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Resolution of Chern–Simons–Higgs Vortex Equations

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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.

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Authors and Affiliations

  1. Institute of Contemporary Mathematics, School of Mathematics and Statistics, Henan University, Kaifeng, 475004, Henan, People’s Republic of China

    Xiaosen Han & Yisong Yang

  2. Taida Institute for Mathematical Sciences, National Taiwan University, Taipei, 10617, Taiwan, ROC

    Chang-Shou Lin

  3. Department of Mathematics, Tandon School of Engineering, New York University, Brooklyn, NY, 11201, USA

    Yisong Yang

  4. NYU–ECNU Institute of Mathematical Sciences, New York University-Shanghai, 3663 North Zhongshan Road, Shanghai, 200062, People’s Republic of China

    Yisong Yang

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  1. Xiaosen Han
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  2. Chang-Shou Lin
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Correspondence to Yisong Yang.

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Communicated by H.-T. Yau

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Han, X., Lin, CS. & Yang, Y. Resolution of Chern–Simons–Higgs Vortex Equations. Commun. Math. Phys. 343, 701–724 (2016). https://doi.org/10.1007/s00220-016-2571-5

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