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Chen S, Yang Y. Existence of multiple vortices in supersymmetric gauge field theory. Proc Math Phys Eng Sci 2012. [DOI: 10.1098/rspa.2012.0159] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
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
n
1
,…,
n
N
, respectively, prescribed vortices arising in the
N
species of the Higgs fields is established under the explicitly stated necessary and sufficient condition
where
e
and
g
are the
U
(1) electromagnetic and
SU
(
N
) chromatic coupling constants,
v
measures the energy scale of broken symmetry and
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.
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Affiliation(s)
- Shouxin Chen
- Institute of Contemporary Mathematics, College of Mathematics and Information Science, Henan University, Kaifeng, Henan 475001, People's Republic of China
| | - Yisong Yang
- Institute of Contemporary Mathematics, College of Mathematics and Information Science, Henan University, Kaifeng, Henan 475001, People's Republic of China
- Department of Mathematics, Polytechnic Institute of New York University, Brooklyn, NY 11201, USA
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