Line Operators of Gauge Theories on Non-Spin Manifolds

TL;DR

The paper provides a comprehensive framework for classifying line operators in 4D gauge theories on oriented non-spin manifolds, including spins, mutual locality, and fusion rules, for all simple Lie algebras. It introduces a spin/charge assignment rule and a complete Lagrangian description with discrete theta terms, clarifying how line operators transform under S- and T-transformations and how theta parameters relate to spin data. The authors systematically catalog all allowed line-operator sets (including spins) across A–E series and detail how these sets manifest in UV completions and in the presence of background fields, revealing how one-form symmetries and their 't Hooft anomalies organize the space of consistent theories. The work also elucidates how gauging one-form symmetries and performing S-duality map between different theories with the same gauge algebra, offering a unifying picture of non-spin gauge theories and their dualities with abelian theories via discrete and continuous theta angles. Overall, this advances the understanding of global structures, spin-constraints, and anomaly physics in 4D gauge theories beyond spin manifolds.

Abstract

We study four-dimensional gauge theories on oriented and non-spin spacetime manifolds. On such manifolds, each line operator arises only either as a boson or a fermion. Based on physical arguments, a method of systematically assigning spin labels to line operators is proposed, and several consistency checks are performed. This is used to classify all possible sets of allowed line operators -- including their spins -- for gauge theories with simple Lie algebras. The Lagrangian descriptions of the theories with these sets of allowed line operators are given. Finally, the one-form symmetries of these theories are studied by coupling to background gauge fields, and their 't Hooft anomalies are computed.

Figures (4)

• Figure 1: The relation between $\Gamma_\mathrm{e},\Gamma_\mathrm{m}$, the (co)weight lattices $\Lambda_G,\Lambda_{\mathrm{c} G}$, and the one-form symmetries and charges of the plain $G=\tilde{G}/\Gamma_\mathrm{m}$ gauge theory. In particular $\Gamma_\mathrm{e}=Z(G)$ is the electric one-form symmetry group (also known as the center symmetry), and $\Gamma_\mathrm{m}=\pi_1(G)$ is the group of magnetic charges carried by the 't Hooft lines, which was denoted $\Gamma$ in Gaiotto:2014kfa.
• Figure 2: $SU(2)$ lattices
• Figure 3: $SO(3)$ lattices
• Figure 4: Table of gauge groups, $\theta$ periodicities and $T$-transformations