A Remark on Kac-Wakimoto Hierarchies of D-type

TL;DR

This paper computes the coefficients of Hirota bilinear equations for the Kac-Wakimoto hierarchy of type D and confirms their match with Givental's total descendant potential of the D singularity, supporting a conjecture.

Contribution

It explicitly calculates the Hirota coefficients for the D-type Kac-Wakimoto hierarchy and verifies their equivalence with Givental's potential, confirming a key conjecture.

Findings

Coefficients of Hirota bilinear equations are explicitly computed.

The bilinear equations match Givental's total descendant potential for D singularity.

Supports the conjecture linking Kac-Wakimoto hierarchies and Givental's potentials.

Abstract

For the Kac-Wakimoto hierarchy constructed from the principal vertex operator realization of the basic representation of the affine Lie algebra $D_n^{(1)}$, we compute the coefficients of the corresponding Hirota bilinear equations, and verify the coincidence of these bilinear equations with the ones that are satisfied by Givental's total descendant potential of the $D_n$ singularity, as conjectured by Givental and Milanov in \cite{GM}