{ "id": "https://doi.org/10.48660/18050002", "doi": "10.48660/18050002", "url": "https://pirsa.org/18050002", "types": { "ris": "MPCT", "bibtex": "misc", "citeproc": "article", "schemaOrg": "MediaObject", "resourceType": "Video Recording", "resourceTypeGeneral": "Audiovisual" }, "creators": [ { "nameType": "Personal", "givenName": "David", "familyName": "Jordan", "affiliation": [ { "name": "University of Edinburgh" } ] } ], "titles": [ { "title": "Braided tensor categories and the cobordism hypothesis" } ], "publisher": { "name": "Perimeter Institute" }, "subjects": [ { "subject": "Mathematical physics" } ], "dates": [ { "date": "2018-05-07T18:00:00", "dateType": "Created" } ], "publicationYear": 2018, "language": "en", "identifiers": [ { "identifier": "https://pirsa.org/18050002", "identifierType": "PURL" } ], "formats": [ "video/mp4" ], "descriptions": { "description": "
The cobordism hypothesis gives a functorial bijection between oriented
\r\n\r\nn-dimensional fully local topological field theories, valued in some
\r\n\r\nhigher category C, and the fully dualizable object of C equipped with
\r\n\r\nthe structure of SO(n)-fixed point. In this talk I'll explain recent
\r\n\r\nworks of Haugseng, Johnson-Freyd and Scheimbauer which construct a
\r\n\r\nMorita 4-category of braided tensor categories, and I'll report on joint
\r\n\r\nwork with Brochier and Snyder which identifies two natural subcategories
\r\n\r\ntherein which are 3- and 4-dualizable. These are the rigid braided
\r\n\r\ntensor categories with enough compact projectives, and the braided
\r\n\r\nfusion categories, respectively. I'll also explain work in progress by
\r\n\r\nus to construct SO(3)- and SO(4)-fixed point structures in each case,
\r\n\r\nstarting from ribbon and pre-modular categories, respectively.
\r\n\r\n\r\n\r\n
Applying the cobordism hypothesis, we obtain 3- and 4-dimensional fully
\r\n\r\nlocal TFT's, which extend the 2-dimensional TFT's we constructed with
\r\n\r\nBen-Zvi and Brochier, and which conjecturally relate to a number of
\r\n\r\nconstructions in the literature, including: skein modules, quantum
\r\n\r\nA-polynomials, Crane-Kauffmann-Yetter invariants; hence our construction
\r\n\r\nputs these on firm foundational grounds as fully local TFT's. A key
\r\n\r\nfeature of our construction in dimension 3 is that we require the input
\r\n\r\nbraided tensor category neither to be finite, nor semi-simple, so this
\r\n\r\nopens up new examples -- such as non-modularized quantum groups at roots
\r\n\r\nof unity -- which were not obtainable by earlier methods.
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