Some title containing the words "homotopy" and "symplectic", e.g. this one

TL;DR

This paper explores the extension of Lie algebroids with higher homotopy structures, connecting them to symplectic geometry and higher-dimensional Hamiltonian mechanics, revealing new insights into their interrelations.

Contribution

It introduces a framework for Lie algebroids with non-trivial higher homotopy and links these structures to symplectic and Poisson geometry, expanding the scope of geometric mechanics.

Findings

Higher homotopy classes lead to central extensions of loop groups

Integration of Lie biagebroids yields double symplectic groupoids

Connections established between higher algebroids and variational problems

Abstract

Using a basic idea of Sullivan's rational homotopy theory, one can see a Lie groupoid as the fundamental groupoid of its Lie algebroid. This paper studies analogues of Lie algebroids with non-trivial higher homotopy. Using various homotopy classes one can obtain e.g. central extensions of loop groups, or one can integrate a Lie biagebroid to a double symplectic groupoid. When combined with symplectic geometry, this idea leads to an infinite sequence of notions, starting with sympectic manifolds, Poisson manifolds and Courant algebroids. They are interrelated with higher-dimensional variational problems, and one can use them to define higher-dimensional Hamiltonian mechanics.