Abstract
We explore the geometric interpretation of the twisted index of 3d mathcal{N} = 4 gauge theories on S1 × Σ where Σ is a closed Riemann surface. We focus on a rich class of supersymmetric quiver gauge theories that have isolated vacua under generic mass and FI parameter deformations. We show that the path integral localises to a moduli space of solutions to generalised vortex equations on Σ, which can be understood algebraically as quasi-maps to the Higgs branch. We show that the twisted index reproduces the virtual Euler characteristic of the moduli spaces of twisted quasi-maps and demonstrate that this agrees with the contour integral representation introduced in previous work. Finally, we investigate 3d mathcal{N} = 4 mirror symmetry in this context, which implies an equality of enumerative invariants associated to mirror pairs of Higgs branches under the exchange of equivariant and degree counting parameters.
Highlights
Introduction and summaryThe Witten index [1] of supersymmetric quantum mechanics, I = TrH(−1)F, (1.1)is a powerful tool to study geometric aspects of supersymmetric theories
We explore the geometric interpretation of the twisted index of 3d N = 4 gauge theories on S1 × Σ where Σ is a closed Riemann surface
We show that the twisted index reproduces the virtual Euler characteristic of the moduli spaces of twisted quasi-maps and demonstrate that this agrees with the contour integral representation introduced in previous work
Summary
Is a powerful tool to study geometric aspects of supersymmetric theories. For example, in a 1d N = (0, 2) sigma model to a compact target M endowed with a holomorphic vector bundle E, the Witten index can be identified with the holomorphic Euler characteristic χ M, KM1/2 ⊗ E = A(T M ) ch(E). In this limit, the virtual Euler characteristic is independent of q and reduces to the equivariant Rozansky-Witten invariants [22] of MH , associated with the three-manifold S1 × Σ, IC t→1 =. We study the massless fluctuations of the bosonic and fermionic fields at a point on the moduli space M, from which we construct the virtual tangent bundle T vir over M From this discussion, we provide a geometric interpretation of the contour integral formula as the virtual Euler characteristics constructed from T vir.
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