Homotopy quantum symmetries, monoidal categories and formality

Project Information

hqsmcf

Grant agreement ID: 274032

Project closed

Start date 1 January 2012

End date 31 December 2014

Funded under

• Specific programme "People" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013)

Total cost € 225 233,60

EU contribution € 225 233,60

Coordinated by

University of Zurich
Switzerland

Final Report Summary - HQSMCF (Homotopy quantum symmetries, monoidal categories and formality)

This project was hosted at the MIT (outgoing host) and at the University of Zürich (return host). Its main objectives were :
(I) to develop a theory of homotopy quantum groups. This can be understood as the natural the- ory that should sit at the intersection of four important disciplines of mathematics and physics : monoidal categories, homotopy theory, quantum groups and higher categories.
(II) to prove the formality of the homotopy Lie algebra governing simultaneous deformations of a Poisson manifold and its coisotropic submanifolds. This is a key step in solving the problem of quantization of symmetries from the point of view of deformation quantization, giving interdisciplinary applications.
The work done during the first period of the mobility was mainly knowledge acquisition, in particular in the fields of homotopy theory, algebraic topology and higher categories, by taking courses and following seminars in these research fields at Harvard and MIT. We have organized a working group at the Max Planck Institute, a special session at the Conference of the Americas, two workshops in Lens a set of lectures at MIT by an invited specialist, L. Schnepps, on the theme of Grothendieck-Teichmüller group and an online graduate seminar at the university of Zürich. We have invited 7 research visitors, to benefit from their expertise on domains connected to this research project.
We have moreover participated in a semester program on this theme at the Newton institute, in Cambridge and attended 16 conferences and schools on connected subjects. We have worked in disseminating the research results of this project by giving 19 oral communications in 15 research seminars, 3 conferences and 1 summer school. We have taken part as a mentor in the Primes program at MIT (for mentoring research projects with talented high school students) and mentored at the MPIM in Bonn four winners of the German math competition.
The current results of this research project have led to 6 (pre)-publications and an habilitation thesis.
The first main result consists in a method to produce homotopy Lie algebras governing simultaneous deformations. We apply this method to old problems in algebra, recovering known results, but with easier technique, and to new problems in geometry obtaining new results which were out of reach for the algebraic operadic techniques. These results are detailed in the two preprints Simultaneous deformations and Poisson geometry and Simultaneous deformations of algebras and morphisms via derived brackets. In particular we are able to build homotopy Lie algebras governing simultaneous deformations of Poisson manifolds and their coisotropic submanifolds, which was the first half of our research objective II).
The second main result, contained in the preprint Homotopy moment maps consists of introducing and studying the concept of moment map in the context of homotopy Lie algebras and n-plectic geometry. In particular we give criteria and obstructions for existence of homotopy moment maps, give examples and a method to obtain examples out of cocycles in equivariant cohomology. Improvements of some of these results are contained in the preprint „A cohomological framework for homotopy moment maps“.
The third main result is contained in the published article „Non abelian cohomology of extensions
of algebras as Deligne groupoid“ (Journal of Algebra). We show that classes of extensions of g by h, that are known to be classified in terms of nonabelian cohomology H2 (g,h), can be understood as elements of the iso-classes of a certain groupoid.
The fourth main result is the subject of the preprint Lower Central Series Ideal Quotients Over Z and Fp which studies dimensions of some modules associated to the descending central series for algebras with relations.
We have made progress in our understanding of sigma model techniques hoping to be able to fulfill the objective II) by in a near future, i.e. prove the formality of the homotopy Lie algebra that we have obtained. Concerning bjective I), the progress that we have made in our knowledge of the current developments in the theory of associators and Grothendieck-Teichmüller theory seem to indicate that it takes part of a very challenging domain, and it is likely that the full achievement of this project will go beyond the scope of this mobility.
The immediate socio-economic impact of this mobility are the strengthening of Europe's research expertise and knowledge, together with advances in the understanding of quantization of symmetries. This impact in particular benefits from the transfer of knowledge through the doctoral course and graduate seminar given at UZH, the conferences and workshops that have been organized in the framework of this research project, the mentoring of interns at MPIM and master-classes at ESI that we have offered.
Also, we hope that the connections that have been established with top scientists at the University of Zürich, ETH, MIT, Harvard and other American institutions will enhance in a sustainable way the current European scientific network.