Supersymmetric partition functions and the three-dimensional A-twist

TL;DR

The paper develops a comprehensive framework for 3d $\mathcal{N}=2$ gauge theories on Seifert-type manifolds $\mathcal{M}_{g,p}$, showing that supersymmetric observables reduce to data from a 2d $A$-twisted theory on $\Sigma_g$. Central to the construction are the fibering operator $\mathcal{F}$ and the handle-gluing operator $\mathcal{H}$, enabling a Bethe-vacua sum that captures the partition function and Wilson-loop correlators as $Z_{\mathcal{M}_{g,p}}(\nu)=\sum_{\hat{u}\in \mathcal{S}_{BE}} \mathcal{F}(\hat{u},\nu)^p \mathcal{H}(\hat{u},\nu)^{g-1} \prod_{\alpha} \Pi_{\alpha}(\hat{u},\nu)^{\mathfrak{n}_{\alpha}}$. The work provides two derivations (Bethe-vacua and UV localization), clarifies the role of Chern–Simons couplings and parity-violating contact terms, and demonstrates consistency with S^3$ and lens-space backgrounds, F-maximization, and 3d dualities. The S^3 partition function emerges as an on-shell fibering operator expectation value, $Z_{S^3}=\langle \mathcal{F}\rangle_{S^2\times S^1}$, and the framework yields a powerful avenue to study dualities by matching on-shell twisted superpotentials across dual pairs. Overall, the results reveal a deep link between 3d $\mathcal{N}=2$ dynamics, 2d A-twisted structures, and topological sectors across manifold topologies, with practical implications for F-maximization and dualities.

Abstract

We study three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories on $\mathcal{M}_{g,p}$, an oriented circle bundle of degree $p$ over a closed Riemann surface, $Σ_g$. We compute the $\mathcal{M}_{g,p}$ supersymmetric partition function and correlation functions of supersymmetric loop operators. This uncovers interesting relations between observables on manifolds of different topologies. In particular, the familiar supersymmetric partition function on the round $S^3$ can be understood as the expectation value of a so-called "fibering operator" on $S^2 \times S^1$ with a topological twist. More generally, we show that the 3d $\mathcal{N}=2$ supersymmetric partition functions (and supersymmetric Wilson loop correlation functions) on $\mathcal{M}_{g,p}$ are fully determined by the two-dimensional A-twisted topological field theory obtained by compactifying the 3d theory on a circle. We give two complementary derivations of the result. We also discuss applications to F-maximization and to three-dimensional supersymmetric dualities.

Figures (7)

• Figure 1: Poles of integrand for $U(1)_{-1/2}$ with one chiral, for $g=0,\;p=1$. Poles due to the positively charged chiral multiplet are shown in blue, and the "poles at infinity" from the negatively charged monopole, $T_+$, are denoted by the red line.
• Figure 2: The $p \neq 0$ JK contour, for $\eta >0$ and $\eta<0$, respectively. For $\eta>0$ the contour surrounds the poles due to positively charged chirals in an anti-clockwise manner, and for $\eta<0$ it surrounds the poles due to negatively charged chirals in a clockwise manner. Only the part of the contour at infinity that satisfies the condition $\mathop{\mathrm{sign}}\nolimits\left(\text{Im}(\partial _u \mathcal{W})\right) =- \mathop{\mathrm{sign}}\nolimits(\eta)$ should be included in the respective contours.
• Figure 3: JK contour $\mathcal{C}_{-1}^\eta$ shown for $\eta>0$ in blue, and $\mathcal{C}_{0}^\eta$ for $\eta<0$ in red. Here we assume $\text{Im}(\tau)<0$, which implies that the contribution at $\text{Im}(u) \rightarrow \infty$ is included in the $\eta>0$ contour.
• Figure 4: Taking $\eta<0$ for $n<0$ and $\eta>0$ for $n \geq 0$, we see the contours do not enclose any poles, and so the bulk contributions vanish, leaving only the boundary contributions. These sum to form the contour $\mathcal{C}_\sigma$. Here we have taken $\text{Im}(\tau)<0$.
• Figure 5: Here we take $\text{Im}(\tau)>0$, and note the corresponding contours, $\mathcal{C}_n^{\eta>0}$ and $\mathcal{C}_n^{\eta<0}$. Summing these as above to obtain the $\sigma$-contour, we see it now runs off to $\text{Re}(u)<0$ as $\text{Im}(u) \rightarrow \infty$.