Critical points of functions, sl_2 representations, and Fuchsian differential equations with only univalued solutions

TL;DR

This paper establishes a connection between the number of specific Fuchsian differential equations with univalued solutions and the multiplicity of certain sl_2 representations, using critical point analysis of a key function.

Contribution

It introduces a novel counting method linking Fuchsian equations to sl_2 representation theory through critical points analysis.

Findings

Number of Fuchsian equations equals the multiplicity of an sl_2 representation

Bethe vectors form a basis in the sl_2 inhomogeneous Gaudin model

Critical points count matches representation multiplicities

Abstract

Let a second order Fuchsian differential equation with only univalued solutions have finite singular points at z_1, ..., z_n with exponents (a_1,b_1), ..., (a_n,b_n). Let the exponents at infinity be (A,B). Then for fixed generic z_1,...,z_n, the number of such Fuchsian equations is equal to the multiplicity of the irreducible sl_2 representation of dimension |A-B| in the tensor product of irreducible sl_2 representations of dimensions |a_1-b_1|, >..., |a_n-b_n|. To show this we count the number of critical points of a suitable function which plays the crucial role in constructions of the hypergeometric solutions of the sl_2 KZ equation and of the Bethe vectors in the sl_2 Gaudin model. As a byproduct of this study we conclude that the Bethe vectors form a basis in the space of states for the sl_2 inhomogeneous Gaudin model.