The 2-component BKP Grassmanian and simple singularities of type D

TL;DR

This paper explicitly identifies the Grassmanian point corresponding to the total descendant potential of type D singularities and shows that Gaussian tau-functions are parametrized by the miniversal unfolding base.

Contribution

It explicitly finds the Grassmanian point for the descendant potential and links Gaussian tau-functions to the unfolding space of type D singularities.

Findings

Explicit Grassmanian point for descendant potential of type D.

Parametrization of Gaussian tau-functions by unfolding space.

Connection between singularity theory and BKP hierarchy.

Abstract

It was proved in 2010 that the principal Kac--Wakimoto hierarchy of type $D$ is a reduction of the 2-component BKP hierarchy. On the other hand, it is known that the total descendant potential of a singularity of type $D$ is a tau-function of the principal Kac--Wakimoto hierarchy. We find explicitly the point in the Grassmanian of the 2-component BKP hierarchy (in the sense of Shiota) that corresponds to the total descendant potential. We also prove that the space of tau-functions of Gaussian type is parametrized by the base of the miniversal unfolding of the simple singularity of type $D$.