Twisted indices, Bethe ideals and 3d $\mathcal{N}=2$ infrared dualities

Abstract

We study the topologically twisted index of 3d $\mathcal{N}=2$ supersymmetric gauge theories with unitary gauge groups. We implement a Gröbner basis algorithm for computing the ${\Sigma }_g\times S^1$ index explicitly and exactly in terms of the associated Bethe ideal, which is defined as the algebraic ideal associated with the Bethe equations of the corresponding 3d $A$-model. We then revisit recently discovered infrared dualities for unitary SQCD with gauge group $U(N_c{)}_{k,k+l{N}_c}$ with $l\ne 0$, namely the Nii duality that generalises the Giveon-Kutasov duality, the Amariti-Rota duality that generalises the Aharony duality, and their further generalisations in the case of arbitrary numbers of fundamental and antifundamental chiral multiplets. In particular, we determine all the flavour Chern-Simons contact terms needed to make these dualities work. This allows us to check that the twisted indices of dual theories match exactly. We also initiate the study of the Witten index of unitary SQCD with $l\ne 0$.

arXiv comment46 pages plus appendix; v2: fixed typos and added references; v3: small corrections in section 2. JHEP version

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• Spin-$s$ Rational $Q$-system
• On the Witten index of 3d $\mathcal{N}=2$ unitary SQCD with general CS levels
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices and CS theories
• Bridging 4D QFTs and 2D VOAs via 3D high-temperature EFTs
• Decomposition squared
• 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