Projective symmetries of three-dimensional TQFTs
Abstract
Quantum field theory has various projective characteristics which are captured by what are called anomalies. This paper explores this idea in the context of fully-extended three-dimensional topological quantum field theories (TQFTs). Given a three-dimensional TQFT (valued in the Morita 3-category of fusion categories), the anomaly identified herein is an obstruction to gauging a naturally occurring orthogonal group of symmetries. In other words, the classical symmetry group almost acts: There is a lack of coherence at the top level. This lack of coherence is captured by a "higher (central) extension" of the orthogonal group, obtained via a modification of the obstruction theory of Etingof-Nikshych-Ostrik-Meir [ENO10]. This extension tautologically acts on the given TQFT/fusion category, and this precisely classifies a projective (equivalently anomalous) TQFT. We explain the sense in which this is an analogue of the classical spin representation. This is an instance of a phenomenon emphasized by Freed [Fre23]: Quantum theory is projective. We also establish a general relationship between the language of projectivity/anomalies and the language of topological symmetries. We also identify a universal anomaly associated with any theory which is appropriately "simple".
Keywords
Cite
@article{arxiv.2311.01637,
title = {Projective symmetries of three-dimensional TQFTs},
author = {Jackson Van Dyke},
journal= {arXiv preprint arXiv:2311.01637},
year = {2025}
}
Comments
57 pages; Section C.3 ("Projective theories") has been moved to Section 2. A previous version included "Hypothesis P" which has now been proven in Proposition 3.4. Section 4.3 (Previously Section 2.3) contains some new content. A new subsection (4.4.7) contains a new proposition (4.47). This implies a new corollary (5.4) in Section 5.1. A new result appears in Theorem 5.7