Smooth generalized symmetries of quantum field theories

Dynamical quantum field theories (QFTs), such as those in which spacetimes are equipped with a metric and/or a field in the form of a smooth map to a target manifold, can be formulated axiomatically using the language of infinity-categories. According to a geometric version of the cobordism hypothesis, such QFTs collectively assemble themselves into objects in an infinity-topos of smooth spaces. We show how this allows one to define and study generalized global symmetries of such QFTs. The symmetries are themselves smooth, so the 'higher -form' symmetry groups can be endowed with, e.g., a Lie group structure. Among the more surprising general implications for physics are, firstly, that QFTs in spacetime dimension d, considered collectively, can have d -form symmetries, going beyond the known (d - 1) -form symmetries of individual QFTs and, secondly, that a global symmetry of a QFT can be anomalous even before we try to gauge it, due to a failure to respect either smoothness (in that a symmetry of an individual QFT does not smoothly extend to QFTs collectively) or locality (in that a symmetry of an unextended QFT does not extend to an extended one). Smoothness anomalies are shown to occur even in 2 -state systems in quantum mechanics (here formulated axiomatically by equipping d =1 spacetimes with a metric, an orientation, and perhaps some unitarity structure). Locality anomalies are shown to occur even for invertible QFTs defined on d =1 spacetimes equipped with an orientation and a smooth map to a target manifold. These correspond in physics to topological actions for a particle moving on the target and the relation to an earlier classification of such actions using invariant differential cohomology is elucidated. (c) 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).