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2020 Fiscal Year Annual Research Report

Diagramatic construction of non-semisimple TQFT

Research Project

Project/Area Number 19F19765
Research InstitutionWaseda University

Principal Investigator

村上 順  早稲田大学, 理工学術院, 教授 (90157751)

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
Project Period (FY) 2019-11-08 – 2021-03-31
KeywordsQuantum Topology / Quantum Invariants / TQFT's / Skein Algebras / Mapping class groups
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.

Research Progress Status

令和2年度が最終年度であるため、記入しない。

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。

  • Research Products

    (4 results)

All 2021 2020 Other

All Int'l Joint Research (2 results) Journal Article (1 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 1 results) Presentation (1 results)

  • [Int'l Joint Research] Universie de Paris/Universite de Bretagne-Sud/Universite de Tours(フランス)

    • Country Name
      FRANCE
    • Counterpart Institution
      Universie de Paris/Universite de Bretagne-Sud/Universite de Tours
  • [Int'l Joint Research] Utah State University(米国)

    • Country Name
      U.S.A.
    • Counterpart Institution
      Utah State University
  • [Journal Article] Non-Semisimple Quantum Invariants and TQFTs from Small and Unrolled Quantum Groups2021

    • Author(s)
      M. De Renzi, N. Geer, B. Patureau-Mirand
    • Journal Title

      Algebraic & Geometric Topology

      Volume: 掲載予定 Pages: 未定

    • Peer Reviewed / Int'l Joint Research
  • [Presentation] TQFTs en dimension 3 a partir de categories modulaires non semi-simples2020

    • Author(s)
      DE RENZI, Marco
    • Organizer
      Geometry and Topology Seminar, Paul Sabatier University, Toulouse (France)

URL: 

Published: 2021-12-27  

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