arXiv subjects

Ann du Crest de Villeneuve

Publications and source records attributed to Ann du Crest de Villeneuve.

3 recordsLinked to original sources

arXiv2025

Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials

For a simple Lie algebra $\mathfrak{g}$ and an irreducible faithful representation $π$ of $\mathfrak{g}$, we introduce the Schur polynomials of $(\mathfrak{g},π)$-type. We then derive the Sato-Zhou type formula for tau functions of the Drinfeld-Sokolov (DS) hierarchy of $\mathfrak{g}$-type. Namely, we show that the tau functions are linear combinations of the Schur polynomials of $(\mathfrak{g},π)$-type with the coefficients being the Plücker coordinates. As an application, we provide a way of computing polynomial tau functions for the DS hierarchy. For $\mathfrak{g}$ of low rank, we give several examples of polynomial tau functions, and use them to detect bilinear equations for the DS hierarchy.

math-ph↗

arXiv2019

From the Adler--Moser polynomials to the polynomial tau functions of KdV

In 1978, M. Adler and J. Moser proved that there exists a unique change of variables that transforms the Adler-Moser polynomials into the polynomial tau functions of the KdV hierarchy. In this paper we exhibit this change of variables.

math-ph↗

arXiv2018

Quantum ${D_4}$ Drinfeld-Sokolov hierarchy and quantum singularity theory

In this paper we compute explicitly the double ramification hierarchy and its quantization for the $D_4$ Dubrovin-Saito cohomological field theory obtained applying the Givental-Teleman reconstruction theorem to the $D_4$ Coxeter group Frobenius manifold, or equivalently the $D_4$ Fan-Jarvis-Ruan-Witten cohomological field theory (with respect to the non-maximal diagonal symmetry group $\langle J\rangle = \mathbb{Z}_3$). We then prove its equivalence to the corresponding Dubrovin-Zhang hierarchy, which was known to coincide with the $D_4$ Drinfeld-Sokolov hierarchy. Our techniques provide hence an explicit quantization of the $D_4$ Drinfeld-Sokolov hierarchy. Moreover, since the DR hierarchy is well defined for partial CohFTs too, our approach immediately computes the DR hierarchies associated to the invariant sectors of the $D_4$ CohFT with respect to folding of the Dynkin diagram, the $B_3$ and $G_2$ Drinfeld-Sokolov hierarchies.

math-ph↗