Quantization of Contact 3-Manifolds and the Reeb Gravitational Field

Abstract

We present a canonical geometric quantization scheme for closed contact 3-manifolds $(M,\xi)$ using the corresponding explicit embedding $M \hookrightarrow \C^3$ as we constructed in our earlier papers. The contact structure becomes holomorphic along the chosen link $L \subset M$ of complex tangents, of which we may also consider this link as the binding of a supporting open book for $\xi$. Stein extension yields a unique holomorphic line bundle $L_\xi \to \C^3$ whose restriction to $L$ will define the quantum Hilbert space $\Hilb_\xi = H^0(L, L_\xi|_L \otimes \kappa^{1/2})$, which we find will be finite-dimensional. The Reeb vector field $R_\alpha$ of a chosen compatible contact form $\alpha$ is shown to be geodesic with a time Killing Reeb field under further assumption that the manifold is Sasakian, and under this assumption will model Einstein gravity over $M$. This construction depends only on $(M,\xi,\alpha)$ and a choice of supporting open book for $\xi$. This establishes a unified geometric framework in which the contact structure encodes quantum mechanics and whose Reeb field will encode gravity assuming that the manifold is Sasakian. In addition, we use my previous paper in which we get a related invariant $\mu_M(\xi) = [L_\xi] \in \Pic_\C(M)$ which is useful in that it distinguishes between different tight contact structures on $\T^3$ in a novel way. We also show how this will have implications for the quantum model and serve as a quantum invariant there-in.