Projective and anomalous representations of categories and their linearizations

TL;DR

This paper explores the relationship between projective and anomalous category representations, introducing a construction that generalizes classical group representation extensions to categories, with applications in physics and linearization of anomalous theories.

Contribution

It presents a novel extension construction for categories associated with anomalies, generalizing the classical relation between projective and linear group representations.

Findings

Establishes an equivalence between anomalous representations and linear functors on an extended category.

Introduces a subcategory where the anomaly acts as scalars, simplifying the representation theory.

Generalizes the classical group extension approach to categorical settings with anomalies.

Abstract

We invesigate the relation between projective and anomalous representations of categories, and show how to any anomaly $J\colon \mathcal{C}\to 2\mathrm{Vect}$ one can associate an extension $\mathcal{C}^J$ of $\mathcal{C}$ and a subcategory $\mathcal{C}^J_{\mathrm{ST}}$ of $\mathcal{C}^J$ with the property that: (i) anomalous representations of $\mathcal{C}$ with anomaly $J$ are equivalent to $\mathrm{Vect}$-linear functors $E\colon \mathcal{C}^J\to \mathrm{Vect}$, and (ii) these are in turn equivalent to linear representations of $\mathcal{C}^J_{\mathrm{ST}}$ 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 $\alpha$, and linear representations of the central extension $G^\alpha$ of $G$ associated with $\alpha$.