Geometry and Energy of Non-abelian Vortices

Abstract

We study pure Yang--Mills theory on $\Sigma \times S^2$, where $\Sigma$ is a compact Riemann surface, and invariance is assumed under rotations of $S^2$. It is well known that the self-duality equations in this set-up reduce to vortex equations on $\Sigma$. If the Yang--Mills gauge group is $\SU{2}$, the Bogomolny vortex equations of the abelian Higgs model are obtained. For larger gauge groups one generally finds vortex equations involving several matrix-valued Higgs fields. Here we focus on Yang--Mills theory with gauge group $\SU{N}/\ZZ_N$ and a special reduction which yields only one non-abelian Higgs field. One of the new features of this reduction is the fact that while the instanton number of the theory in four dimensions is generally fractional with denominator $N$, we still obtain an integral vortex number in the reduced theory. We clarify the relation between these two topological charges at a bundle geometric level. Another striking feature is the emergence of non-trivial lower and upper bounds for the energy of the reduced theory on $\Sigma$. These bounds are proportional to the area of $\Sigma$. We give special solutions of the theory on $\Sigma$ by embedding solutions of the abelian Higgs model into the non-abelian theory, and we relate our work to the language of quiver bundles, which has recently proved fruitful in the study of dimensional reduction of Yang--Mills theory.

22 pages, 1 figure

References in corpus (4)

• Non-Abelian Vortices on Riemann Surfaces: an Integrable Case
• SU(3)-Equivariant Quiver Gauge Theories and Nonabelian Vortices
• Dimensional Reduction, Monopoles and Dynamical Symmetry Breaking
• Maximally Non-Abelian Vortices from Self-dual Yang--Mills Fields

Cited by in corpus (4)

• Vortex counting from field theory
• Covariant Quiver Gauge Theories
• A ladder of topologically non-trivial non-BPS states
• Non-abelian vortices on CP^1 and Grassmannians