Asymptotic solutions to the sl_2 KZ equation and the intersection of Schubert classes

TL;DR

This paper proves that the number of critical point orbits of a master function related to the sl_2 KZ equation equals the intersection number of certain Schubert classes, linking solutions of the KZ equation to Schubert calculus.

Contribution

It provides a new, less technical proof connecting critical point orbits to Schubert calculus, confirming the conjecture for the sl_2 KZ equation.

Findings

Number of critical point orbits equals Schubert intersection number

New proof based on polynomial Wronski determinants

Application to sl_p KZ equations discussed

Abstract

The hypergeometric solutions to the KZ equation contain a certain symmetric ``master function'', [SV]. Asymptotics of the solutions correspond to critical points of the master function and give Bethe vectors of the inhomogeneous Gaudin model, [RV]. The general conjecture is that the number of orbits of critical points equals the dimension of the relevant vector space, and that the Bethe vectors form a basis. In [ScV], a proof of the conjecture for the sl_2 KZ equation was given. The difficult part of the proof was to count the number of orbits of critical points of the master function. Here we present another, ``less technical'', proof based on a relation between the master function and the map sending a pair of polynomials into the Wronski determinant. Within these frameworks, the number of orbits becomes the intersection number of appropriate special Schubert classes. Application of the Schubert calculus to the sl_p KZ equation is discussed.