Symplectic groupoids and quantization of Poisson manifolds

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Groups typically emerge as symmetries associated with a specific object. However, the concept of a groupoid allows for more general symmetries that act on collections of objects rather than just a single one. This project tackles fundamental questions of importance in differential geometry and the mathematical theory of quantization by employing a combination of groupoid techniques and semi-classical analysis. The project's significance lies in its potential to uncover new fundamental aspects in geometry and unravel mysteries surrounding the nature of quantization in physics, which are crucial for understanding the geometric aspects of our universe. The project involves collaborations with researchers from Europe and South America. Additionally, the project's lead investigator will be involved in the organization of international meetings, summer courses, and a weekly seminar at the University of Illinois, Urbana-Champaign. The project consists of three main tasks. The first task introduces a novel approach to non-formal deformation quantization of Poisson manifolds, where the star products have kernels defined by semi-classical Fourier integral operators. This approach has two objectives: establishing a connection to integrability through symplectic groupoids and proving the existence of star products for a wide range of Poisson manifolds. The second task focuses on the PI's research on Poisson manifolds of compact type (PMCT), which are central objects in Poisson geometry analogous to compact Lie algebras in Lie Theory. The current goals are to develop the theory in the non-regular case and study Hamiltonian spaces of PMCTs, extending classical results for Lie group actions. This includes investigating Hamiltonian spaces of symplectic torus bundles, which generalize symplectic toric manifolds. The third task continues the exploration of an approach to classification problems of geometric structures using Cartan's realizations and employing Lie groupoid techniques, pioneered by the PI and collaborators. The PI will expand previous work on finite-dimensional families to include the infinite-dimensional case. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.