Supersymmetric Kinks and real algebraic curves †

The kinks of the (1+1)-dimensional Wess-Zumino model with polynomic superpotential are investigated and shown to be related to real algebraic curves. 1 The dimensional reduction of the (3+1)-dimensional Wess-Zumino model, produces an interesting (1+1)-dimensional Bose-Fermi system; this field theory enjoys N=2 extended supersymmetry provided that the interactions are introduced via a real harmonic superpotential, see [1]. In a recent paper [2] Gibbons and Townsend have shown the existence of domain-wall intersections in the (3+1)D WZ model, the authors relying on the supersymmetry algebra of the (2+1)D dimensional reduction of the system. Although the domain-wall junctions are two-dimensional structures, their properties are reminiscent of the one-dimensional kinks from which they are made. In this letter we shall thus describe the kinks of the underlying (1+1)-dimensional system. The basic fields of the theory are: • Two real bosonic fields, φ a (x µ), a = 1, 2 that can be assembled in the complex field: φ(x µ) = φ 1 (x µ) + iφ 2 (x µ) ∈ Maps(R 1,1 , C). x µ = (x 0 , x 1) are local coordinates in the R 1,1 Minkowski space, where we choose the metric g µν , g 00 = −g 11 = 1, g 12 = g 21 = 0. • Two Majorana spinor fields ψ a (x µ), a = 1, 2. We work in a Majorana representation of the Clifford algebra {γ µ , γ ν } = 2g µν , γ 0 = σ 2 , γ 1 = iσ 1 , γ 5 = γ 0 γ 1 = σ 3 where σ 1 , σ 2 , σ 3 are the Pauli matrices, such that ψ a * = ψ a. We also define the adjoint spinor as ¯ ψ(x µ) = ψ t (x µ)γ 0 and consider Majorana-Weyl spinors: ψ a ± (x µ) = 1±γ 5 2 ψ a (x µ) with only one non-zero component. Interactions are introduced through the holomorphic superpotential: W (φ) = 1 2 W 1 (φ 1 , φ 2) + iW 2 (φ 1 , φ 2). One could in principle start from the supercharges: ˆ Q BC ± = dx 1 a,b f B ab (∂ 0 φ a ∓ ∂ 1 φ a)ψ b ± ± c f C bc ∂W C ∂φ c ψ a ∓ where W B , B = 1, 2, …