Critical points and mKdV hierarchy of type $C^{(1)}_n$

Alexander Varchenko (Department of Mathematics, University of North Carolina, Chapel Hill, N.C., U.S.A.; and Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, Moscow, Russia)

Tyler Woodruff (Department of Mathematics, University of North Carolina, Chapel Hill, N.C., U.S.A.)

Abstract

We consider the population of critical points, generated from the critical point of the master function with no variables, which is associated with the trivial representation of the twisted affine Lie algebra ${C_{n}^{(1)}}$. The population is naturally partitioned into an infinite collection of complex cells ${\mathbb{C}^{m}}$, where m are positive integers. For each cell we define an injective rational map ${\mathbb{C}^{m}}\to M({C_{n}^{(1)}})$ of the cell to the space $M({C_{n}^{(1)}})$ of Miura opers of type ${C_{n}^{(1)}}$. We show that the image of the map is invariant with respect to all mKdV flows on $M({C_{n}^{(1)}})$ and the image is point-wise fixed by all mKdV flows $\frac{\partial }{\partial {t_{r}}}$ with index r greater than $2m$.

Keywords

critical points, master functions, mKdV hierarchies, Miura opers, affine Lie algebras

2010 Mathematics Subject Classification

Primary 37K20. Secondary 17B80, 81R10.

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The first-named author was supported in part by NSF grants DMS-1665239, DMS-1954266.

Received 3 November 2018

Accepted 8 April 2019

Published 13 November 2020