2015/03/28 by Xiaosen Han, Chang-Shou Lin, Yisong Yang · 30 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Cartan matrix #Cartan subalgebra #Dimension (graph theory) #Homotopy and Cohomology in Algebraic Topology #Matrix (chemical analysis) #Nonlinear Waves and Solitons #Resolution (logic) #Simple (philosophy) #Subalgebra #Vortex #hep-th #math-ph #math.AP #math.MP #msc:35J20 #msc:35J50 #msc:35Q #msc:58E15 #msc:81T13
paper · pdf · doi:10.1007/s00220-016-2571-5
published in Communications in Mathematical Physics 343(2), 701-724 (Springer Science+Business Media) · 28 pages
arxiv created 2015/03/28 · openalex publication_date 2016/01/29 · openalex created_date 2016/06/24 · arxiv updated 2017/09/14 · openalex updated_date 2026/08/06
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.