Laplacian gauge and instantons

Topological excitations produce obstructions to making the gauge field smooth everywhere. Therefore, they should appear as singularities in an otherwise smooth gauge. This offers the possibility of identifying such excitations via gaugefixing. After gauge-fixing, the gauge field becomes singular, and the gauge ill-defined, on a sub-manifold characterizing the topological excitations. Even though the precise location of this manifold typically depends on the specific gauge condition chosen, its existence does not. This gauge-fixing approach was suggested by ’t Hooft to identify chromo–magnetic monopoles [1]. It has been recognized recently that both monopoles and center vortices appear together as gauge singularities of co-dimension 3 and 2 respectively, when one tries to enforce a smooth gauge for the adjoint SU(N)/ZN field [2]. It is then natural to also study what happens when one tries to enforce a smooth gauge for the SU(N) field. As we show below, point-like (co-dimension 4) singularities appear, coming from the topological charge of the Yang-Mills field. Thus, gaugefixing allows a unifying perspective on all 3 kinds of topological Yang-Mills excitations: center vortices, monopoles and instantons.