Cluster State as a Noninvertible Symmetry-Protected Topological Phase.

We show that the standard 1+1D Z_{2}×Z_{2} cluster model has a noninvertible global symmetry, described by the fusion category Rep(D_{8}). Therefore, the cluster state is not only a Z_{2}×Z_{2} symmetry protected topological (SPT) phase, but also a noninvertible SPT phase. We further find two new commuting Pauli Hamiltonians for the other two Rep(D_{8}) SPT phases on a tensor product Hilbert space of qubits, matching the classification in field theory and mathematics. We identify the edge modes and the local projective algebras at the interfaces between these noninvertible SPT phases. Finally, we show that there does not exist a symmetric entangler that maps between these distinct SPT states.

Quantization of Axion-Gauge Couplings and Noninvertible Higher Symmetries.

We derive model-independent quantization conditions on the axion couplings (sometimes known as the anomaly coefficients) to the standard model gauge group [SU(3)×SU(2)×U(1)_{Y}]/Z_{q} with q=1, 2, 3, 6. Using these quantization conditions, we prove that any QCD axion model to the right of the E/N=8/3 line on the |g_{aγγ}|-m_{a} plot must necessarily face the axion domain wall problem in a postinflationary scenario. We further demonstrate the higher-group and noninvertible global symmetries in the standard model coupled to a single axion. These generalized global symmetries lead to universal bounds on the axion string tension and the monopole mass. If the axion were discovered in the future, our quantization conditions could be used to constrain the global form of the standard model gauge group.

Noninvertible Time-Reversal Symmetry.

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 Letter, we point out that in both the free Maxwell theory and massive QED, there is a noninvertible time-reversal symmetry at every rational θ angle, i.e., θ=πp/N. The noninvertible time-reversal symmetry is implemented by a conserved, antilinear operator without an inverse. It is a composition of the naive time-reversal transformation and a fractional quantum Hall state. We also find similar noninvertible time-reversal symmetries in non-Abelian gauge theories, including the N=4 SU(2) super Yang-Mills theory along the locus |τ|=1 on the conformal manifold.

Noninvertible Global Symmetries in the Standard Model.

We identify infinitely many noninvertible generalized global symmetries in QED and QCD for the real world in the massless limit. In QED, while there is no conserved Noether current for the U(1)_{A} axial symmetry because of the Adler-Bell-Jackiw anomaly, for every rational angle 2πp/N, we construct a conserved and gauge-invariant topological symmetry operator. Intuitively, it is a composition of the axial rotation and a fractional quantum Hall state coupled to the electromagnetic U(1) gauge field. These conserved symmetry operators do not obey a group multiplication law, but a noninvertible fusion algebra. They act invertibly on all local operators as axial rotations, but noninvertibly on the 't Hooft lines. We further generalize our construction to QCD, and show that the coupling π^{0}F∧F in the effective pion Lagrangian is necessary to match these noninvertible symmetries in the UV. Therefore, the conventional argument for the neutral pion decay using the ABJ anomaly is now rephrased as a matching condition of a generalized global symmetry.