Title: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 $f:M\to M'$ 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 $N\to N'$. 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.