Calabi-Yau structure and Bargmann type transformation on the Cayley projective plane

Abstract

Our purpose is to show the existence of a Calabi-Yau structure on the punctured cotangent bundle $T^{*}_{0}(P^2\mathbb{O})$ of the Cayley projective plane $P^{2}\mathbb{O}$ and to construct a Bargmann type transformation from a space of holomorphic functions on $T^{*}_{0}(P^2\mathbb{O})$ to $L_{2}$-space on $P^{2}\mathbb{O}$. The space of holomorphic functions corresponds to the Fock space in the case of the original Bargmann transformation. A K\"ahler structure on $T^{*}_{0}(P^{2}\mathbb{O})$ was shown by identifying it with a quadrics in the complex space $\mathbb{C}^{27}\backslash\{0\}$ and the natural symplectic form of the cotangent bundle $T^{*}_{0}(P^2\mathbb{O})$ is expressed as a K\"ahler form. Our method to construct the transformation is the pairing of polarizations, one is the natural Lagrangian foliation given by the projection map ${\bf q}:T^{*}_{0}(P^2\mathbb{O})\longrightarrow P^{2}\mathbb{O}$ and the polarization given by the K\"ahler structure. The transformation gives a quantization of the geodesic flow in terms of one parameter group of elliptic Fourier integral operators whose canonical relations are defined by the graph of the geodesic flow action at each time. It turn out that for the Cayley projective plane the results are not same with other cases of the original Bargmann transformation for Euclidean space, spheres and other projective spaces.