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Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups I

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  • © 2023

Overview

Authors:
  1. Simon Lentner

    1. Department of Mathematics, University of Hamburg, Hamburg, Germany

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  2. Svea Nora Mierach

    1. Dept of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Canada

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  3. Christoph Schweigert

    1. Department of Mathematics, Universität Hamburg, Hamburg, Germany

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  4. Yorck Sommerhäuser

    1. Dept of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Canada

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  • Discusses conformal field theory in a non-semisimple setting
  • Uses Lyubashenko’s approach via nets and ribbon graphs
  • Describes the relation to Hochschild cohomology of Hopf algebras

Part of the book series: SpringerBriefs in Mathematical Physics (BRIEFSMAPHY, volume 44)

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Table of contents (3 chapters)

  1. Front Matter

    Pages I-IX
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  2. Mapping Class Groups

    • Simon Lentner, Svea Nora Mierach, Christoph Schweigert, Yorck Sommerhäuser
    Pages 1-25

  3. Tensor Categories

    • Simon Lentner, Svea Nora Mierach, Christoph Schweigert, Yorck Sommerhäuser
    Pages 27-47

  4. Derived Functors

    • Simon Lentner, Svea Nora Mierach, Christoph Schweigert, Yorck Sommerhäuser
    Pages 49-66

  5. Back Matter

    Pages 67-68
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Keywords

About this book

The book addresses a key question in topological field theory and logarithmic conformal field theory: In the case where the underlying modular category is not semisimple, topological field theory appears to suggest that mapping class groups do not only act on the spaces of chiral conformal blocks, which arise from the homomorphism functors in the category, but also act on the spaces that arise from the corresponding derived functors. It is natural to ask whether this is indeed the case. The book carefully approaches this question by first providing a detailed introduction to surfaces and their mapping class groups. Thereafter, it explains how representations of these groups are constructed in topological field theory, using an approach via nets and ribbon graphs. These tools are then used to show that the mapping class groups indeed act on the so-called derived block spaces. Toward the end, the book explains the relation to Hochschild cohomology of Hopf algebras and the modular group.



Authors and Affiliations

  • Department of Mathematics, University of Hamburg, Hamburg, Germany

    Simon Lentner

  • Dept of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Canada

    Svea Nora Mierach, Yorck Sommerhäuser

  • Department of Mathematics, Universität Hamburg, Hamburg, Germany

    Christoph Schweigert

Bibliographic Information

  • Book Title: Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups I

  • Authors: Simon Lentner, Svea Nora Mierach, Christoph Schweigert, Yorck Sommerhäuser

  • Series Title: SpringerBriefs in Mathematical Physics

  • DOI: https://doi.org/10.1007/978-981-19-4645-5

  • Publisher: Springer Singapore

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023

  • Softcover ISBN: 978-981-19-4644-8Published: 26 July 2023

  • eBook ISBN: 978-981-19-4645-5Published: 25 July 2023

  • Series ISSN: 2197-1757

  • Series E-ISSN: 2197-1765

  • Edition Number: 1

  • Number of Pages: IX, 68

  • Number of Illustrations: 2 b/w illustrations, 14 illustrations in colour

  • Topics: Mathematical Physics, Algebraic Topology, Category Theory, Homological Algebra

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