2020 Fiscal Year Annual Research Report
Diagramatic construction of non-semisimple TQFT
| Project/Area Number |
19F19765
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| Research Institution | Waseda University |
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Principal Investigator |
村上 順 早稲田大学, 理工学術院, 教授 (90157751)
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| Co-Investigator(Kenkyū-buntansha) |
DE RENZI MARCO 早稲田大学, 理工学術院, 外国人特別研究員
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| Project Period (FY) |
2019-11-08 – 2021-03-31
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| Keywords | Quantum Topology / Quantum Invariants / TQFT's / Skein Algebras / Mapping class groups |
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| Outline of Annual Research Achievements |
The first result is a construction of a combinatorial model for non-semisimple quantum invariants and TQFTs by developing a skein theoretic formulation of non-semisimple TQFTs associated with the small quantum group Uq(sl2) when q is a root of unity of odd order, by analogy with the construction of Blanchet, Habegger, Masbaum, and Vogel in the semisimple case. With Christian Blanchet and Jun Murakami, we developed a diagrammatic construction of representations of the small quantum group Uq(sl2) when q is a root of unity of odd order. Then, with Jun Murakami, we obtained a fully combinatorial reformulation of the non-semisimple quantum invariants associated with Uq(sl2) when q is a root of unity of odd order. More precisely, we defined an extended version of the Temperley-Lieb category when q is a root of unity of odd order.
The second result is a construction of non-semisimple TQFTs and mapping class group representations from modular categories. This is a joint effort with Nathan Geer, Bertrand Patureau, Azat Gainutdinov, and Ingo Runkel. Together, we developed a renormalized version of the quantum invariants of Lyubashenko, which we extended to full TQFTs. The theory of modified traces then makes it possible to renormalize the construction in order to define fully monoidal functors, as we have already done in the case of Hennings invariants with Geer and Patureau. We also study the projective quantum representations of mapping class groups produced by this construction, and show that they recover Lyubashenko’s ones.
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| Research Progress Status |
令和2年度が最終年度であるため、記入しない。
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| Strategy for Future Research Activity |
令和2年度が最終年度であるため、記入しない。
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