Non-Invertible Gauss Law and Axions

TL;DR

This work identifies and constructs non-invertible global symmetries in the 3+1d axion-Maxwell theory at minimal axion-photon coupling, turning the naive shift and center symmetries into non-invertible 0- and 1-form defects. It develops two complementary constructions, via explicit defects and via half higher gauging, and shows how these defects act differently on axion/Wilson data versus monopole/axion-string data, yielding a non-invertible Gauss law and new selection rules tied to the Witten effect. The authors further show that these non-invertible symmetries mix with invertible higher-form symmetries in a higher-structure-like way, and derive universal inequalities on the energy scales at which various IR symmetries emerge, with implications for the Weak Gravity Conjecture and the Completeness Hypothesis in quantum gravity. Collectively, the results illuminate how non-invertible generalized global symmetries constrain IR dynamics, defect interplays, and quantum-gravity consistency conditions in axion–Maxwell systems.

Abstract

In axion-Maxwell theory at the minimal axion-photon coupling, we find non-invertible 0- and 1-form global symmetries arising from the naive shift and center symmetries. Since the Gauss law is anomalous, there is no conserved, gauge-invariant, and quantized electric charge. Rather, using half higher gauging, we find a non-invertible Gauss law associated with a non-invertible 1-form global symmetry, which is related to the Page charge. These symmetries act invertibly on the axion field and Wilson line, but non-invertibly on the monopoles and axion strings, leading to selection rules related to the Witten effect. We also derive various crossing relations between the defects. The non-invertible 0- and 1-form global symmetries mix with other invertible symmetries in a way reminiscent of a higher-group symmetry. Using this non-invertible higher symmetry structure, we derive universal inequalities on the energy scales where different infrared symmetries emerge in any renormalization group flow to the axion-Maxwell theory. Finally, we discuss implications for the Weak Gravity Conjecture and the Completeness Hypothesis in quantum gravity.

Figures (13)

• Figure 1: Non-invertible Gauss law implemented by the non-invertible 1-form symmetry surface operator ${\cal D}^{(1)}_{p/N}$ labeled by $p/N\in \mathbb{Q}/\mathbb{Z}$. Here $W$ and $H$ stand for the minimally charged Wilson and 't Hooft lines, located at a point in the 3-dimensional space and extended in the time direction (the time direction is not shown in the figure). The non-invertible 1-form symmetry ${\cal D}^{(1)}_{p/N}$ measures the electric charge of the Wilson line invertibly by a phase $e^{2\pi ip/N}$, much as an ordinary Gauss law operator $e^{ i \alpha Q}$ (with $\alpha=2\pi p/N$) does. However, it annihilates the 't Hooft line, and is therefore a non-invertible operator.
• Figure 2: In a general QFT, higher gauging of a higher-form symmetry on a closed submanifold generates a nontrivial condensation defect. Half higher gauging corresponds to higher gauging the symmetry on a submanifold with a topological Dirichlet boundary condition imposed on the boundary. This generates a topological twist defect which lives on the boundary of the condensation defect. If the theory is self-dual under the higher gauging, then the resulting condensation defect is trivial, and the half higher gauging generates a genuine topological defect not attached to anything.
• Figure 3: The action of the non-invertible 1-form symmetry defect $\mathcal{D}^{(1)}_{p/N}$ on extended operators/defects. $W$, $H$, $S$ are the minimal Wilson line, 't Hooft line, and axion string worldsheet, respectively. For simplicity, we assume that the spacetime manifold is locally $\mathbb{R}^4$. The directions along which various operators extend are summarized in the table, and the $x^4$-direction is suppressed in the drawing. The non-invertible 1-form symmetry $\mathcal{D}^{(1)}_{p/N}$ acts on the Wilson line $W$ invertibly by a phase, but it acts on the 't Hooft line $H$ and the axion string worldsheet $S$ non-invertibly.
• Figure 4: The non-invertible Gauss law. The non-invertible operator ${\cal D}^{(1)}_{p/N}$ measures the electric charge invertibly on the Wilson line $W^q$. On the other hand, the dyonic line $H_{m,q}$ is annihilated when measured by $\mathcal{D}^{(1)}_{p/N}$ if $m\neq 0$ mod $N$. Physically, it means that the electric charge $q$ of a dyon $H_{m,q}$ with magnetic charge $m$ is only well-defined modulo $m$, i.e., $q\sim q+m$. Similarly, the axion string worldsheet $S_w$ is annihilated by ${\cal D}^{(1)}_{p/N}$ if $w\neq 0$ mod $N$.
• Figure 5: (a) Non-invertible action of $\mathcal{D}^{(1)}_{p/N}$ on the 't Hooft line $H_m$ for $m\neq 0$ mod $N$. We first pull $\mathcal{D}^{(1)}_{p/N}$ past $H_m$, and then shrink the former to a point. This generates a putative topological endpoint for the topological line $\eta^{\text{(w)}}_{2\pi pm/N}$. However, the line $\eta^{\text{(w)}}_{2\pi pm/N}$ does not admit any topological endpoint, since it acts faithfully on the other operators Chang:2018iay. Therefore, such a configuration results in a vanishing correlation function. (b) Non-invertible action of $\mathcal{D}^{(1)}_{p/N}$ on the axion string worldsheet $S_w$ for $w\neq 0$ mod $N$. Similar to before, we first pull $\mathcal{D}^{(1)}_{p/N}$ past $S_w$, and then shrink the former to a line. This generates a putative topological boundary line for the topological surface $\eta^{\text{(m)}}_{2\pi pw/N}$, which does not exist. Therefore, such a configuration leads to a vanishing correlation function. Both the 't Hooft line and the axion string worldsheet are extended in time, and they are point and line in space as shown in the figure, respectively.