Non-invertible Time-reversal Symmetry

TL;DR

The work reveals that non-invertible time-reversal symmetries emerge at every rational θ-angle in abelian gauge theories, realized by composing naive time reversal with a fractional quantum Hall interface and, equivalently, by gauging magnetic one-form symmetries. These constructions extend to non-Abelian theories such as N=4 SU(2) SYM along the locus |τ|=1, where duality defects combine with time-reversal to yield anti-linear, non-invertible symmetries whose fusion is governed by condensation defects. The authors analyze mixed anomalies that control the existence and nature of these symmetries, illustrate RG-consistent realizations in massive QED, and discuss the implications for trivially gapped phases via CK SPT invariance, including PSU(N) generalizations. The results open avenues to study generalized symmetries, their tangential structures, and partition functions on unoriented manifolds, with potential connections to the strong CP problem and beyond.

Abstract

In gauge theory, it is commonly stated that time-reversal symmetry only exists at $θ=0$ or $π$ for a $2π$-periodic $θ$-angle. In this paper, we point out that in both the free Maxwell theory and massive QED, there is a non-invertible time-reversal symmetry at every rational $θ$-angle, i.e., $θ= πp/N$. The non-invertible time-reversal symmetry is implemented by a conserved, anti-linear operator without an inverse. It is a composition of the naive time-reversal transformation and a fractional quantum Hall state. We also find similar non-invertible time-reversal symmetries in non-Abelian gauge theories, including the $\mathcal{N}=4$ $SU(2)$ super Yang-Mills theory along the locus $|τ|=1$ on the conformal manifold.

Figures (3)

• Figure 1: Some (anti-)linear (non-)invertible symmetries of the Maxwell theory. The locus where the theory has an invertible time-reversal symmetry is indicated by red lines. At $\tau =i$ and at $\tau = e^{i\pi/3}$, we have enhanced symmetries $D_8^{\mathsf{T}}$ and $D_{12}^{\mathsf{T}}$, respectively. At every rational value of the $\theta$-angle with $\theta=\pi p/N$, there is a non-invertible time-reversal symmetry $\mathsf{T}^{\theta=\frac{\pi p}{N}}$, which is indicated by the vertical blue lines. When a blue line intersects with the $|\tau|=1$ locus at a purple dot, the non-invertible time-reversal symmetry factorizes into the invertible time-reversal symmetry $\mathsf{T}^{|\tau|=1}$ and a linear, non-invertible symmetry \ref{['Nality']}. Some of the linear non-invertible symmetries of the Maxwell theory are also indicated. At the green dots, i.e., at $\tau = iN$, the theory realizes a linear, non-invertible duality defect Choi:2021kmx. At the cyan dots, i.e., at $\tau = Ne^{2\pi i/3}$, the theory realizes a linear, non-invertible triality defect Choi:2022zal. At these green/cyan points, there are non-invertible time-reversal symmetries obtained by composing the duality/triality defects with the invertible time-reversal symmetries. Finally, there is a $\mathbb{Z}_2^{(0)}$ charge conjugation symmetry and a $U(1)^{(1)}\times U(1)^{(1)}$ one-form symmetry everywhere. The red dashed lines and the hollow dots are outside the fundamental domain $\cal F$ and are related to the solid redlines and the solid dots, respectively, by duality transformations.
• Figure 2: The massless QED has an infinite linear non-invertible symmetry generated by $\mathcal{D}_\frac{p}{N}$Choi:2022jqyCordova:2022ieu, and an invertible time-reversal symmetry generated by $K$. Once we turn on a complex mass term $me^{-i\pi p/N}$, both of these symmetries are explicitly broken. The defect $\mathcal{D}_\frac{p}{N}$ now becomes a topological interface separating two massive QED theories with different mass parameters $me^{-i\pi p/N}$ and $me^{+i\pi p/N}$, and similarly for $K$. The composition of these two symmetries is a non-invertible time-reversal symmetry operator $\mathsf{T}^{\theta=\frac{\pi p}{N}}$ preserved by the massive QED at $\theta= \pi p/N$.
• Figure 3: Invertible and non-invertible time-reversal symmetries of QED on the complex mass plane. If the mass $m e^{-i\theta}$ is real (i.e., if $\theta=0,\pi$), the theory has an invertible time-reversal symmetry. When the phase of the mass term is rational, i.e., $\theta=\pi p/N$, we have the non-invertible time-reversal symmetry generated by $\mathsf{T}^{\theta=\frac{\pi p}{N}}$. At zero mass, we have the infinite linear non-invertible symmetry generated by $\mathcal{D}_{\frac{p}{N}}$ for all rational numbers $p/N$ and the invertible time-reversal symmetry.