Kahler toric manifolds from dually flat spaces

Abstract

We present a correspondence between real analytic Kähler toric manifolds and dually flat spaces, similar to Delzant correspondence in symplectic geometry. This correspondence gives rise to a lifting procedure: if f : M → M ′ is an affine isometric map between dually flat spaces and if N and N ′ are Kähler toric manifolds associated to M and M ′ , respectively, then there is an equivariant Kähler immersion N → N ′ . For example, we show that the Veronese and Segre embeddings are lifts of inclusion maps between appropriate statistical manifolds. We also discuss applications to Quantum Mechanics.