,k,l,n_f$ 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 $n_f\ne n_a$, generalising previously known results for either $l=0$ or $n_f=n_a$. 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 $U(N)\times U(N^{\prime })$ gauge group.

37 pages+appendix; v2: corrected typos and references; v3: small clarifications after feedback from SciPost referees

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Cited by in corpus (3)

• One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices and CS theories
• Schubert line defects in 3d GLSMs, part II: Partial flag manifolds and parabolic quantum polynomials
• Schubert line defects in 3d GLSMs, part I: Complete flag manifolds and quantum Grothendieck polynomials