Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 007, 19 pages      arXiv:1306.3072      https://doi.org/10.3842/SIGMA.2014.007
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

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

Johan van de Leur
Mathematical Institute, University of Utrecht, P.O. Box 80010, 3508 TA Utrecht, The Netherlands

Received September 23, 2013, in final form January 09, 2014; Published online January 15, 2014

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.

Key words: affine Kac-Moody algebra; loop group orbit; Kac-Wakimoto hierarchy; isotropic Grassmannian; total descendent potential; W constraints.

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