Self-self-dual spaces of polynomials

TL;DR

This paper introduces the concept of self-self-dual polynomial spaces, explores their geometric and algebraic properties, and connects them to G_2-populations and the Bethe Ansatz in integrable models.

Contribution

It defines self-self-dual spaces of polynomials, establishes their relation to G_2 structures, and links these to Bethe Ansatz solutions and G_2 flag varieties.

Findings

Self-self-dual spaces have a natural non-degenerate skew-symmetric 3-form.

They correspond to G_2-populations related to the Bethe Ansatz of the Gaudin model.

A G_2-population is isomorphic to the G_2 flag variety.

Abstract

A space of polynomials V of dimension 7 is called self-dual if the divided Wronskian of any 6-subspace is in V. A self-dual space V has a natural inner product. The divided Wronskian of any isotropic 3-subspace of V is a square of a polynomial. We call V self-self-dual if the square root of the divided Wronskian of any isotropic 3-subspace is again in V. We show that the self-self-dual spaces have a natural non-degenerate skew-symmetric 3-form defined in terms of Wronskians. We show that the self-self-dual spaces correspond to G_2-populations related to the Bethe Ansatz of the Gaudin model of type G_2 and prove that a G_2-population is isomorphic to the G_2 flag variety.