Resolution of Chern--Simons--Higgs Vortex Equations

TL;DR

This paper proves a general existence theorem for non-Abelian Chern--Simons--Higgs vortex equations with complex constraints, resolving a longstanding open problem by employing a degree-theorem argument and properties of the Cartan matrix.

Contribution

It introduces a novel method to handle multiple constraints in vortex equations, extending existence results to general simple Lie algebra cases.

Findings

Established a comprehensive existence theorem for the equations.

Utilized a degree-theorem approach with Cartan matrix positivity.

Resolved a long-standing open problem in the field.

Abstract

It is well known that the presence of multiple constraints of non-Abelian relativisitic Chern--Simons--Higgs vortex equations makes it difficult to develop an existence theory when the underlying Cartan matrix $K$ of the equations is that of a general simple Lie algebra and the strongest result in the literature so far is when the Cartan subalgebra is of dimension 2. In this paper we overcome this difficulty by implicitly resolving the multiple constraints using a degree-theorem argument, utilizing a key positivity property of the inverse of the Cartan matrix deduced in an earlier work of Lusztig and Tits, which enables a process that converts the equality constraints to inequality constraints in the variational formalism. Thus this work establishes a general existence theorem which settles a long-standing open problem in the field regarding the general solvability of the equations.