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3d-TQFTs are one of the best toy models to understand the mathematics behind Quantum Field theory.

The best-understood 3d Topological Quantum Field Theories (TQFT) are Chern-Simons Theories (CS) studied by Witten, which can be obtained via the Reshetikhin-Turaev (RT) construction. This construction use quantum groups associated to Lie algebras to define 3-manifold invariants, known as quantum invariants. The algebraic ingredients of RT-constructions come from the finite dimensional representation category of the quantum group Uq(sl(2)) at roots of unity. It was shown that the relevant algebraic structure was that of a modular category that is finite and semisimple. In trying to obtain new types of 3d TQFTs one can attempt to weaken these requirements:

by

1)dropping finiteness      (-------------------> Teichmüller TQFT)

or

2)dropping semisimplicity (-----------------> non semisimple TQFT)

 

In both cases, a qualitatively new feature as compared to the finitely semisimple case is the appearance of continuous parameters in the description of representations. These new types of 3d TQFTs are much less known and this is one of the motivations behind my research projects.

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In particular, I am interested in studying different examples of 3 manifold invariants using mapping class group (MCG) representations and until now my focus was

1)on super algebra like OSP(1|2) which is an example of infinite dim representation and I constructed the super quantum Teichmüller theory 

and MCG rep for that, 

and 

2)on super algebra like gl(1|1) which is an example of nonsemisimple representation and using the combinatorial quantization approach

of the associated Chern-Simons theory, I constructed the MCG rep.

 

Keywords: quantum groups and Representation theory, Super Lie algebra, Teichmüller TQFT, non-semi-simple TQFT, combinatorial quantisation,