Higher anomalies, compressing SPTs and cohomology operations

Abstract

We explicitly compute and tabulate the suspension map for cohomology operations, applying it to the dimensional reduction of anomalies and higher group structures of generalised symmetries in quantum field theories and lattice systems. In a quantum system with a discrete $p$-form symmetry, we can restrict the symmetry defects to a codimension-$q$ subspace of spacetime, or equivalently confine the gauge field to a slab whose thickness approaches zero. This results in a $(p-q)$-form symmetry on the subspace. We determine the fate of two important properties of symmetries under this process. First, we compute the anomaly of the reduced symmetry, or the higher anomaly of the original symmetry, which acts as an obstruction to higher gauging and onsiteability. Second, for two symmetries forming a higher-group structure, encoded as a Postnikov class, we compute the reduced higher-group structure, which may or may not be trivial. Both cases are captured by the $q$-fold iteration of the cohomology suspension $Ω$, also known as the loop functor or transgression, in the cohomology of Eilenberg--MacLane spaces. Classical results state that $Ω$ is an isomorphism in a stable range and annihilates mixed anomalies. To calculate $Ω$ over a base ring which is not a field, we make use of small models for the chain DGAs of Eilenberg--MacLane spaces, pioneered by Cartan and Moore, whose results we review and extend. The critical added step is the improvement of a mod $p$ chain contraction to the $p$-local integers, obtained via a $p$-adic series. We automate the computations in a software package \texttt{emcm}, and provide extensive tables of the results. Finally, to make contact with explicit cochain formulations of cohomology operations, we demonstrate how to calculate cup products and certain cup-$i$ products in this framework.