EVENT DETAILS AND ABSTRACT

Title: Quantization of symplectic vector spaces: An algebra-geometric approach
Speaker: Ronnie Hadani
Speaker Info: University of Chicago
Brief Description:
Special Note:
Abstract:

Quantization is a fundamental procedure in mathematics and in physics. From the physical side, quantization is the procedure by which one associates to a classical mechanical system its quantum counterpart. From the mathematical side, it seems that quantization is a way to construct interesting Hilbert spaces out of symplectic manifolds, suggesting a method for constructing representations of the corresponding groups of symplectomorphisms.

In my lecture, I will consider the problem of quantization in the simplified setting of symplectic vector spaces over the finite field $\mathbb{F}_{p}$. Specifically, I will construct a quantization functor, $\mathcal{H}$, associating a Hilbert space $\mathcal{H}_{V}$, to a finite dimensional symplectic vector space $\left( V,\omega \right) $ over $\mathbb{F}_{p}$. As a result, we will obtain a canonical model for the Weil representation of the symplectic group $Sp\left( V\right) $.

The main technical result, is a proof of a strong form of the Stone-von Neumann theorem for the Heisenberg group over $\mathbb{F}_{p}$. This result, roughly, concerns the existence of a canonical flat connection on a certain vector bundle $\mathcal{H}$, defined on $Lag\left( V\right) $. In this terminology, the space $\mathcal{H}_{V}$ is obtained as the space of horizontal sections in $\mathcal{H}$.

The connection is constructed as follows: It is given explicitly as a system of isomorphisms between pairs of fibers, $F_{M,L}:\mathcal{H}_{|L}\overset{% \simeq }{\rightarrow }\mathcal{H}_{|M}$, for $M,L\in Lag\left( V\right) $ which are in transversal position, i.e., $M\cap L=0$. The construction of $F_{M,L}$, for general $M,L\in Lag\left( V\right) $, is obtained from the transversal formulas using the algebra-geometric operation of (perverse) extension.