On the geometric quantization of contact manifolds

TL;DR

This paper develops a geometric quantization framework for contact manifolds with group actions, defining a transversally elliptic Dirac operator and extending symplectic quantization concepts to contact geometry.

Contribution

It introduces a contact analog of the Kostant-Souriau quantization approach and constructs a transversally elliptic Dirac operator for contact manifolds with symmetry.

Findings

Defined a $G$-transversally elliptic Dirac operator on contact manifolds.

Established a contact version of geometric quantization analogous to symplectic case.

Provided a formula for the equivariant index as a generalized function on $G$.

Abstract

Suppose that $(M,E)$ is a compact contact manifold, and that a compact Lie group $G$ acts on $M$ transverse to the contact distribution $E$. In an earlier paper, we defined a $G$-transversally elliptic Dirac operator $\dirac$, constructed using a Hermitian metric $h$ and connection $\nabla$ on the symplectic vector bundle $E\rightarrow M$, whose equivariant index is well-defined as a generalized function on $G$, and gave a formula for its index. By analogy with the geometric quantization of symplectic manifolds, the $\mathbb{Z}_2$-graded Hilbert space $Q(M)=\ker \dirac \oplus \ker \dirac^{*}$ can be interpreted as the "quantization" of the contact manifold $(M,E)$; the character of the corresponding virtual $G$-representation is then given by the equivariant index of $\dirac$. By defining contact analogues of the algebra of observables, pre-quantum line bundle and polarization, we further extend the analogy by giving a contact version of the Kostant-Souriau approach to quantization, and discussing the extent to which this approach is reproduced by the index-theoretic method.