Projective geometry of second-order supersymmetry

Abstract

We show that anomaly-free second-order supersymmetry (SUSY) in quantum mechanics admits a projective-geometric formulation in terms of a projective connection and a vector field preserving it. Once the physical coordinate and energy origin are fixed, these data determine the partner Hamiltonians and supercharges; the field's quadratic invariant fixes the central parameter and candidate singlet energies. The fixed-point structure of this field yields explicit criteria for smooth continuation of the quantum operators through superpotential zeros. The classically fictitious similarity transformation selects affine data whose covariant factors generate the Schwarzian quantum correction. Dimensional reduction of a Riemannian model reproduces these affine data through the volume measure, while a coordinate adapted to the selected field implements coupling-constant metamorphosis. A one-gap Lame pair illustrates the relation between the SUSY auxiliary geometry and the Gelfand-Dikii-Virasoro structure governing physical solution products. Its Lax-Novikov integral acts on Schrodinger solutions through the first-order action of a projective stabilizer; in the reflectionless limit it becomes proportional, in the physical coordinate, to the auxiliary third Bol operator.