Lagrangian Grassmannians, CKP hierarchy and hyperdeterminantal relations

TL;DR

This paper explores the geometric relationship between Lagrangian Grassmannians and the CKP integrable hierarchy, extending finite-dimensional concepts to infinite dimensions and deriving hyperdeterminantal relations for tau-functions.

Contribution

It extends the Lagrangian Grassmannian framework to infinite-dimensional Hilbert spaces and derives hyperdeterminantal relations relevant to the CKP hierarchy.

Findings

Hyperdeterminantal relations characterize the CKP hierarchy.

Infinite-dimensional Lagrangian Grassmannian framework is established.

Tau-functions satisfy multiparametric hyperdeterminantal relations.

Abstract

This work concerns the relation between the geometry of Lagrangian Grassmannians and the CKP integrable hierarchy. The Lagrange map from the Lagrangian Grassmannian of maximal isotropic (Lagrangian) subspaces of a finite dimensional symplectic vector space $V\oplus V^*$ into the projectivization of the exterior space $\Lambda V$ is defined by restricting the Pl\"ucker map on the full Grassmannian to the Lagrangian sub-Grassmannian and composing it with projection to the subspace of symmetric elements under dualization $V \leftrightarrow V^*$. In terms of the affine coordinate matrix on the big cell, this reduces to the principal minors map, whose image is cut out by the $2 \times 2 \times 2$ quartic {\em hyperdeterminantal} relations. To apply this to the CKP hierarchy, the Lagrangian Grassmannian framework is extended to infinite dimensions, with $V\oplus V^*$ replaced by a polarized Hilbert space $ {\mathcal H} ={\mathcal H}_+\oplus {\mathcal H}_-$, with symplectic form $\omega$. The image of the Plucker map in the fermionic Fock space ${\mathcal F}= \Lambda^{\infty/2}{\mathcal H}$ is identified and the infinite dimensional Lagrangian map is defined. The linear constraints defining reduction to the CKP hierarchy are expressed as a fermionic null condition and the infinite analogue of the hyperdeterminantal relations is deduced. A multiparametric family of such relations is shown to be satisfied by the evaluation of the $\tau$-function at translates of a point in the space of odd flow variables along the cubic lattices generated by power sums in the parameters.