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(𝑃²𝕆) of the Cayley projective plane 𝑃²𝕆 and to construct a Bargmann type transformation from a space of holomorphic functions on 𝑇^*₀(𝑃²𝕆) to 𝐿₂-space on 𝑃²𝕆. The space of holomorphic functions corresponds to the Fock space in the case of the original Bargmann transformation. A Kähler structure on 𝑇^*₀(𝑃²𝕆) was given by identifying it with a quadric in the complex space ℂ²⁷ ∖{0} and the natural symplectic form of the cotangent bundle 𝑇^*₀(𝑃²𝕆) is expressed as a Kähler form. Our construction of the transformation is the pairing of polarizations, one is the natural Lagrangian foliation given by the projection map 𝒒 : 𝑇^*₀(𝑃²𝕆) → 𝑃²𝕆 and the other is the polarization given by the Kähler 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 turns 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.