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Super Andersen-Kashaev Topological Quantum Field Theory

Projektbeschreibung

Ausarbeitung einer topologischen Feldtheorie anhand der Quanten-Teichmüller-Theorie

Eine topologische Quantenfeldtheorie ist eine Quantenfeldtheorie mit einem Schwerpunkt auf topologische Invarianten. Obwohl diese Theorien im Rahmen der Physik entstanden, sind sie auch für die Mathematik von Interesse, da sie unter anderem mit der Quantentopologie zusammenhängen. Bislang mangelte es an einem mathematischen Verständnis von topologischen Quantenfeldtheorien, die mit nicht kompakten Super-Eich-Gruppen in Verbindung stehen. Das EU-finanzierte Projekt SAKTQFT zielt darauf ab, topologische Quantenfeldtheorien auf der Grundlage der Superquanten-Teichmüller-Theorie zu entwickeln. Von den Ergebnissen des Projekts erwartet man sich, dass sie eine Brücke zwischen verschiedenen Bereichen der Mathematik schlagen und möglicherweise neue Forschungsgebiete eröffnen werden, die voll und ganz von der Physik inspiriert sind.

Ziel

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.

Koordinator

SYDDANSK UNIVERSITET
Netto-EU-Beitrag
€ 207 312,00
Adresse
CAMPUSVEJ 55
5230 Odense M
Dänemark

Auf der Karte ansehen

Region
Danmark Syddanmark Fyn
Aktivitätstyp
Higher or Secondary Education Establishments
Links
Gesamtkosten
€ 207 312,00