A Generalized Crystalline Equivalence Principle
Devon Stockall, Matthew Yu
TL;DR
This work generalizes the crystalline equivalence principle (CEP) by formulating an equivalence between $n$-dimensional crystalline topological phases on a $G$-space $\\mathcal X$ valued in a $G$-category $\\mathbf{\\Theta}$ and $n$-dimensional TQFTs with internal $\\mathcal X//G$-symmetry valued in $\\mathbf{Un}(\\mathbf{\\Theta})$. The authors develop a fibrational, $(\\infty,n)$-categorical framework based on straightening/unstraightening and the homotopy quotient to rigorously relate spatially dependent theories to internal-symmetry theories, and they extend this to fermionic settings and potential noninvertible/categorical symmetries. They introduce a comprehensive anomaly theory for $\\infty$-groupoid symmetries, showing that anomalies are classified by bundles over the symmetry space and that anomalous crystalline theories can be viewed as relative TQFTs with a bulk interpretation in one higher dimension. A linearized example clarifies the relative-theory perspective in the $(n+1)$-Vect / $n$-Vect setting, illustrating how categorical anomalies arise and how they translate under the CEP. Overall, the paper supplies a robust bridge between lattice/spacetime-symmetry phases and internal-symmetry TQFTs, enabling systematic classification of anomalies and potentially accommodating noninvertible and categorical symmetries in a unified framework.
Abstract
We prove a general version of the crystalline equivalence principle which gives an equivalence of categories between a category of TQFTs defined on a generic space with $G$-symmetry, and a category of TQFTs with internal symmetry. We give a definition and classification of anomalies associated to TQFTs in the presence of spatial symmetry, which we then generalize to a definition of an anomaly for a categorical symmetry.