Anomalies, higher symmetries, and homotopy theory I: crossed precosheaves and a nonabelian Cech construction

@misc{kapustin2026anomalies,
  title = {{Anomalies, higher symmetries, and homotopy theory I: crossed precosheaves and a nonabelian Cech construction}},
  author = {Anton Kapustin and Shixiong Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19004},
  eprint = {2610.19004},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19004},
}

TY  - UNPB
TI  - Anomalies, higher symmetries, and homotopy theory I: crossed precosheaves and a nonabelian Cech construction
AU  - Anton Kapustin
AU  - Shixiong Xu
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.19004
UR  - https://arxiv.org/abs/2610.19004
AB  - Symmetries of a quantum field theory or a quantum lattice model are expected to be described not by a group but by a homotopy type, whose homotopy groups are the higher symmetry groups and whose Postnikov invariants encode 't Hooft anomalies. We propose a homotopy-theoretic definition of 't Hooft anomalies based on the structure of localized symmetries in the operator-algebraic approach. The input is a crossed precosheaf: a collection of groups attached to regions of space, equipped with pairings recording that the commutator of two transformations localized in two regions is localized in their intersection. To a crossed precosheaf and a finite family of regions we attach a pointed connected homotopy type, the Cech-Loday classifying space. As the family of regions ranges over finite covers of space, the classifying spaces form a pro-object in the homotopy category. There is a canonical map from the ordinary classifying space of the symmetry group to this pro-type. Composing this map with a group action gives an anomaly whose nontriviality obstructs gauging. In favorable cases, the Cech-Loday space is essentially independent of the cover. In such cases, the homotopy fiber of the canonical map is a candidate for the classifying space of the higher group of symmetries. We give examples of crossed precosheaves arising from gauge transformations in continuum field theory and from quantum cellular automata and finite-depth circuits. For quantum spin systems in one and two dimensions, our construction recovers the Else-Nayak index and its generalizations to two dimensions.
ER  -