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 $f:M\to M^{\prime }$ is an affine isometric map between dually flat spaces and if $N$ and $N^{\prime }$ are Kähler toric manifolds associated to $M$ and $M^{\prime }$, respectively, then there is an equivariant Kähler immersion $N\to N^{\prime }$. 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.