Kahler toric manifolds from dually flat spaces

TL;DR

This paper establishes a new correspondence between Kähler toric manifolds and dually flat spaces, enabling a lifting procedure for affine isometric maps that connects geometric structures with applications in quantum mechanics.

Contribution

It introduces a novel correspondence similar to Delzant's in symplectic geometry, linking Kähler toric manifolds with dually flat spaces and providing a method to lift affine isometric maps to Kähler immersions.

Findings

Veronese and Segre embeddings are lifts of inclusion maps between statistical manifolds.

The correspondence facilitates a new perspective on the structure of Kähler toric manifolds.

Applications to quantum mechanics are discussed.

Abstract

We present a correspondence between real analytic K\"{a}hler 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 K\"{a}hler toric manifolds associated to $M$ and $M'$, respectively, then there is an equivariant K\"{a}hler 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.