Seminarium KMMF "Teoria Dwoistości"

Loosley, a Courant algebroid is a vector bundle with a Loday-Leibniz bracket and a nondegenerate bilinear form on its space of sections together with some compatibility conditions. Following Roytenburg it is known that Courant algebroids have a neat supermanifold formulation as 'symplectic Lie 2-algebroids'. Without details, we have a graded symplectic supermanifold and an odd Hamiltonian that is homological, i.e., {\theta, \theta} =0. In this talk, I will show how pre-Courant algebroids, so 'Courant algebroids' without the Jacobi identity, have a very similar formulation as 'symplectic almost Lie 2-algebroids'. In particular we condition on the odd Hamiltonian is relaxed to be {{\theta, \theta},f} =0 for weight/degree zero functions. We will explore some of the consequences of this reformulation, including how to describe pre-Courant algebroids with an additional compatible non-negative grading. Examples of what we refer to as weighted pre-Courant algebroids include Courant algebroids in the category of vector bundles.As a side remark, Courant algebroids and their relatives have made a resurgence in theoretical physics via double field theory.Based on Andrew James Bruce & Janusz Grabowski, Pre-Courant algebroids, Journal of Geometry and Physics 142 (2019) 254-273. To attend our online seminar, please use the Zoom identificator: https://us02web.zoom.us/j/8145917621?pwd=bVVKend0SHFKNUNyUUQ4cWNRK3laZz09