Topological Operators and Completeness of Spectrum in Discrete Gauge Theories

TL;DR

The paper reexamines the link between spectrum completeness and symmetry in discrete gauge theories, showing that in 3d completeness is equivalent to the absence of Gukov-Witten topological lines, and extends the analysis to 4d where completeness implies no GW surfaces (though some topological operators may persist). It leverages representation theory (Burnside) and modular tensor category (Müger) insights to prove the equivalence and uses explicit examples like S_3 to illustrate the landscape of endable vs. topological lines. The authors argue that topological operators, not just ordinary global symmetries, are the relevant objects to constrain in quantum gravity, proposing a generalized swampland perspective where such operators should be absent or highly constrained. These results refine the conventional lore linking completeness to no electric one-form symmetry and suggest new avenues for understanding symmetry, topology, and gravity in higher-dimensional gauge theories.

Abstract

In many gauge theories, the existence of particles in every representation of the gauge group (also known as completeness of the spectrum) is equivalent to the absence of one-form global symmetries. However, this relation does not hold, for example, in the gauge theory of non-abelian finite groups. We refine this statement by considering topological operators that are not necessarily associated with any global symmetry. For discrete gauge theory in three spacetime dimensions, we show that completeness of the spectrum is equivalent to the absence of certain Gukov-Witten topological operators. We further extend our analysis to four and higher spacetime dimensions. Since topological operators are natural generalizations of global symmetries, we discuss evidence for their absence in a consistent theory of quantum gravity.

Figures (10)

• Figure 1: Topological line operator $U_g(M^{(1)})$ supported on a manifold with boundary. Physical observables are invariant under deformation of the manifold $M^{(1)}$ stretching between $x$ and $y$ provided the boundary $\partial M = \{ x, y\}$ is held fixed.
• Figure 2: The duality line $T$ in the two-dimensional Ising CFT defines a (non-invertible) map on the local operators by encircling a local operator ${V}(x)$ and shrinking the circle. This defines an action on the Hilbert space of states via operator-state correspondence.
• Figure 3: Fusion of lines $L_a$ and $L_b$.
• Figure 4: We can surround a topological line $L_a$ around another line $L_b$, and then shrink $L_a$ to a point. This defines an action of $L_a$ on $L_b$ denoted as $L_a \cdot L_b = B^a_b L_b$ with $B^a_b$ the braiding coefficient.
• Figure 5: Fusion of endable lines $L_a$ and $L_b$. If $L_a$ and $L_b$ are both endable, then any line $L_c$ with $N_{ab}^c \neq 0$ must also be endable.