TOPOLOGICAL INTERPRETATION OF ELECTRIC CHARGE, DUALITY AND CONFINEMENT IN 2+1 DIMENSIONS

Abstract

Electric charge in QED is topological in the sense that the electric current is a curl of a local gauge-invariant field — the dual electromagnetic field strength. In 2+1 dimensions it can be explicitly represented as the winding number in terms of a local field V describing the Nielsen-Olesen magnetic vortices: [Formula: see text]. Electrically charged particles are then visualized as topological solitons of V corresponding to elements of the homotopy group π1(S1)=Z. The quantization of electric charge and the universality of the electromagnetic coupling are thus given a topological interpretation. The low energy physics is described by a “dual” Lagrangian written in terms of the field V only. This dual Lagrangian furnishes the Landau-Ginzburg description of the Coulomb-Higgs phase transition with V as a pertinent local order parameter. The symmetry which is associated with the phase transition is the magnetic flux symmetry. The nonzero VEV of V in the Coulomb phase breaks the magnetic flux spontaneously, leading to the appearance of a massless Goldstone boson—the photon. The vortex field V is also constructed in non-Abelian gauge theories. Here the solitons of V are identified with the “constituent” quarks. The dual Lagrangian contains explicit flux-symmetry-breaking terms. As a result the topological solitons (quarks) are linearly confined. We describe in detail the derivation of the dual Lagrangian and the topological mechanism of confinement in non-Abelian theories with adjoint and fundamental matter. The extension of the above description to the (3+1)-dimensional gauge theories is briefly discussed. Several notions are easily generalizable but the picture is far from being complete.