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Lie theory, associators and Topological Quantum Field Theory

Applicant Alekseev Anton
Number 165666
Funding scheme Project funding (Div. I-III)
Research institution Section de Mathématiques Université de Genève
Institution of higher education University of Geneva - GE
Main discipline Mathematics
Start/End 01.04.2016 - 31.03.2018
Approved amount 554'247.00
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All Disciplines (2)

Discipline
Mathematics
Theoretical Physics

Keywords (3)

Drinfeld associators; Lie theory; Topological Quantum Field Theory

Lay Summary (French)

Lead
La théorie topologique des champs est un domaine qui se trouve à cheval entre la physique théorique et les mathématiques. Des modèles étudiés en théorie topologique sont en relation étroite avec des modèles qui décrivent la nature. Par exemple, la théorie de Yang-Mills, qui encode les interactions entre des particules quantiques, admet une version topologique. Les questions posées dans cette théorie sont motivées par la recherche en mathématiques. Des exemples classiques sont des problèmes de classification des noeuds et des variétés en dimensions 3 et 4 qui sont reliés à plusieurs modèles en théorie des champs.
Lay summary
Dans ce projet, nous nous proposons d'étudier les relations entre la théorie topologique des champs et la notion de symétrie. Nos outils principaux seront les groupes et algèbres de Lie qui encodent de telles symétries dans le langage mathématique. Les buts majeurs du projet sont les suivants:

 1) De développer la théorie des surfaces de Wilson en théorie de jauge topologiques et physiques. Les surfaces de Wilson représentent un nouveau type d'observables 2-dimensionnels en théories de jauge relié aux lignes de Wilson (des observables unidimensionnels).

 2) De comprendre les liens entre les associateurs de Drinfeld et la théorie de Rouvière qui porte sur les espaces homogènes.

3) De construire le sigma-modèle de Poisson en tant que théorie topologique des champs étendue (en dimensions deux, un et zéro).

4) De développer les liens entre la théorie topologique des champs et la géométrie dérivée.

Ce projet supporte plusieurs postes pour des doctorants et postdoctorants.

 
Direct link to Lay Summary Last update: 01.04.2016

Responsible applicant and co-applicants

Employees

Publications

Publication
Quantization of anomaly coefficients in 6D N=(1,0) supergravity
Monnier Samuel, Moore Gregory W., Park Daniel S. (2018), Quantization of anomaly coefficients in 6D N=(1,0) supergravity, in J. High Energy Phys., February 2018(2018:20), 1-40.
Chern-Simons, Wess-Zumino and other cocycles from Kashiwara-Vergne and associators
Alekseev Anton, Naef Florian, Xu Xiaomeng, Zhu Chenchang (2018), Chern-Simons, Wess-Zumino and other cocycles from Kashiwara-Vergne and associators, in Lett. Math. Phys., 108(3), 757-778.
The Goldman-Turaev Lie bialgebra in genus zero and the Kashiwara-Vergne problem
Alekseev Anton, Kawazumi Nariya, Kuno Yusuke, Naef Florian (2018), The Goldman-Turaev Lie bialgebra in genus zero and the Kashiwara-Vergne problem, in Adv. Math., 326, 1-53.
Goldman-Turaev formality from the Knizhnik-Zamolodchikov connection
Alekseev Anton, Naef Florian (2017), Goldman-Turaev formality from the Knizhnik-Zamolodchikov connection, in C. R. Math. Acad. Sci. Paris, 355(11), 1138-1147.
Higher genus Kashiwara-Vergne problems and the Goldman-Turaev Lie bialgebra
Alekseev Anton, Kawazumi Nariya, Kuno Yusuke, Naef Florian (2017), Higher genus Kashiwara-Vergne problems and the Goldman-Turaev Lie bialgebra, in C. R. Math. Acad. Sci. Paris, 355(2), 123-127.

