# Two approaches to contact quantization

## GAP Seminar

## Meeting Details

For more information about this meeting, contact Ping Xu, Mathieu Stiénon, Nigel Higson.

**Speaker:** Sean Fitzpatrick, Mount Allison University

**Abstract:** Geometric quantization is a familiar problem in symplectic geometry, and one that is well-understood in many settings. In the case of a compact KÃ¤hler manifold there are two well-known approaches: the traditional Souriau-Kostant quantization, and quantization in terms of the index of the Dolbeault-Dirac operator; given certain assumptions the two quantizations agree.
I will discuss contact analogues of both approaches to quantization, for the case of Sasakian manifolds, where the existence of a compatible Cauchy-Riemann structure allows us to make use of tools from CR geometry. In the first approach I will identify analogues of the Poisson algebra, prequantum line bundle and polarizations. I will then show how to construct a differential operator similar to the Dolbeault-Dirac operator that is not elliptic, but transversally elliptic, and compute its index using the Paradan-Vergne index theorem. Time permitting, I'll comment on why the two approaches give slightly different answers in the Sasakian setting.

## Room Reservation Information

**Room Number:** 106 McAllister

**Date:** 03/30/2010

**Time:** 2:30pm - 3:30pm