Descripción del proyecto
Desarrollo de una teoría topológica de campos a partir de la teoría cuántica de Teichmüller
Una teoría cuántica topológica de campos (TTCC o TQFT, por sus siglas en inglés) es una teoría cuántica de campos centrada en invariantes topológicos. Aunque los físicos concibieron estas teorías, los matemáticos también están interesados en ellas, ya que están relacionadas, entre otras cosas, con la topología cuántica. Hasta ahora, se desconocía el principio matemático de las TTCC relacionadas con grupos no compactos con supersimetría gauge. El objetivo del proyecto financiado con fondos europeos SAKTQFT es concebir TTCC basadas en la teoría supercuántica de Teichmüller. Los resultados del proyecto podrían tender puentes entre diferentes áreas de las matemáticas y abrir nuevas líneas de investigación completamente inspiradas por la física.
Objetivo
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)Coordinador
5230 Odense M
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