Projective and anomalous representations of categories and their linearizations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913870608596992 |
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| author | Fiorenza, Domenico Vuppulury, Chetan |
| author_facet | Fiorenza, Domenico Vuppulury, Chetan |
| contents | 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 $α$, and linear representations of the central extension $G^α$ of $G$ associated with $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01521 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Projective and anomalous representations of categories and their linearizations Fiorenza, Domenico Vuppulury, Chetan Category Theory Mathematical Physics Quantum Algebra Representation Theory 18D25 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 $α$, and linear representations of the central extension $G^α$ of $G$ associated with $α$. |
| title | Projective and anomalous representations of categories and their linearizations |
| topic | Category Theory Mathematical Physics Quantum Algebra Representation Theory 18D25 |
| url | https://arxiv.org/abs/2506.01521 |