Seifert fibering operators in 3d $\mathcal{N}=2$ theories

TL;DR

<3-5 sentence high-level summary>The paper provides a comprehensive, exact framework for computing supersymmetric partition functions of 3d ${\mathcal N}=2$ gauge theories on general Seifert manifolds ${\mathcal M}_3$ by recasting the problem in terms of a 2d ${\mathcal N}=(2,2)$ A-model on the base ${\hat\Sigma}$ and introducing geometry-changing line defects called fibering operators. It introduces the $(q,p)$-fibering operators that insert exceptional Seifert fibers and shows how the full ${Z_{\mathcal M_3}}$ is obtained as a finite-sum over Bethe vacua of the UV theory, with explicit contributions from CS terms, chiral multiplets, and vector multiplets. The authors perform both a 3d A-model Bethe-vacua computation and a Coulomb-branch localization derivation, and verify nontrivial infrared dualities by matching fibering operators across dual theories, including numerical checks for numerous $(q,p)$ and dual pairs. They connect the formalism to lens spaces and holomorphic blocks, clarifying the relationship between fibering operators and holomorphic blocks in the rational-squashing limit, and discuss extensions to 3d CS theories and potential uplifts to higher dimensions. The results provide a robust, modular toolkit for exploring supersymmetric observables on a broad class of Seifert geometries and for probing dualities in 3d ${\mathcal N}=2$ gauge theories.

Abstract

We study 3d $\mathcal{N}=2$ supersymmetric gauge theories on closed oriented Seifert manifold---circle bundles over an orbifold Riemann surface---, with a gauge group G given by a product of simply-connected and/or unitary Lie groups. Our main result is an exact formula for the supersymmetric partition function on any Seifert manifold, generalizing previous results on lens spaces. We explain how the result for an arbitrary Seifert geometry can be obtained by combining simple building blocks, the "fibering operators." These operators are half-BPS line defects, whose insertion along the $S^1$ fiber has the effect of changing the topology of the Seifert fibration. We also point out that most supersymmetric partition functions on Seifert manifolds admit a discrete refinement, corresponding to the freedom in choosing a three-dimensional spin structure. As a strong consistency check on our result, we show that the Seifert partition functions match exactly across infrared dualities. The duality relations are given by intricate (and seemingly new) mathematical identities, which we tested numerically. Finally, we discuss in detail the supersymmetric partition function on the lens space $L(p,q)_b$ with rational squashing parameter $b^2 \in \mathbb{Q}$, comparing our formalism to previous results, and explaining the relationship between the fibering operators and the three-dimensional holomorphic blocks.

Figures (4)

• Figure 1: Solid fibered tori $T(q,t)$. The central fiber, shown in red, is exceptional if $q>1$. Generic fibers are shown in black.
• Figure 2: On the left, the contour $\Gamma_\alpha$ is shown for finite $\tau$, with towers of poles separated by $\tau$. As $\tau$ becomes small, the contributions from $\hat{u}$ and its images are dominant, shown in red. On the right, we take the $\tau \rightarrow 0$ limit, where we may approximate the answer by the contribution from $u=\hat{u}$. Here the towers of poles have collapsed to form the branch cuts of $\mathcal{W}(u)$.
• Figure 3: Contour $\Gamma_\alpha$ corresponding to a block $B_{\widetilde{g}}$ with non trivial $(q,p)$, at finite $\tau$.
• Figure 4: In the limit $\tau \rightarrow 0$, the dominant contributions to the integral over $\Gamma_\alpha$ come from the regions around $\hat{u}$ and their images. These can be reassembled into a series of $q$ shifted copies of contours passing through the critical points $u=q\tilde{u} = \hat{u}, \hat{u}+1,...,\hat{u}+q-1$.