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Construction of the Witten–Reshetikhin–Turaev TQFT from conformal field theory

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Abstract

In Andersen and Ueno (J Knot Theory Ramif 16:127–202, 2007) we constructed the vacua modular functor based on the sheaf of vacua theory developed in Tsuchiya et al. (Adv Stud Pure Math 19:459–566, 1989) and the abelian analog in Andersen and Ueno (Int J Math 18:919–993, 2007). We here provide an explicit isomorphism from the modular functor underlying the skein-theoretic model for the Witten–Reshetikhin–Turaev TQFT due to Blanchet, Habbeger, Masbaum and Vogel to the vacua modular functor. This thus provides a geometric construction of the TQFT first proposed by Witten and constructed first by Reshetikhin–Turaev from the quantum group \(U_q(\text{ sl }(N))\).

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Notes

  1. Please see the discussion following this theorem for the proper interpretation of this property.

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Acknowledgments

We thank Christian Blanchet, Yukihiro Kanie, Gregor Masbaum, Akihiro Tsuchiya and Yasuhiko Yamada for valuable discussions and the referee for several helpful comments which improved the paper.

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Authors and Affiliations

  1. Center for Quantum Geometry of Moduli Spaces, University of Aarhus, 8000, Aarhus, Denmark

    Jørgen Ellegaard Andersen

  2. Faculty of Science and Engineering, Department of Industrial and System Engineering, Hosei University, Koganei, Tokyo, 184-0002, Japan

    Kenji Ueno

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  1. Jørgen Ellegaard Andersen
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Correspondence to Jørgen Ellegaard Andersen.

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Supported in part by the center of excellence grant “Center for quantum geometry of Moduli Spaces” from the Danish National Research Foundation and by Grant-in-Aid for Scientific Research No. 19340007 from JSPS (Japan Society for the Promotion of Science).

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Andersen, J.E., Ueno, K. Construction of the Witten–Reshetikhin–Turaev TQFT from conformal field theory. Invent. math. 201, 519–559 (2015). https://doi.org/10.1007/s00222-014-0555-7

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  • DOI: https://doi.org/10.1007/s00222-014-0555-7

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