# Critical points of master functions and mKdV hierarchy of type $A^{(2)}_{2n}$

## TL;DR

This paper studies the structure of critical points related to the master function of a twisted affine Lie algebra and shows their invariance under the mKdV hierarchy, revealing a deep connection between algebraic geometry and integrable systems.

## Contribution

It constructs a partition of critical points into complex cells and defines rational maps to Miura opers, proving their invariance under the mKdV flows for the first time.

## Key findings

- Critical points form complex cells parametrized by ^m.
- Constructed injective rational maps to Miura opers.
- Images are invariant under all mKdV flows and fixed by flows with index > 4m.

## 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 $A^{(2)}_{2n}$. The population is naturally partitioned into an infinite collection of complex cells $\mathbb{C}^m$, where $m$ are some positive integers. For each cell we define an injective rational map $\mathbb{C}^m \to M(A^{(2)}_{2n})$ of the cell to the space $M(A^{(2)}_{2n})$ of Miura opers of type $A^{(2)}_{2n}$. We show that the image of the map is invariant with respect to all mKdV flows on $M(A^{(2)}_{2n})$ and the image is point-wise fixed by all mKdV flows $\frac\partial{\partial t_r}$ with index $r$ greater than $4m$.