Collaboration

Group / person Country
Types of collaboration
The University of Tokyo Japan (Asia)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
University of Strasbourg France (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Exchange of personnel
University of Notre Dame United States of America (North America)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
- Exchange of personnel
University of Göttingen Germany (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Department of Mathematics, University of Toronto Canada (North America)
- in-depth/constructive exchanges on approaches, methods or results
- Exchange of personnel

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
Conformal and Symplectic Geometry Talk given at a conference Moduli of flat connections, Kashiwara-Vergne and Batalin-Vilkovisky 05.02.2018 Auckland, New Zealand Alekseev Anton;
London Analysis and Probability seminar Individual talk Momentum maps and stochastic processes 19.11.2017 London, Great Britain and Northern Ireland Alekseev Anton;
Derived Algebraic Geometry and Interactions Talk given at a conference mini-course "Shifted Poisson Geometry" 12.06.2017 Toulouse, France Safronov Pavel;
Seminar in Mathematical Physics, Universiyt of Vienna Individual talk Path integrals for Wilson lines and Wilson surfaces 16.05.2017 Vienna, Austria Alekseev Anton;
Hot Topics: Galois Theory of Periods and Applications Talk given at a conference The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem 27.03.2017 MSRI, Berkeley, United States of America Alekseev Anton;
Winter School in Mathematical Physics Talk given at a conference mini-course "Non-commutative differential calculus and its applications in Lie theory and Poisson geometry" 08.01.2017 Les Diablerets, Switzerland Alekseev Anton;


Self-organised

Title Date Place
Winter School in Mathematical Physics 07.01.2018 Les Diablerets, Switzerland

Associated projects

Number Title Start Funding scheme
152812 Equivariant cohomology, gauge theory and higher associators 01.04.2014 Project funding (Div. I-III)
141869 NCCR SwissMAP: The Mathematics of Physics (phase I) 01.07.2014 National Centres of Competence in Research (NCCRs)
178794 Linearization, Poisson structures and Quantization 01.04.2018 Project funding (Div. I-III)

Abstract

This project is at the interface between Mathematics and Theoretical Physics. The motivations often come from the Physics side while the methods and the framework are in most cases placed within Mathematics. The project covers several themes which are unified by the use of Drinfeld associators and by the notions of Topological Quantum Field Theory (TQFT).The project addresses various questions which range from relatively easy (and more realistic) ones to more difficult and more ambitious ones. The following problems/themes are of importance within the project:1) To use the language of derived geometry for better understanding of Lagrangian TQFTs.2) To develop the theory of higher associators and Grothendieck-Teichmüller groups, and to relate it to the LMO functor.3) To study uniqueness of Rouvière's e-function in the theory of symmetric spaces, and to obtain explicit formulas for the e-function in terms of Drinfeld associators.4) To construct quantum Poisson sigma models as fully extended TQFTs.5) To develop the theory of Wilson surfaces in 3D Chern-Simons theory and in the 4D topological Yang-Mills theory.These topics are rooted in my previous research on group valued moment maps and on the Kashiwara-Vergne Conjecture, and they develop it in various directions. The project is built around the activity of my research group at the University of Geneva. It includes funding for 3 graduate students: Olga Chekeres and Elise Raphael are working within the previous project, and their Ph.D. studies will enter the decisive phase under this project. Donald Youmans is currently a Ph.D. student at MPI Bonn, and he is scheduled to move to Geneva in September 2016. The project also includes funding for a postdoctoral fellow. Dr. Philippe Humbert is currently a fellow with the previous project and he will be completing his work during the first 6 months of the current project. He will then be replaced by Dr. Pavel Safronov. Safronov is a rising star in his field. A possibility to attract him to Geneva is a great opportunity for our research group. This hiring has a very high priority within the project.The project relies on a close collaboration with the groups of Dr. Pavel Mnev (MPI Bonn) and Prof. B. Enriquez (Strasbourg). It includes funding for travel of collaborators to conferences and summer/winter schools and for running a research seminar.
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