The formal moment map geometry of the space of symplectic connections
Symplectic connections are one of the main ingredients in the construction of Fedosov star products. The space of symplectic connections is an infinite dimensional symplectic manifold on which acts the group of Hamiltonian diffeomorphisms with moment map computed in [2]. Though different, this moment map is known to share similar properties with the scalar curvature of Kähler manifolds [5].
In the lectures, based on the preprint [6], I will give a formal analogue of the work of Foth-Uribe [4]. Namely, I will describe a bundle of Fedosov star product algebras on the space of symplectic connections. Such a bundle admits a canonical formal connection adapted to the star product on the fibers. I will study the curvature of the formal connection. I will show the star product trace of the curvature gives a formal symplectic form on the space of symplectic connections which is invariant by the action of the group of Hamiltonian diffeomorphisms. I will show the star product trace gives a formal moment map for this action. Finally, I will discuss applications of the above picture to the study of automorphisms of star product and Hamiltonian diffeomorphisms.
In the lectures, based on the preprint [6], I will give a formal analogue of the work of Foth-Uribe [4]. Namely, I will describe a bundle of Fedosov star product algebras on the space of symplectic connections. Such a bundle admits a canonical formal connection adapted to the star product on the fibers. I will study the curvature of the formal connection. I will show the star product trace of the curvature gives a formal symplectic form on the space of symplectic connections which is invariant by the action of the group of Hamiltonian diffeomorphisms. I will show the star product trace gives a formal moment map for this action. Finally, I will discuss applications of the above picture to the study of automorphisms of star product and Hamiltonian diffeomorphisms.
Lecturer
Laurent La Fuente-Gravy
Date
6th ~ 10th September, 2021
Prerequisite
Basic knowledge of differential and symplectic geometry (vector bundles, connections, symplectic forms,...)
Reference
[1] J.E. Andersen, P. Masulli, F. Schätz, Formal connections for families of star products, Comm. Math. Physics 342 (2), 739–768 (2016).
[2] M. Cahen, S. Gutt, Moment map for the space of symplectic connections, Liber Amoricorum Delanghe, F. Brackx and H. De Schepper eds., Gent Academia Press, 2005, 27–36.
[3] B.V. Fedosov, A simple geometrical construction of deformation quantization, Journal of Differential Geometry 40, 213-238 (1994).
[4] T. Foth, A. Uribe, The manifold of compatible almost complex structures and geometric quantization, Comm. Math. Phys. 274 (2), 357–379 (2007).
[5] A. Futaki, L. La Fuente-Gravy, Kähler geometry and deformation quantization with moment maps, ICCM proceedings 2018, 31–66 (2020).
[6] L. La Fuente-Gravy, The formal moment map geometry of the space of symplectic connections, arXiv:2106.13608.
[2] M. Cahen, S. Gutt, Moment map for the space of symplectic connections, Liber Amoricorum Delanghe, F. Brackx and H. De Schepper eds., Gent Academia Press, 2005, 27–36.
[3] B.V. Fedosov, A simple geometrical construction of deformation quantization, Journal of Differential Geometry 40, 213-238 (1994).
[4] T. Foth, A. Uribe, The manifold of compatible almost complex structures and geometric quantization, Comm. Math. Phys. 274 (2), 357–379 (2007).
[5] A. Futaki, L. La Fuente-Gravy, Kähler geometry and deformation quantization with moment maps, ICCM proceedings 2018, 31–66 (2020).
[6] L. La Fuente-Gravy, The formal moment map geometry of the space of symplectic connections, arXiv:2106.13608.
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Notes Public
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