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Super Andersen-Kashaev Topological Quantum Field Theory

Project description

Constructing a topological field theory from quantum Teichmüller theory

A topological quantum field theory (TQFT) is a quantum field theory that focusses on topological invariants. Although these theories were invented by physicists, they are also of mathematical interest, being related, amongst other things, to quantum topology. Until now, there has been a lack of mathematical understanding on TQFTs associated with non-compact super gauge groups. The EU-funded SAKTQFT project aims to construct TQFTs based on super quantum Teichmüller theory. The project's results are expected to build a bridge across different areas of mathematics, possibly opening new research gates completely inspired by physics.

Objective

A topological quantum field theory (TQFT) is a model of quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathematical interest because of their relation to quantum topology. Particularly interesting examples of 3d TQFTs arise from Chern-Simons (CS) theory with non-compact gauge groups. A connected component of the phase space of PSL(2,R) CS theory is identified with Teichmüller space, and its quantum theory corresponds to a specic class of unitary mapping class group representations in infinite dimensional Hilbert spaces. By using quantum Teichmüller Theory (qTT), Andersen and Kashaev construct a one parameter family of TQFT's dened on certain shaped triangulated pseudo 3-manifolds. On the other hand, Teichmüller theory has an interesting generalization originating from the deformation theory of super Riemann surfaces which initially was motivated by super string perturbation theory. In the super generalized case, the gauge groups is replaced by the super group OSP(1|2). Recently, qTT of super Riemann surfaces has been constructed by using coordinates associated to the ideal triangulations of super Riemann surfaces. To this date, we lack mathematical understanding of TQFT's with non-compact super gauge groups. This proposal aims to construct and study TQFT's based on super qTT as follows: Construction of the tetrahedral partition functions from the mapping class groupiod representations of super qTT using a version of charging, establish tetrahedral symmetries and prove well-denedness and topological invariance for a certain class of triangulated shaped pseudo 3-manifolds and finally make calculations for different examples. These results are expected to build a bridge between very different areas of mathematics, thus possibly opening new research gates completely inspired by physics.

Coordinator

SYDDANSK UNIVERSITET
Net EU contribution
€ 207 312,00
Address
CAMPUSVEJ 55
5230 Odense M
Denmark

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Region
Danmark Syddanmark Fyn
Activity type
Higher or Secondary Education Establishments
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Total cost
€ 207 312,00