The KdV vacuum operator and its Katz extension

2026/06/22 by Martin T. Luu · 1 voice
Physics and Astronomy · Mathematics · #nlin.SI #math-ph #math.RT

paper · pdf · doi:10.1016/j.geomphys.2025.105517

Abstract

We define a connection on the formal disc that can be used to single out the vacuum of the Drinfeld-Sokolov KdV hierarchy associated to a simple complex finite-dimensional Lie algebra. As a connection, it has a canonical Katz extension from the disc to the sphere. We express this Katz extension in terms of the Kac coordinates of a suitable Weyl group conjugacy class. As a consequence, we show that the Katz extension has meaning in the context of the integrable hierarchy: It describes an additional symmetry.

Citations

• From conjugacy classes in the Weyl group to semisimple conjugacy classes
• Bethe Ansatz and the Spectral Theory of Affine Lie Algebra-Valued Connections I. The simply-laced Case
• Tau Functions and Virasoro Symmetries for Drinfeld-Sokolov Hierarchies
• A rigid irregular connection on the projective line
• Computing with nilpotent orbits in simple Lie algebras of exceptional type
• Regular Conjugacy Classes in the Weyl Group and Integrable Hierarchies
• Additional Symmetries of Generalized Hierarchies

Discussions

• Martin T. Luu: The KdV vacuum operator and its Katz extension https://arxiv.org/abs/2606.23878 https://arxiv.org/pdf/2606.23878 https://arxiv.org/html/2606.23878 [bsky, 0 points, 0 comments]

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