Virtual Monopole Geometry and Confinement

Generalizing the geometry of the gauge covariant variables in Yang-Mills theory proposed by Johnson and Haagensen, the 4-d geometry associated with a monopole is defined for SU(2). There are three relevant geometries: AdS$_2\times S^2$, $R^2\times S^2$ and $H_+\times S^2$, depending on the asymptotic behavior of the torsion. Using this geometry, the Wilson loop average is computed {\it \`{a} la} Nambu-Goto action. In case of AdS$_2\times S^2$, it satisfies the area law.