Electric magnetic duality of the confining flux tube in Abelian-projected gluodynamics∗

The structure of the Abelian-projected (AP) flux tube in SU(2) gauge theory in the maximally Abelian gauge is studied by applying the Hodge decomposition to the Abelian Wilson loop. It is shown that the profile of the AP flux tube has the same structure as the classical flux tube solution in the dual Abelian Higgs model, which is composed of a Coulombic and a solenoidal electric field linked by monopole supercurrent. It is known that the profiles of the classical flux-tube solution in the dual Abelian Higgs (DAH) model and the Abelian projected (AP) flux tube, observed in lattice simulations within the Abelian projection scheme in the maximally Abelian gauge (MAG), look quite similar [1, 2, 3]. We then ask how these two kinds of profiles can be related each other. This question would be non-trivial since the AP flux tube contains the quantum effects at work in non-Abelian lattice gauge simulations while the DAH flux tube is a classical solution obtained by solving the field equations. To answer the question, it is important to clarify the structure of both flux tubes [4, 5]. The DAH model, after applying the Hodge decomposition to the dual gauge field, is written, in terms of its magnetic and electric parts, B and B = 2π∆−1δ ∗ Σ, as SDAH = βm 2 (dB + 2π ∗ C) +|(d + i(B + Bele))χ|2 + λ(|χ|2 − v) , (1) where ∗Cele = ∆−1δ ∗ j with an external electric current j = −δΣele, and χ denotes the complex scalar monopole field. Σ is the electric Dirac string. By solving the field equations taking into account the boundary conditions of fields determined by B, we see that the flux tube is led by the superposition of two ∗Talk given at “Confinement V”, Gargnano, Italy, 10-14 Sep. 2002. 1We use differential form notations.