The Twisted Index and Topological Saddles

Abstract

The twisted index of 3d N = 2 gauge theories on S 1 × Σ has an algebro-geometric interpretation as the Witten index of an effective supersymmetric quantum mechanics. In this paper, we consider the contributions to the supersymmetric quantum mechanics from topological saddle points in supersymmetric localization of abelian gauge theories. Topological saddles are configurations where the matter fields vanish and the gauge symmetry is unbroken, which exist for non-vanishing effective Chern-Simons levels. We compute the contributions to the twisted index from both topological and vortex-like saddles points and show that their combination recovers the Jeffrey-Kirwan residue prescription for the twisted index and its wall-crossing.

Supersymmetric Ground States of 3d $\mathcal{N}=4$ Gauge Theories on a Riemann Surface

“…The aim of this section is use supersymmetric localisation to reduce the A-twisted theory on R × Σ to an explicit N = 4 supersymmetric quantum mechanics that captures the space of supersymmetric ground states H. The method is supersymmetric localisation. Following our previous work [9][10][11], we will choose a Higgs branch type localisation scheme leading to an algebro-geometric interpretation of the space of supersymmetric ground states.…”

“…This provides a non-trivial consistency check on our computations. Alternative localisation schemes that lead to geometric interpretation of the supersymmetric twisted index will underpin the constructions in this paper [9][10][11].…”

The colored Jones polynomials as vortex partition functions

“…In this paper, for simplicity we identify the stability parameters with the FI parameters ξ a = Im(τ a ) (see [28] for the difference between them). Although the partition function has a boundary contribution at u a = ±∞, we assume that the boundary contribution is irrelevant to the vortex partition functions in this paper and only focus on the bulk (first) factor (see [34] for an interpretation of the boundary contribution as topological saddles of an effective supersymmetric quantum mechanics).…”

Quantum K theory of symplectic Grassmannians

“…Another such uplift has recently attracted attention: quantum K theory [2][3][4][5][6][7][8][9] arises in descriptions of three-dimensional supersymmetric gauge theories, 'uplifting' Gromov-Witten invariants of two-dimensional (2,2) supersymmetric theories. In particular, physics computations of the quantum K theory invariants, analogues of Gromov-Witten invariants, have been discussed in [10][11][12][13][14][15] (see also [16][17][18][19][20][21][22][23][24][25][26][27][28][29]).…”