One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices and CS theories

TL;DR

This work develops a comprehensive framework to compute topologically twisted indices of 3d ${ m N}=2$ Chern-Simons-matter theories with discrete one-form symmetries by working in the 3d ${ m A}$-model on $oldsymbol{ extSigma_g imes S^1}$ and analyzing Bethe vacua on $T^2$. It provides explicit constructions of symmetry operators, studies their ’t Hooft anomalies, and implements gauging of both one-form and zero-form sectors, yielding genus-$g$ twisted indices and genus-1 Witten indices for ${ m SU}(N)_K$ and its quotients ${ m SU}(N)/oldsymbol{Z}_r$; the results are expressed via refinements of Jordan’s totient functions and, in some cases, modular anomaly data on $T^3$. A central achievement is the detailed accounting of Bethe vacua under higher-form symmetries, including exact formulas for PSU and twisted sectors, and the demonstration that index integrality is tied to the infrared 3d TQFT being bosonic. The paper also clarifies the role of gravitational mixed anomalies in shaping the allowed discrete gaugings and their impact on state counting, and it connects the field-theoretic data to number-theoretic structures, opening routes to further mathematical and physical exploration. Significance lies in providing a rigorous, semiclassical-to-IR bridge for generalized global symmetries in 3d gauge theories and in delivering concrete, testable predictions for twisted indices across a broad class of theories and gaugings.

Abstract

We study three-dimensional $\mathcal{N}=2$ supersymmetric Chern-Simons-matter gauge theories with a one-form symmetry in the $A$-model formalism on $Σ_g\times S^1$. We explicitly compute expectation values of topological line operators that implement the one-form symmetry. This allows us to compute the topologically twisted index on the closed Riemann surface $Σ_g$ for any real compact gauge group $G$ as long as the ground states are all bosonic. All computations are carried out in the effective $A$-model on $Σ_g$, whose $S^1$ ground states are the so-called Bethe vacua. We discuss how the 3d one-form symmetry acts on the Bethe vacua, and also how its 't Hooft anomaly constrains the vacuum structure. In the special case of the $\text{SU}(N)_K$ $\mathcal{N}=2$ Chern-Simons theory, we obtain results for the $(\text{SU}(N)/\mathbb Z_r)^θ_K$ $\mathcal{N}=2$ Chern-Simons theories, for all non-anomalous $\mathbb Z_r \subseteq \mathbb Z_N$ subgroups of the centre of the gauge group, and with a $\mathbb Z_r$ $θ$-angle turned on. In the special cases with $N$ even, $\frac{N}{r}$ odd and $\frac{K}{r}$ even, we find a mixed 't Hooft anomaly between gravity and the $\mathbb Z_r^{(1)}$ one-form symmetry of the $\text{SU}(N)_K$ theory, and the infrared 3d TQFT after gauging is spin. In all cases, we count the Bethe states and the higher-genus states in terms of refinements of Jordan's totient function. This counting gives us the twisted indices if and only if the infrared 3d TQFT is bosonic. Our results lead to precise conjectures about integrality of indices, which appear to have a strong number-theoretic flavour. Note: this paper directly builds upon unpublished notes by Brian Willett from 2020.

Figures (8)

• Figure 1: Depiction of the $\Gamma^{(1)}$ and $\Gamma^{(0)}$ symmetry operators $\mathcal{U}^\gamma(S_A^1)\equiv \Pi^\gamma$ and $\mathcal{U}^\gamma(\mathcal{C})$, respectively, on $\Sigma\times S^1_A$. Here, $\mathcal{U}^\gamma(S_A^1)$ wraps the $S^1_A$ factor in the 3d geometry, depicted here by a small circle, and it is thus a local operator on $\Sigma$. The topological line $\mathcal{U}^\gamma(\mathcal{C})$ is supported on a cycle $\mathcal{C}$ on $\Sigma$, in general. Here $\Sigma$ is the cylinder, which is the relevant configuration to discuss the Hilbert space $\mathscr{H}_{S^1}$ of the $A$-model.
• Figure 2: The 2d SPT phase arising from the intersection of the topological lines $\mathcal{U}^\gamma$ and $\mathcal{U}^\delta$ on the cylinder, see \ref{['phases SPT']}. The phase is obtained by resolving the intersection into two trivalent junctions, where we assign the phase $b_2(\gamma, \delta)$ to each junction (with some customary orientation). That $b_2$ is a group cohomology class follows from the associativity of fusion and from gauge invariance.
• Figure 3: $a$- and $b$-cycles on the Riemann surface $\Sigma_g$, here depicted for genus $g=3$.
• Figure 4: Operators in 2d TQFT corresponding to the cap, cylinder, and pair of pants. We can think as $\mathcal{H}^{{\frac{1}{2}}}$ as a formal square root of the handle-gluing operator $\mathcal{H}$, keeping in mind that we always obtain integer powers of $\mathcal{H}$ when computing observables on a closed $\Sigma$.
• Figure 5: Operators in the TQFT corresponding to the cylinder and pair of pants with topological lines inserted. We could also consider the insertion of a line at the boundary of a cap, but this is topologically trivial; in this formalism, this follows from the fact that $\mathcal{H}(\hat{u}+\gamma)= \mathcal{H}(\hat{u})$, so that formally $\sum_{\hat{u}} \mathcal{H}^{-{\frac{1}{2}}} \mathcal{U}^\gamma \lvert\hat{u}\rangle= \sum_{\hat{u}} \mathcal{H}^{-{\frac{1}{2}}}\lvert\hat{u}\rangle$.

Theorems & Definitions (4)

• Proposition 1
• Lemma 1: Lemma 19 of Grunberg:2003aa
• proof
• Proposition 2