### 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 language | English (US) |
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Pages (from-to) | 1-24 |

Number of pages | 24 |

Journal | Communications in Mathematical Physics |

DOIs | |

State | Accepted/In press - Jan 29 2016 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*, 1-24. https://doi.org/10.1007/s00220-016-2571-5

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

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, pp. 1-24. https://doi.org/10.1007/s00220-016-2571-5

}

TY - JOUR

T1 - Resolution of Chern–Simons–Higgs Vortex Equations

AU - Han, Xiaosen

AU - Lin, Chang Shou

AU - Yang, Yisong

PY - 2016/1/29

Y1 - 2016/1/29

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84955609239&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84955609239&partnerID=8YFLogxK

U2 - 10.1007/s00220-016-2571-5

DO - 10.1007/s00220-016-2571-5

M3 - Article

SP - 1

EP - 24

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

ER -