Geometric Quantization and Equivariant Cohomology

Abstract

Let G be a real Lie group acting on a C ^∞ even dimensional oriented manifold M. In many cases it is possible to associate to a G-equivariant Hermitian vector bundle ε over M, equipped with a G-invariant Hermitian connection A, a canonical virtual unitary representation Q(M, ε, A) of G in a virtual Hilbert space H(M, ε, A). The space H(M, ε, A) will be referred to as the quantized space of (M, ε, A). The meaning of canonical is the following. Although the geometric model for the quantized space H(M, ε, A) may be elusive, we can give a canonical character formula for Q(M, ε, A): there exists an admissible bouquet of equivariant cohomology classes bch(ε, A) on M such that

$$Tr\,\left( {Q\left( {M,\,\mathcal{E},\,\mathbb{A}} \right)} \right)\, = \,{i^{ - \frac{{\dim M}}{2}}}\,\int_b {bch\left( {\mathcal{E},\mathbb{A}} \right)} $$

((F):)

as an equality of generalized functions on the group G. The notion of admissible bouquet and the notion of integration ∫^b of such bouquets will be described in this article. The bouquet bch(ε, A) will be called the bouquet of Chern characters of the bundle ε with connection A (in fact, we will have to modify the notion of G-equivariant vector bundle to the notion of G-equivariant quantum bundle in order for bch(ε, A) to be an admissible bouquet).