H2020Индивидуална стипендия2020–2023

SAKTQFT · Super Andersen-Kashaev Topological Quantum Field Theory

„Хоризонт 2020“ — Действия „Мария Склодовска-Кюри“

Период: 2020-11-01 → 2023-10-23
Финансиране от ЕС: 207 312 €
Участници: 1
Схема: MSCA-IF

Линиите свързват координатора с партньорите.

Накратко на български

Математическата физика тук изследва супер-обобщена версия на топологичните квантови полеви теории, като изчислява конкретни функции за геометрии на триизмерни многообразия. Това помага за по-доброто разбиране на връзката между квантовата топология и теорията на суперстрингите.

Този кратък обзор е генериран от изкуствен интелект

Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.

Резултати накратко

Super Andersen-Kashaev Topological Quantum Field Theory

The interaction between mathematics and theoretical physics led to the creation of a new interdisciplinary research topic called mathematical physics where mathematical rigour and physical intuition merge in a natural way. Topological Quantum Field Theory (TQFT) is an excellent example of this fruitful interplay. 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 specific class of unitary mapping class group representations is infinite-dimensional Hilbert spaces, in contrast to more standard TQFTs. This TQFT was previously formulated in the mathematics literature by Andersen and Kashaev. The goal of this project was to consider the gauge groups which is replaced by the super group OSP(1|2) and study the super generalized version of TQFT. In this project, we had some results in super Teichmüller spin TQFT. We obtained concrete expressions for the partition functions of the super Teichmüller spin TQFT for a class of spin 3-manifold geometries. We then compute the perturbative expansions of the partition functions, to obtain perturbative invariants of spin 3-manifolds. The mentioned super generalization originated from the deformation theory of super Riemann surfaces which initially was motivated by superstring perturbation therefore in the second part, we focused on 2d-TQFT aspect. In particular, we focused on the mathematical techniques related to that so-called Topological Recursion(TR). We studied the spectral curve of the Super TR, so-called super spectral curve and the super geometry behind this curve. We focused on the OSP(1|2) super gauge group, which was the main interest in our original proposal.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз

Цел на проекта

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.

Оригинален текст от CORDIS (на английски).

Участници

SYDDANSK UNIVERSITET · Odense MКоординаторДания