Three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories and partition functions on Seifert manifolds: A review

TL;DR

This review surveys exact results for 3D $ ext{N}=2$ supersymmetric gauge theories on half-BPS Seifert manifolds, foregrounding localization techniques that reduce path integrals to finite-dimensional sums and integrals. It develops curved-space SUSY via rigid supersymmetry, THFs, and the 3D A-twist, then details Coulomb-branch localization, one-loop determinants, and contour constructions for partition functions like $Z_{S^3_b}$ and the topologically twisted index. A unifying 3D–2D picture—the A-model on a base Riemann surface—yields Bethe vacua and line-operator algebras, enabling direct tests of IR dualities such as level/rank, elementary mirror symmetry, and Aharony duality, including their influence on handle-gluing and flux operators. The framework provides precise checks of dualities and links to holographic contexts (AdS$_4$/CFT$_3$) and the 3D/3D correspondence, with substantial computational machinery for Seifert geometries and beyond.

Abstract

We give a pedagogical introduction to the study of supersymmetric partition functions of 3D $\mathcal{N}{=}2$ supersymmetric Chern-Simons-matter theories (with an $R$-symmetry) on half-BPS closed three-manifolds---including $S^3$, $S^2 \times S^1$, and any Seifert three-manifold. Three-dimensional gauge theories can flow to non-trivial fixed points in the infrared. In the presence of 3D $\mathcal{N}{=}2$ supersymmetry, many exact results are known about the strongly-coupled infrared, due in good part to powerful localization techniques. We review some of these techniques and emphasize some more recent developments, which provide a simple and comprehensive formalism for the exact computation of half-BPS observables on closed three-manifolds (partition functions and correlation functions of line operators). Along the way, we also review simple examples of 3D infrared dualities. The computation of supersymmetric partition functions provides exceedingly precise tests of these dualities.

Figures (3)

• Figure 1: Any gauge flux (dynamical or flavor) through $\Sigma_g$ (as depicted on the left-hand-side figure, for some random flux profile) can be concentrated to a point (as shown on the right), which can then be interpreted as a 2D local operator. For the 3D theory on $\Sigma_g \times S^1$, the flux operator, $\Pi$, is a line operator wrapped over the $S^1$ (with this circle shown here as well).
• Figure 2: Shrinking an handle to a point, we obtain the handle-gluing operator, $\mathcal{H}$, as a quasi-local operator of the $A$-model.
• Figure 3: On the left: Dehn surgery, cutting out a disk $D_2$ on the base, and gluing back the cap $D_2 \times S^1$ with an $SL(2, \mathbb{Z})$ twist. This introduce an exceptional Seifert fiber at the center of the disk. On the right: Shrinking the radius of $D_2$ to zero size, the introduction of the exceptional fiber is viewed a local defect operator, $\mathcal{G}_{q,p}$, on the base (and wrapping the $S^1$).