Minimal nondegenerate extensions
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866912909958840320 |
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| author | Johnson-Freyd, Theo Reutter, David |
| author_facet | Johnson-Freyd, Theo Reutter, David |
| contents | We prove that every slightly degenerate braided fusion category admits a minimal nondegenerate extension, and hence that every pseudo-unitary super modular tensor category admits a minimal modular extension. This completes the program of characterizing minimal nondegenerate extensions of braided fusion categories.
Our proof relies on the new subject of fusion 2-categories. We study in detail the Drinfel'd centre Z(Mod-B) of the fusion 2-category Mod-B of module categories of a braided fusion 1-category B. We show that minimal nondegenerate extensions of B correspond to certain trivializations of Z(Mod-B). In the slightly degenerate case, such trivializations are obstructed by a class in $\mathrm{H}^5(K(\mathbb{Z}_2, 2); k^\times)$ and we use a numerical invariant -- defined by evaluating a certain two-dimensional topological field theory on a Klein bottle -- to prove that this obstruction always vanishes.
Along the way, we develop techniques to explicitly compute in braided fusion 2-categories which we expect will be of independent interest. In addition to the model of Z(Mod-B) in terms of braided B-module categories, we develop a computationally useful model in terms of certain algebra objects in B. We construct an S-matrix pairing for any braided fusion 2-category, and show that it is nondegenerate for Z(Mod-B). As a corollary, we identify components of Z(Mod-B) with blocks in the annular category of B and with the homomorphisms from the Grothendieck ring of the Müger centre of B to the ground field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_15167 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Minimal nondegenerate extensions Johnson-Freyd, Theo Reutter, David Quantum Algebra Category Theory Representation Theory 18M20, 18M15, 18N10 We prove that every slightly degenerate braided fusion category admits a minimal nondegenerate extension, and hence that every pseudo-unitary super modular tensor category admits a minimal modular extension. This completes the program of characterizing minimal nondegenerate extensions of braided fusion categories. Our proof relies on the new subject of fusion 2-categories. We study in detail the Drinfel'd centre Z(Mod-B) of the fusion 2-category Mod-B of module categories of a braided fusion 1-category B. We show that minimal nondegenerate extensions of B correspond to certain trivializations of Z(Mod-B). In the slightly degenerate case, such trivializations are obstructed by a class in $\mathrm{H}^5(K(\mathbb{Z}_2, 2); k^\times)$ and we use a numerical invariant -- defined by evaluating a certain two-dimensional topological field theory on a Klein bottle -- to prove that this obstruction always vanishes. Along the way, we develop techniques to explicitly compute in braided fusion 2-categories which we expect will be of independent interest. In addition to the model of Z(Mod-B) in terms of braided B-module categories, we develop a computationally useful model in terms of certain algebra objects in B. We construct an S-matrix pairing for any braided fusion 2-category, and show that it is nondegenerate for Z(Mod-B). As a corollary, we identify components of Z(Mod-B) with blocks in the annular category of B and with the homomorphisms from the Grothendieck ring of the Müger centre of B to the ground field. |
| title | Minimal nondegenerate extensions |
| topic | Quantum Algebra Category Theory Representation Theory 18M20, 18M15, 18N10 |
| url | https://arxiv.org/abs/2105.15167 |