Descrizione del progetto
Costruire una teoria dei campi topologici a partire dalla teoria quantistica di Teichmüller
Una teoria quantistica dei campi topologici (topological quantum field theory, TQFT) è una teoria quantistica dei campi incentrata sugli invarianti topologici. Sebbene queste teorie siano state inventate dai fisici, sono anche di interesse matematico, essendo legate, tra le altre cose, alla topologia quantistica. Fino ad oggi c’è stata una mancanza di comprensione matematica sui TQFT associati a gruppi di super gauge non compatti. Il progetto SAKTQFT, finanziato dall’UE, mira a costruire TQFT basati sulla teoria dei super quantum Teichmüller. Si prevede che i risultati del progetto possano gettare un ponte tra le diverse aree della matematica, aprendo eventualmente nuove strade di ricerca completamente ispirate alla fisica.
Obiettivo
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.
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MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinatore
5230 Odense M
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