Hochschild cohomology, modular tensor categories, and mapping class groups I / Simon Lentner, Svea Nora Mierach, Christoph Schweigert, Yorck Sommerhäuser.
Lentner, Simon, author.; Mierach, Svea Nora, author.; Schweigert, Christoph, author.; Sommerhäuser, Yorck, 1966- author.
2023
QC20.7.H65
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Title Hochschild cohomology, modular tensor categories, and mapping class groups I / Simon Lentner, Svea Nora Mierach, Christoph Schweigert, Yorck Sommerhäuser.
Author Lentner, Simon, author.
ISBN 9789811946455 (electronic bk.)
9811946450 (electronic bk.)
9789811946448
9811946442
DOI https://doi.org/10.1007/978-981-19-4645-5
Published Singapore : Springer, [2023]
Copyright ©2023
Language English
Description 1 online resource (ix, 68 pages) : illustrations (chiefly color).
Item Number 10.1007/978-981-19-4645-5 doi
Call Number QC20.7.H65
Dewey Decimal Classification 530.15423
Summary 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.
Bibliography, etc. Note Includes bibliographical references.
Access Note Access limited to authorized users.
Source of Description Online resource; title from PDF title page (SpringerLink, viewed August 1, 2023).
Added Author Mierach, Svea Nora, author.
Schweigert, Christoph, author.
Sommerhäuser, Yorck, 1966- author.
Series SpringerBriefs in mathematical physics ; v. 44.
Available in Other Form Print version: 9789811946448
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Intro
Contents
Introduction
Chapter 1 Mapping Class Groups
1.1 The classification of surfaces
1.2 The fundamental group
1.3 Mapping class groups
1.4 Dehn twists
1.5 Braidings
1.6 The action on the fundamental group
1.7 Dehn twists for special curves
1.8 Dehn twists related to two boundary components
1.9 The capping homomorphism
1.10 The Birman sequence
1.11 Singular homology
1.12 The mapping class group of the torus
1.13 The mapping class group of the sphere
Chapter 2 Tensor Categories
2.1 Finiteness
2.2 Factorizable Hopf algebras

2.3 Coends
2.4 Coends from Hopf algebras
2.5 The block spaces
2.6 Mapping class group representations
2.7 Modular functors
2.8 The case of the torus
Chapter 3 Derived Functors
3.1 Projective resolutions
3.2 Derived block spaces
3.3 The case of the sphere
3.4 Hochschild cohomology
References

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