Projective and anomalous representations of categories and their linearizations

2025/06/02 by Domenico Fiorenza, Fiorenza, Domenico, Chetan Vuppulury +1 · 1 citation
Mathematics · #18D25 #Advanced Topics in Algebra #Category Theory (math.CT) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2506.01521

openalex publication_date 2025/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We invesigate the relation between projective and anomalous representations of categories, and show how to any anomaly J\colon C→ 2Vect one can associate an extension C^J of C and a subcategory C^J_ST of C^J with the property that: (i) anomalous representations of C with anomaly J are equivalent to Vect-linear functors E\colon C^J→ Vect, and (ii) these are in turn equivalent to linear representations of C^J_ST where "J acts as scalars". This construction, inspired by and generalizing the technique used to linearize anomalous functorial field theories in the physics literature, can be seen as a multi-object version of the classical relation between projective representations of a group G, with given 2-cocycle α, and linear representations of the central extension G^α of G associated with α.

Citations

• Relative field theories via relative dualizability
• The Chiral Anomaly of the Free Fermion in Functorial Field Theory
• Modular categories as representations of the 3-dimensional bordism 2-category
• Anomalies and Invertible Field Theories
• Relative quantum field theory
• Supersymmetric field theories and generalized cohomology
• Constructing symmetric monoidal bicategories
• On the Classification of Topological Field Theories

Cited by

• Invertible projective 2-representations from invertible 2d TQFTs with defects

Related

• Anomalous Actions of Groups on Tensor Categories
• Modular categories as representations of the 3-dimensional bordism 2-category
• Local versus global subtleties of projective representations
• Tensor categories attached to exceptional cells in Weyl groups
• Inductive systems of the symmetric group, polynomial functors and tensor categories