2019 Fiscal Year Annual Research Report
Diagramatic construction of non-semisimple TQFT
| Project/Area Number |
19F19765
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| Research Institution | Waseda University |
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Principal Investigator |
村上 順 早稲田大学, 理工学術院, 教授 (90157751)
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| 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
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| Project Period (FY) |
2019-11-08 – 2022-03-31
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| Keywords | Quantum Topology / Quantum Invariants / TQFTs / Skein Algebras / Mapping Class Groups |
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| Outline of Annual Research Achievements |
In collaboration with Christian Blanchet and Jun Murakami, we gave a diagrammatic description of the monoidal category generated by the fundamental representation of the small quantum group of sl(2) at a root of unity q of odd order. More precisely, we defined an extended version of the Temperley-Lieb category of parameter -q-1/q obtained by adding generators and relations at the level of morphisms. This extension is inspired by crucial differences between the category of representations of small quantum sl(2) and that of Lusztig’s divided power version, which corresponds directly to the standard Temperley-Lieb category. The definition is based on a generalized version of Jones-Wenzl idempotents which realizes projectors on indecomposable projective representations. We proved there exists a full monoidal functor from the extended Temperley-Lieb category to the category of representations of small quantum sl(2) which sends the monoidal generator to the fundamental representation.
In parallel, in collaboration with Azat Gainutdinov, Nathan Geer, Bertrand Patureau, and Ingo Runkel, we defined a renormalized version of Lyubashenko’s non-semisimple quantum invariants of closed 3-manifolds, which we extended to TQFTs. Our construction uses the theory modified traces to define a quantum invariant for each finite, non-degenerate, unimodular ribbon category. Using the universal construction, we were able to extend the renormalized Lyubashenko invariant associated with a finite factorizable ribbon category to a symmetric monoidal functor on the category of admissible cobordisms.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
In collaboration with Christian Blanchet and Jun Murakami, we are currently extending the graphical calculus derived from Kauffman’s bracket polynomial in a bichrome sense, by analogy with the algebraic approach. In order to do this, we are using the diagrammatic description of the monoidal category generated by the fundamental representation of small quantum sl(2) we previously obtained in terms of the extended Temperley-Lieb category. A key step of the project consists in figuring out a diagrammatic translation of the very rich structure of a crucial object of the category of representation of small quantum sl(2), called the coend, which is embodied by the adjoint representation. The construction of a family of quantum invariants of closed 3-manifolds will naturally be based on a decomposition of the symmetrized integral, another fundamental ingredient form the algebraic viewpoint, in terms of traces and pseudo-traces, as well as on the theory of modified traces. The extension to a family of non-semisimple TQFTs will then be done in the usual way, through the universal construction. The idea is to obtain a diagrammatic construction of non-semisimple quantum invariants and TQFTs associated with small quantum sl(2) which does not require any knowledge of its representation theory.
In parallel, in collaboration with Azat Gainutdinov, Nathan Geer, Bertrand Patureau, and Ingo Runkel, we are proving the mapping class group representations issued by our TQFT construction are equivalent to Lyubashenko’s one.
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| Strategy for Future Research Activity |
In the coming months, I plan to complete the diagrammatic construction of non-semisimple quantum invariants of closed 3-manifolds associated with small quantum sl(2), as well as their extension to TQFTs, as explained above.
In a second moment, my goal is to use this model for the study of associated geometric problems. For example, this combinatorial construction would naturally induce actions of skein algebras on non-semisimple state spaces of closed surfaces. Representations of these algebraic structures have a geometric interest, and they already attracted considerable attention in the semi-simple case. A combinatorial model for non-semisimple TQFTs would then immediately induce new families of representations of skein algebras. The diagrammatic approach would also be interesting in order to generalize Witten’s asymptotic conjecture, which connects Witten-Reshetikhin-Turaev invariants, and in particular their asymptotic behavior with respect to the order of the root of the unity, with gauge theoretical quantities like the Chern-Simons invariant and the Reidemeister torsion. In the semisimple case, the skein module of a 3-manifold can be obtained as a deformation of the coordinate ring of its SL(2)-character variety. An analogous interpretation in the non-semisimple case would allow us to look for asymptotic relations between actions of mapping class groups of different nature: quantum ones (on non-semisimple skein modules) on one side, and geometric ones (on spaces of square-integrable functions associated with character varieties) on the other.
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