On the geometric quantization of contact manifolds

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 $E\to M$, 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.

25 pages, references added, several corrections and clarifications and some reorganization of content

References in corpus (2)

• Quantization of Contact Manifolds and Thermodynamics
• Index of transversally elliptic operators

Cited by in corpus (7)

• Contact Hamiltonian Mechanics
• Contact variational integrators
• Contact Geometry and Quantum Mechanics
• Exact Baker-Campbell-Hausdorff formula for the contact Heisenberg algebra
• The Quantum Darboux Theorem,
• Canonical quantization for Equilibrium Thermodynamics
• On transversally elliptic operators and the quantization of manifolds with $f$-structure