Abstract
This paper studies supersymmetric ground states of 3d \mathcal{N}=4𝒩=4 supersymmetric gauge theories on a Riemann surface of genus gg. There are two distinct spaces of supersymmetric ground states arising from the AA and BB type twists on the Riemann surface, which lead to effective supersymmetric quantum mechanics with four supercharges and supermultiplets of type \mathcal{N}=(2,2)𝒩=(2,2) and \mathcal{N}=(0,4)𝒩=(0,4) respectively. We compute the space of supersymmetric ground states in each case, graded by flavour and R-symmetries and in different chambers for real mass and FI parameters, for a large class of supersymmetric gauge theories. The results are formulated geometrically in terms of the Higgs branch geometry. We perform extensive checks of compatibility with the twisted index and mirror symmetry.
Highlights
This paper studies the supersymmetric ground states of 3d N = 4 gauge theories on × Σ, where Σ is a Riemann surface of genus g
We introduce real mass parameters for Higgs branch global symmetry corresponding to tri-hamiltonian isometries of X
Let Ω denote the full Hilbert space of the effective N = 4 supersymmetric quantum mechanics. This transforms as a unitary representation of the R-symmetry U(1)H × U(1)C and global symmetry TH × TC left unbroken by generic real mass and FI parameters
Summary
We further assume that for generic real mass parameters, the fixed locus of corresponding ∗ actions on X consists of isolated fixed points Under these assumptions, we will be able to determine the effective supersymmetric quantum mechanics exactly and compute the spaces of supersymmetric ground states, graded by a R-symmetries and global symmetries. The relationship is subtle due to non-compactness of the target space Our construction overcomes this difficulty by introducing real FI and mass parameters that break the R-symmetry necessary to perform a full A- and B-type topological twist on a generic three-manifold M3 but are perfectly compatible on the specific background M3 = × Σ
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