Vladimir Salnikov

Generalized geometry in relations to physics and mechanics

Contents: Poisson and Dirac geometry, some related Q-manifolds constructions for sigma models, structure preserving numerical methods in mechanics, Poisson and Dirac integrators.

In this minicourse I will make an overview of some recent advances in applying differential and generalized geometry to exhibiting intrinsic properties of physical and mechanical systems. I will start by recalling the standard constructions from Poisson geometry and Dirac structures and fit them to the description using differential graded geometry. I will then define the so-called equivariant Q-cohomology and explain the link of it to the gauging procedure. The two main examples will be the Poisson sigma model (the most general AKSZ one in space-time dimension 2) and the Dirac sigma model (the most general one obtained from the gauging procedure of a 3d Wess-Zumino term). As for mechanics, I will explain how Dirac structures appear naturally in dynamics of systems with constraints or coupled systems. We will also address the question of defining dynamics on Dirac structures via some variational principles. Time permitting, I will conclude by mentioning some consequences of these results for construction of geometric integrators - numerical methods preserving the geometric structure of the equations and qualitative properties of respective mechanical systems.

Some references:

V. Salnikov, T. Strobl, Dirac Sigma Models from Gauging, Journal of High Energy Physics, 11/2013 ; 2013(11). DOIs:10.1007/JHEP11(2013)110
A. Kotov, V. Salnikov, T. Strobl, 2d Gauge Theories and Generalized Geometry, Journal of High Energy Physics, 2014:21, 2014, DOI:10.1007/JHEP08(2014)021
A. Kotov, V. Salnikov, The category of Z-graded manifolds: what happens if you do not stay positive. Preprint: arXiv:2108.13496
V. Salnikov, A. Hamdouni, D. Loziienko, Generalized and graded geometry for mechanics: a comprehensive introduction, Mathematics and Mechanics of Complex Systems, Vol. 9, No. 1, 2021
O. Cosserat, C. Laurent-Gengoux, A. Kotov, L. Ryvkin, V. Salnikov, On Dirac structures admitting a variational approach, Mathematics and Mechanics of Complex Systems, 2023