Resolution of Chern–Simons–Higgs Vortex Equations

Xiaosen Han, Chang Shou Lin, Yisong Yang

Research output: Contribution to journalArticle

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 that settles a long-standing open problem in the field regarding the general solvability of the equations.

Original languageEnglish (US)
Pages (from-to)1-24
Number of pages24
JournalCommunications in Mathematical Physics
DOIs
StateAccepted/In press - Jan 29 2016

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Cartan Matrix
Vortex
vortices
Cartan Subalgebra
Existence Theory
Simple Lie Algebra
Equality Constraints
existence theorems
Inequality Constraints
Positivity
Existence Theorem
Convert
Solvability
Open Problems
matrices
algebra
theorems
formalism
Theorem

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Resolution of Chern–Simons–Higgs Vortex Equations. / Han, Xiaosen; Lin, Chang Shou; Yang, Yisong.

In: Communications in Mathematical Physics, 29.01.2016, p. 1-24.

Research output: Contribution to journalArticle

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