On the geometric quantization of contact manifolds

Details On the geometric quantization of contact manifolds On the geometric quantization of contact manifolds

2009-09-10

arxiv.org/abs/0909.2023v3

Journal of Geometry and Physics 61 (2011), pp. 2384-2399

Mathematics

Symplectic Geometry

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

Scientific paper

10.1016/j.geomphys.2011.07.011

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.

Affiliated with

Also associated with

No associations

Rating

On the geometric quantization of contact manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the geometric quantization of contact manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the geometric quantization of contact manifolds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-132303