The quantum mass of the susy kink and the BPS bound

Abstract

Recently, the long-standing problem of calculating the quantum mass of supersymmetric and nonsupersymmetric solitons 1+1 dimensions in an unambiguous way, and the issue of whether or not it satisfies the BPS bound at the quantum level, was solved [1–4]. We discuss here the kink. One basic idea is to compute ∂/∂m ∑ω (which converges) instead of ∑ω, and to fix the integration constant by imposing the renormalization condition that the vacuum energies of the topological sector and the nontopological sector become equal when m and other dimensional constants (if present) tend to zero. Another new idea is to derive the boundary conditions from the symmetries of the action and the topology of the classical solution. For the bosonic fluctuations about the kink we then find antiperiodic boundary conditions, while for the fermionic fluctuations we find twisted boundary conditions. The almost universally used periodic or antiperiodic boundary conditions are shown to be inconsistent for fermions (!). The central charge Z in the supersymmetry algebra is classically given by the usual total derivative, but at the quantum level an “anomaly” (an extra term of order ℏ) is found in Z. This term is calculated by using a higher-derivative regularization scheme which preserves supersymmetry. Regulating the supersymmetry generators in this way, the BPS bound remains saturated for the supersymmetric kink, but both the mass and the central charge receive (equal) quantum corrections.