Non-Invertible Gauss Law and Axions

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.

Non-Invertible Gauss Law and Axions

The paper "Non-Invertible Gauss Law and Axions" by Yichul Choi, Ho Tat Lam, and Shu-Heng Shao investigates the phenomenon of non-invertible symmetries within the framework of axion-Maxwell theory at minimal axion-photon coupling. The exploration is motivated by the need to understand novel aspects of global symmetries, specifically non-invertible ones, which have implications for quantum field theory and potentially for quantum gravity.

Overview

The authors focus on axion-Maxwell theory, characterized by a Lagrangian that includes a coupling of the axion field with the electromagnetic field. This theory is known for manifesting non-trivial global symmetries, particularly when considering generalized global symmetries. The paper demonstrates that, at the minimal coupling scenario where the axion-photon interaction quantization parameter $K = 1$, both 0- and 1-form symmetries, typically thought to be broken or absent, actually manifest as non-invertible symmetries. These symmetries are explored through the concept of "half higher gauging" and the construction of non-trivial topological operators.

Key Contributions

1. Non-Invertible Symmetries: The authors establish that axion-Maxwell theory harbors non-invertible 0- and 1-form global symmetries. These arise from naive shift and center symmetries which, contrary to being completely broken, re-emerge in this non-invertible form. The 0-form symmetry is linked to charge shift symmetries while the 1-form symmetry is associated with electric center symmetries relating to the Page charge.
2. Non-Invertible Gauss Law: A crucial insight is that the traditional Gauss law for electric charge is nugatory in the presence of axion-photon coupling owing to anomalies. Instead, the authors propose a non-invertible Gauss law using novel symmetry operators defined on closed manifolds that, while topological and gauge-invariant, do not abide by inverse properties typical of symmetry operators.
3. Implications for Electric Charge Quantization: The exploration implies that the usual concept of a conserved, gauge-invariant, quantized electric charge is not applicable in axion-electromagnetism. Instead, electric charge information is encoded within non-invertible structures that invite reinterpretation in processes such as the Witten effect.
4. Higher Symmetry Structure: The work indicates possible higher structures and non-invertible enhancements of classical symmetry concepts, suggesting more nuanced symmetry group formulations. The fusion of symmetry operators, particularly in intersecting configurations, indicates a similar role to higher groups but within a non-invertible framework.
5. Applications and Consequences: The paper extends its implications to core areas in theoretical physics, for instance, proposing constraints on scales of symmetry breaking that connect to the dynamics of axion strings and monopoles, and discussing elements of the Weak Gravity Conjecture and the Completeness Hypothesis in the context of quantum gravity.

Conclusion and Speculation

This research delineates a refined understanding of symmetries in field theories where classical concepts do not hold. By proposing new structures and demonstrating their properties in a constrained theoretical model, the work sets the stage for broader applications in both theoretical explorations and potentially in string theory scenarios or low-energy field theories emerging from a high-energy effective approach.

Future investigations may explore the categorical symmetries and intersections with supergravity or string-inspired frameworks. Additionally, the potential redefinition of Goldstone boson-related phenomena in this context poses intriguing questions for follow-up theoretical innovation. Through these explorations, non-invertible symmetries continue to enrich our understanding of fundamental symmetries in physics.