A unified view on symmetry, anomalous symmetry and non-invertible gravitational anomaly

TL;DR

The paper argues for a unified viewpoint in which symmetry, 't Hooft anomalies, and non-invertible gravitational anomalies are all aspects of a single framework governed by topological order data in one higher dimension. It develops a multi-component partition function formalism Z^i(M^D,A) indexed by symmetry twists and bulk anyons, and shows how mapping-class-group transformations encode anomalies through cocycles and SPT invariants. By analyzing 1+1D theories, spin chains, and 1+1D boundaries of 2+1D topological orders (Z_2, double-semion, S_3), the work demonstrates precise correspondences between symmetry-twisted boundary data and bulk anyon content, including non-invertible anomalies arising after gauging. The results furnish a holographic and algebraic toolkit (group cohomology, cocycles, G-crossed categories) to diagnose and classify anomalies, relate boundary dynamics to bulk topology, and unify the treatment of invertible and non-invertible cases with potential implications for condensed matter and quantum field theory.

Abstract

In this paper, using 1+1D models as examples, we study symmetries and anomalous symmetries via multi-component partition functions obtained through symmetry twists, and their transformations under the mapping class group of spacetime. This point of view allows us to treat symmetries and anomalous symmetries as non-invertible gravitational anomalies (which are also described by multi-component partition functions, transforming covariantly under the mapping group transformations). This allows us to directly see how symmetry and anomalous symmetry constraint the low energy dynamics of the systems, since the low energy dynamics is directly encoded in the partition functions. More generally, symmetries, anomalous symmetries, non-invertible gravitational anomalies, and their combinations, can all be viewed as constraints on low energy dynamics. In this paper, we demonstrate that they all can be viewed uniformally and systematically as pure (non-invertible) gravitational anomalies.

Figures (20)

• Figure 1: The many-body energy spectrum of $J_1$-$J_2$ Heisenberg model on a ring of 16 sites, where the ground state energy is shifted to 0. The horizontal axis is the crystal momentum $k/2\pi$. The numbers by the bars indicate the $SU(2)$ multiplets (the degeneracies). The energy levels at energy $\approx 3.2$ and $k = \pi$ contain $SU(2)$ multiplets 1, 3, and 5. The energy levels at energy $\approx 0.8$ and $k = 0$ contain $SU(2)$ multiplets 1 and 3 (which correspond to spin $\frac{1}{2} \otimes \frac{1}{2}$).
• Figure 2: The many-body energy spectrum of $J_1$-$J_2$ Heisenberg model on a ring of 17 sites, where the ground state energy is shifted to 0. The horizontal axis is the crystal momentum $k/2\pi$. The numbers by the bars indicate the $SU(2)$ multiplets (the degeneracies).
• Figure 3: The many-body energy spectrum of $J_1$-$J_2$ Heisenberg model on a ring of 18 sites, where the ground state energy is shifted to 0. The horizontal axis is the crystal momentum $k/2\pi$. The numbers by the bars indicate the $SU(2)$ multiplets (the degeneracies). The energy levels at energy $\approx 3.2$ and $k = \pi$ contain $SU(2)$ multiplets 1, 3, and 5. The energy levels at energy $\approx 0.8$ and $k = 0$ contain $SU(2)$ multiplets 1 and 3 (which correspond to spin $\frac{1}{2} \otimes \frac{1}{2}$).
• Figure 4: The many-body energy spectrum of $J_1$-$J_2$ Heisenberg model on a ring of 19 sites, where the ground state energy is shifted to 0. The horizontal axis is the crystal momentum $k/2\pi$. The numbers by the bars indicate the $SU(2)$ multiplets (the degeneracies).
• Figure 5: The many-body energy spectrum of $J_1$-$J_2$ Heisenberg model on a ring of 20 sites, where the ground state energy is shifted to 0. The horizontal axis is the crystal momentum $k/2\pi$. The numbers by the bars indicate the $SU(2)$ multiplets (the degeneracies). The energy levels at energy $\approx 3.0$ and $k = 0$ contain $SU(2)$ multiplets 1, 3, and 5. The energy levels at energy $\approx 0.8$ and $k = \pi$ contain $SU(2)$ multiplets 1 and 3 (which correspond to spin $\frac{1}{2} \otimes \frac{1}{2}$).