Non-invertible 1-form symmetry and Casimir scaling in 2d Yang-Mills theory

TL;DR

The paper addresses why 2d Yang–Mills exhibits Casimir scaling with an infinite set of string tensions not fixed by center symmetry. It introduces a non-invertible 1-form symmetry generated by a topological disorder operator that acts on Wilson loops and distinguishes representations, using the exact heat-kernel solution to derive representation-sensitive Wilson loop behavior. The key results show that Wilson loop expectations follow Casimir scaling and that the disorder operator provides a representation-projection mechanism, with the operator being non-invertible and generalizing center symmetry. The work suggests a symmetry-based explanation for the 2d spectrum and discusses potential implications for confinement in higher dimensions, including possible large-N emergent non-invertible symmetries and dimensional-reduction insights.

Abstract

Pure Yang-Mills theory in 2 spacetime dimensions shows exact Casimir scaling. Thus there are infinitely many string tensions, and this has been understood as a result of non-propagating gluons in 2 dimensions. From ordinary symmetry considerations, however, this richness in the spectrum of string tensions seems mysterious. Conventional wisdom has it that it is the center symmetry that classifies string tensions, but being finite it cannot explain infinitely many confining strings. In this note, we resolve this discrepancy between dynamics and kinematics by pointing out the existence of a non-invertible 1-form symmetry, which is able to distinguish Wilson loops in different representations. We speculate on possible implications for Yang-Mills theories in 3 and 4 dimensions.

Figures (3)

• Figure 1: Characteristic behavior of derivative of inter-quark potential in some representations $\alpha$ in $d=3,4$$\mathrm{SU}(N)$ Yang--Mills theory. Here, we take Young tableaux $\alpha=(1,0,\ldots, 0), (1,0,\ldots,0,1), (2,0,\ldots,0,1)$ for red, blue, and black curves, respectively.
• Figure 2: Expectation value of a Wilson loop $W_\gamma(\Gamma)$ on $S^2$.
• Figure 3: Defect operator $V_{[U]}(x)$ inside and outside of the Wilson loop $W_{\gamma}(\Gamma)$.