Existence of multiple vortices in supersymmetric gauge field theory

Shouxin Chen, Yisong Yang

Research output: Contribution to journalArticlepeer-review


Two sharp existence and uniqueness theorems are presented for solutions of multiple vortices arising in a six-dimensional brane-world supersymmetric gauge field theory under the general gauge symmetry group G =U(1) × SU(N) and with N Higgs scalar fields in the fundamental representation of G. Specifically, when the space of extra dimension is compact so that vortices are hosted in a 2-torus of volume /ω/, the existence of a unique multiple vortex solution representing n1, . . . , nN , respectively, prescribed vortices arising in the N species of the Higgs fields is established under the explicitly stated necessary and sufficient condition ni <g2v2/8πN/ ω/ + 1/N(1 -1/N) [g/e]2)n, i =1, . . . ,N, where e and g are the U(1) electromagnetic and SU(N) chromatic coupling constants, v measures the energy scale of broken symmetry and n =σ Ni=1 ni is the total vortex number; when the space of extra dimension is the full plane, the existence and uniqueness of an arbitrarily prescribed n-vortex solution of finite energy is always ensured. These vortices are governed by a system of nonlinear elliptic equations, which may be reformulated to allow a variational structure. Proofs of existence are then developed using the methods of calculus of variations.

Original languageEnglish (US)
Pages (from-to)3923-3946
Number of pages24
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2148
StatePublished - Dec 8 2012


  • Calculus of variations
  • Non-Abelian gauge theory
  • Nonlinear elliptic equations
  • Vortices

ASJC Scopus subject areas

  • General Mathematics
  • General Engineering
  • General Physics and Astronomy


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