Projective and anomalous representations of categories and their linearizations

Abstract

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

arXiv comment29 pages; exposition abridged and improved; final version, to appear in Theory and Applications of Categories