Twisted indices of 3d $$ \mathcal{N} $$ = 4 gauge theories and enumerative geometry of quasi-maps

Abstract

We explore the geometric interpretation of the twisted index of 3d N = 4 gauge theories on S 1 × 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 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. arXiv:1812.05567v3 [hep-th] 2 Jul 2019 C Characteristic classes on a fixed locus and their integration 60

Factorisation of 3d $$ \mathcal{N} $$ = 4 twisted indices and the geometry of vortex moduli space

“…Protected indices in supersymmetric QFT often have an interpretation in terms of the geometry of an appropriate moduli space of classical solutions. In recent work, Bullimore et al [5,6], have suggested such an interpretation for the twisted index. In this paper we will confirm this proposal and extend it in several ways.…”

“…As mentioned above, Bullimore et al [5,6] have proposed a geometrical interpretation of the twisted index of N = 4 theories in terms of χ t genera of vortex moduli spaces. In this case the relevant configurations are "global" vortices corresponding in the IR to unbased quasi-maps from P 1 to the Higgs branch.…”

“…In recent work [5,6] give a geometric interpretation of the N = 2 twisted index in terms of the χ t genera of vortex moduli spaces -we expand on this geometric interpretation in the N = 4 case with the angular momentum refinement, q, turned on. In this section we describe how holomorphic factorisation relates to the geometry of global and local Laumon space and the localisation computations of [25].…”

Blocks and vortices in the 3d ADHM quiver gauge theory

“…In the limit q → 0, the authors argue in [28], that χ t (Q d ) becomes the Poincaré polynomial of the compact core π −1 (0) ⊂ Q d . In this section we study the same "Poincaré polynomial" limit of the ADHM theory and, by analogy with T ρ [SU(N )], 8 we expect that we are again computing a Poincaré polynomial of the compact core of a putative resolution of the space of quasimaps to Hilb N (C 2 ):…”

“…Three dimensional gauge theories with N = 4 supersymmetry sit at the centre of a remarkable web of connections between physics and mathematics [1][2][3][4][5][6][7][8][9][10]. On the physics side, these theories flow to strongly interacting conformal fixed points in the IR and, in many cases, the resulting conformal theories coincide with worldvolume theories of M-theory membranes [11,12].…”