Holographic theory of non-invertible gravitational anomalies

Abstract

We review a many-body formulation of gravitational anomalies as obstructions to realizing the complete low-energy data by a local {\it bosonic} lattice quantum system of the same dimension. In this formulation, an anomalous theory can be realized at the boundary of a gapped {\it bosonic} lattice bulk, whose topological order gives a one-to-one classification of the anomaly. An invertible bulk reproduces conventional anomaly inflow; a bulk with non-trivial topological excitations yields a non-invertible generalization. To describe the holographic theory of anomalies, we explain the distinction between uniquely defined topological order and many versions of topological quantum field theory and discuss the boundary-to-bulk map. As an application of the holographic theory of anomalies, we describe morphisms between local quantum theories (such as quantum field theories). These morphisms are built from a bulk domain wall and an invertible boundary junction, which give rise to a sandwich picture. Chiral boson/fermion theory, Laughlin edges, and boundaries of $\mathbb{Z}_2$ higher gauge theory with bosonic or fermionic point-like excitations illustrate the different kinds of information about non-invertible anomalies contained in transport coefficients, superselection sectors, and condensation data. In particular, a fermionic theory can be viewed as a theory with a particular non-invertible gravitational anomaly. Those examples also reveal that {\it symmetry $\sim$ gravitational anomaly $\sim$ topological order in one higher dimension}.