Local and global aspects of charge-current generating world-sheet scalar potentials

The electromagnetic interactions of a test string, including in particular the intrinsic self-interactions, are governed by its charge-current two-vector density Q^α (α=τ,σ). Being locally conserved, Q^α is derivable from a parent world-sheet scalar potential V(τ,σ), that is, Q^α = ε^αβV_β. However, to characterize the electromagnetic properties of the string does not mean a priori specifying V(τ,σ) up to a global gauge transformation. In fact, it is only when V(τ,σ) is treated as an additional canonical variable that the superconductivity integrability constraint emerges as an equation of motion. Electric charge quantization then follows, exclusively for closed strings, provided the phase e^iV stays single-valued with respect to the σ-periodicity (Δσ = 2π) in exactly the same way as the global property of the built-in phase e^iV of an order parameter dictates magnetic flux quantization. The pedagogical case of a self-interacting circular loop, for which V=f{hook}(τ)+nσ ⇔ χ = g(τ) + mσ, with n(m) counting the total number of electric charges (magnetic fluxons), is studied in the framework of two reparametrization invariant models. Following a Nielsen-Olsen type model, we advocate a novel approach to unification, with V (rather than χ) serving as the fifth dimension. The alternate model, favored on field-theoretical grounds, conceptually differs from the first one by strictly forbidding the collapse of a self-interacting loop.