Non-semi-simple TQFTs, Reidemeister torsion and Kashaev's invariants

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• 作者：Christian Blanchet^a ; ^{christian.blanchet@imj-prg.fr" class="auth_mail" title="E-mail the corresponding author} ; Francesco Costantino^b ; ^{Francesco.Costantino@math.univ-toulouse.fr" class="auth_mail" title="E-mail the corresponding author} ; Nathan Geer^c ; ^{nathan.geer@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Bertrand Patureau-Mirand^d ; ^{bertrand.patureau@univ-ubs.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Topological quantum field theory ; Reidemeister torsion ; 3-manifold invariants ; Mapping class group representations ; Kashaev invariant ; Representations of quantum groups
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：1 October 2016
• 年：2016
• 卷：301
• 期：Complete
• 页码：1-78
• 全文大小：1423 K

文摘

We construct and study a new family of TQFTs based on nilpotent highest weight representations of quantum sl(2)$\mathfrak{sl}(2)$ at a root of unity indexed by generic complex numbers. This extends to cobordisms the non-semi-simple invariants defined in [9] including the Kashaev invariant of links. Our case is not covered by the modular category framework and so we give a new example of application of the so-called “universal construction”. For each root of unity of order 2r   where r≥2$r\ge 2$ is not divisible by 4, our TQFT provides a monoidal functor from a category of surfaces and their cobordisms into the category of graded finite dimensional vector spaces. The functor is always symmetric monoidal but for even values of r   the braiding on GrVect has to be the super-symmetric one, thus our TQFT may be considered as a super-TQFT. In the special case r=2$r=2$ our construction yields a TQFT for a canonical normalization of Reidemeister torsion and we re-prove the classification of Lens spaces via the non-semi-simple quantum invariants defined in [9]. We prove that the representations of mapping class groups and Torelli groups resulting from our constructions are potentially more sensitive than those obtained from the standard Reshetikhin–Turaev functors; in particular we prove that the action of the bounding pairs generators of the Torelli group has always infinite order.