The (n,1)-Reduced DKP Hierarchy, the String Equation and W Constraints

TL;DR

This paper constructs the (n,1)-reduced DKP hierarchy for simple singularities, linking it to W constraints and the string equation, providing new formulations and insights into the hierarchy's structure.

Contribution

It introduces a novel reduction-based construction of the type D principal hierarchy, offering Lax and Grassmannian formulations and connecting the string equation to W constraints.

Findings

The hierarchy can be derived as a reduction of the DKP hierarchy.

The string equation induces significant W constraints.

Constraints are expressed via Lax and Orlov-Schulman operators.

Abstract

The total descendent potential of a simple singularity satisfies the Kac-Wakimoto principal hierarchy. Bakalov and Milanov showed recently that it is also a highest weight vector for the corresponding W-algebra. This was used by Liu, Yang and Zhang to prove its uniqueness. We construct this principal hierarchy of type D in a different way, viz. as a reduction of some DKP hierarchy. This gives a Lax type and a Grassmannian formulation of this hierarchy. We show in particular that the string equation induces a large part of the W constraints of Bakalov and Milanov. These constraints are not only given on the tau function, but also in terms of the Lax and Orlov-Schulman operators.