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Smooth generalized symmetries of quantum field theories

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Gripaios, B 
Randal-Williams, O 


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 -categories. According to a geometric version of the cobordism hypothesis, such QFTs collectively assemble themselves into objects in an -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.



hep-th, hep-th, math.AT, math.CT

Journal Title

Journal of Geometry and Physics

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Elsevier BV
STFC (ST/T000694/1)
Science and Technology Facilities Council (STFC) consolidated grant ST/T000694/1; U.S. National Science Foundation (NSF) grant PHY-2014071.