Comments on twisted indices in 3d supersymmetric gauge theories

TL;DR

This work develops and systematizes the twisted index for three-dimensional N=2 and N=4 gauge theories on Σ_g × S^1 using supersymmetric localization. By expressing the index as a sum over flux sectors and JK residues, it is shown that the result reduces to a Bethe-sum controlled by the twisted superpotential, enabling exact analyses of dualities, Wilson-loop algebras, and line operators. The authors demonstrate powerful consistency checks, including Aharony and Giveon-Kutasov dualities, Verlinde-type formulas in pure CS theories, and detailed explorations of 3d N=4 mirror symmetry with A-/B-twists and line operators. They further connect genus-zero twisted indices to Coulomb/Higgs branch Hilbert series, providing a bridge between localization, integrable structures, and moduli-space enumerative geometry with broad implications for dualities and operator algebras in 3d SUSY theories.

Abstract

We study three-dimensional ${\mathcal N}=2$ supersymmetric gauge theories on ${Σ_g \times S^1}$ with a topological twist along $Σ_g$, a genus-$g$ Riemann surface. The twisted supersymmetric index at genus $g$ and the correlation functions of half-BPS loop operators on $S^1$ can be computed exactly by supersymmetric localization. For $g=1$, this gives a simple UV computation of the 3d Witten index. Twisted indices provide us with a clean derivation of the quantum algebra of supersymmetric Wilson loops, for any Yang-Mills-Chern-Simons-matter theory, in terms of the associated Bethe equations for the theory on ${\mathbb R}^2 \times S^1$. This also provides a powerful and simple tool to study 3d ${\mathcal N}=2$ Seiberg dualities. Finally, we study A- and B-twisted indices for ${\mathcal N}=4$ supersymmetric gauge theories, which turns out to be very useful for quantitative studies of three-dimensional mirror symmetry. We also briefly comment on a relation between the $S^2 \times S^1$ twisted indices and the Hilbert series of ${\mathcal N}=4$ moduli spaces.

Figures (7)

• Figure 1: A generic $A_L$-type linear quiver with $\mathcal{N}=4$ supersymmetry. The circles and squares stand for $U(N_s)$ gauge groups and $SU(M_s)$ flavor groups ($s=1, \cdots, L$), respectively.
• Figure 2: The vortex operator $V_k$ dual to the Wilson loop $W_k$ in $T[SU(2)]$ theory. The quiver in the dotted box is a one-dimensional $\mathcal{N}=(2,2)$ GLSM consisting of a gauge group $G=U(k)$ with one fundamental, one anti-fundamental and one adjoint multiplet.
• Figure 3: The 1d vortex loops in 3d $\mathcal{N}=4$ theories are invariant under the so-called hopping duality, as shown here for $T[SU(2)]$. This follows from the fact that the D1-brane can freely move along the D3-brane. The figure on the left corresponds to the D1-brane attached to the left $NS5$-brane, and the figure on the right corresponds to the D1-brane attached to the right NS5-brane. See Appendix \ref{['Appendix:loop']}
• Figure 4: The singularities in the $\hat{D}$ plane. When we choose $\delta>0$, the $\hat{D}$ integral from $\Delta_{\epsilon,m}^{(+)}$ are modified to that of $\hat{D}=0$ and a contour that passes negative imaginary axis, where the latter contribution can be deformed away to give a vanishing contribution. For $\Delta_{\epsilon,m}^{(-)}$, the contour can be deformed away to infinity. We set $\beta=1$ for simplicity.
• Figure 5: Brane construction of the charge $k$ Wilson loop for $T[SU(2)]$, and its S-dual configuration. The horizontal segment represents a stretched D3-brane, which is invariant under S-duality.