Group Actions on Deformation Quantizations and an Equivariant Algebraic Index Theorem.

PhD Defense by Niek de Kleijn

Abstract: Deformation quantizations are algebras obtained by formal deformation of the algebra of functions on a smooth manifold. A first order approximation of such algebras provides a Poisson structure on the underlying manifold and this provides a link with the Hamiltonian formalism of classical mechanics and the theory of quantization. In the thesis we study deformation quantizations such that the corresponding Poisson structure is non-degenerate in the presence of a discrete group of symmetries.
The main result of the thesis is an equivariant algebraic index formula. The algebraic index theorem is an algebraic analog of the Atiyah-Singer index theorem where one replaces the pseudo-differential operators (which form a deformation quantization) by any symplectic formal deformation quantization. The theorem boils down to showing that the (unique) trace on the deformation quantization and a certain product of characteristic classes are cohomologous as periodic cyclic cocycles. In the equivariant case we consider a natural invariant extension of the trace to the crossed product of the deformation quantization with a discrete group and show that it is cohomologous to a certain product of equivariant characteristic classes as periodic cyclic cocycles.
To obtain this formula we construct an equivariant deformed version of the Gelfand-Fuks map appearing in Gelfand-Kazdan's theory of formal geometry. To make this construction more transparent and since a thorough account seems to be missing in the literature, we develop this theory of formal geometry from the ground up.
We then consider a deformed version of the theory and show how it ties in to Fedosov's well-known construction of deformation quantizations and the algebraic index theorem. This makes clear that proving an equivariant algebraic index theorem boils down to the right definition of equivariant Gelfand-Fuks map, which we then provide.

Supervisor: Prof. Ryszard Nest, Math, University of Copenhagen

Assessment committee:
Assoc. Prof. Dan Erik Petersen (chairman), MATH, University of Copenhagen
Prof. Stefan Waldman, Wuerzburg University
Prof. Simone Gutt, Universite Libre de Brussels