Critical points of master functions and integrable hierarchies

TL;DR

This paper links critical points of a master function to rational solutions of the mKdV hierarchy for affine Lie algebra sl_N, using tau-functions and Schur polynomials with Wronskian identities.

Contribution

It establishes a novel connection between critical points of master functions and integrable hierarchies, introducing new constructions via tau-functions and Schur polynomials.

Findings

Critical points generate rational solutions of mKdV hierarchy.

Constructs critical points from N-tuples of tau-functions and Schur polynomials.

Proves Wronskian identities for tau-functions and Schur polynomials.

Abstract

We consider the population of critical points generated from the trivial critical point of the master function with no variables and associated with the trivial representation of the affine Lie algebra $\hat{\frak{sl}}_N$. We show that the critical points of this population define rational solutions of the equations of the mKdV hierarchy associated with $\hat{\frak{sl}}_N$. We also construct critical points from suitable $N$-tuples of tau-functions. The construction is based on a Wronskian identity for tau-functions. In particular, we construct critical points from suitable $N$-tuples of Schur polynomials and prove a Wronskian identity for Schur polynomials.