##### What's Hot

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.

Share.