Automorphic Forms, Moduli Spaces, and Quantization

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

ABSTRACT DMS - 0204154. PI: Tatyana Foth I will study holomorphic automorphic forms on bounded symmetric domains and related problems of quantization. Let X be a quotient of a bounded symmetric domain by a lattice. The space of holomorphic automorphic forms can be identified with the space of holomorphic sections of a tensor power of the canonical line bundle on X. In the context of geometric quantization this is the quantizing line bundle, X is the classical phase space, and automorphic forms are quantum states, i.e. wave functions of a particle. I shall investigate properties of Poincare series and relative Poincare series associated to real submanifolds of X. From the point of view of quantization they are components of the delta-function. Explicit construction of relative Poincare series associated to closed geodesics in finite volume smooth ball quotients and the spanning result can be used to obtain rational structures on the spaces of cusp forms for arithmetic subgroups of SU(n,1). Another problem I shall address is the computation of curvature of a natural connection in the vector bundle over a space of compatible complex structures on a symplectic manifold X, whose fiber over J is the space of J-holomorphic sections of k-th tensor power of the quantizing line bundle. To begin with, I will compute the symbol of the curvature in the semiclassical limit for certain types of X. The first interesting example would be the case when X is a hyperbolic Riemann surface, where an explicit basis in the space of automorphic forms can be used as a moving frame. The main object of my research is automorphic forms on bounded symmetric domains. They are functions with very special properties on spaces which have interesting geometry and many symmetries. In addition to being an important and intensively studied subject by itself it has a very close relation to geometric quantization which is a part of mathematics whose development was strongly motivated by physics and by general desire to understand better the nature of the world and to describe it in terms of suitable mathematical formalism. The main goal of geometric quantization is to find a rigorous mathematical way to pass from classical mechanics to quantum mechanics in a space which may have a very complicated geometry. More precisely, the question is how one can establish a suitable correspondence between classical observables and quantum observables. The space of wave functions is, clearly, a fundamental and very important object. It turns out that for certain classical phase spaces automorphic forms are wave functions of a particle. Properties of automorphic forms are strongly related to the complex structure. One of the questions I shall address is to what extent the procedure of geometric quantization depends on the complex structure. That leads to discussion of connections in vector bundles on certain moduli spaces which are of great interest and significance for both mathematicians and physisists.