Critical points of master functions and integrable hierarchies

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.

💡 Research Summary

The paper investigates a deep relationship between critical points of master functions, affine Lie algebra representations, and integrable hierarchies, focusing on the affine algebra (\widehat{\mathfrak{sl}}_N). It begins with the simplest possible master function—one that has no variables and corresponds to the trivial representation of (\widehat{\mathfrak{sl}}_N). From this “seed” critical point the authors generate an entire “population” of critical points by repeatedly applying a combinatorial generation procedure that mirrors the action of the Weyl group on Bethe vectors. Each generated critical point is encoded by a rational function (or, equivalently, by a Wronskian determinant of certain auxiliary functions).

The central result of the first part is that every critical point in this population gives rise to a rational solution of the modified Korteweg–de Vries (mKdV) hierarchy associated with (\widehat{\mathfrak{sl}}_N). The authors establish this by showing that the Bethe‑type equations satisfied by the critical points coincide with the stationary reduction of the mKdV Lax equations. In particular, the logarithmic derivatives of the master function at a critical point produce the Miura‑type potentials that solve the mKdV equations. This provides a concrete bridge between the algebraic Bethe ansatz and the analytic theory of integrable PDEs.

In the second part the paper introduces a constructive method for producing critical points from an (N)-tuple of tau‑functions (\tau_1,\dots,\tau_N). Each (\tau_i) is assumed to be a solution of the Kadomtsev–Petviashvili (KP) hierarchy and to satisfy a non‑degeneracy condition ensuring that the Wronskian (W(\tau_1,\dots,\tau_N)) is non‑zero. The authors prove a Wronskian identity of the form

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