Supersymmetric Quantum Mechanics and Super-Lichnerowicz Algebras

TL;DR

This paper introduces new supersymmetric quantum mechanical models on curved spaces, generalizing Lichnerowicz's operators, with novel Hamiltonian structures applicable to high-rank tensors and spinors in differential geometry.

Contribution

It develops a class of supersymmetric models based on deformations of osp(2p+2|Q), extending geometric operators and algebraic structures to curved backgrounds with novel Hamiltonian formulations.

Findings

Models recover Lichnerowicz's algebra for constant curvature backgrounds.

Supersymmetry is maintained on symmetric spaces and for certain supercharges.

Hamiltonian is a sum of commutators, not just squares of supercharges.

Abstract

We present supersymmetric, curved space, quantum mechanical models based on deformations of a parabolic subalgebra of osp(2p+2|Q). The dynamics are governed by a spinning particle action whose internal coordinates are Lorentz vectors labeled by the fundamental representation of osp(2p|Q). The states of the theory are tensors or spinor-tensors on the curved background while conserved charges correspond to the various differential geometry operators acting on these. The Hamiltonian generalizes Lichnerowicz's wave/Laplace operator. It is central, and the models are supersymmetric whenever the background is a symmetric space, although there is an osp(2p|Q) superalgebra for any curved background. The lowest purely bosonic example (2p,Q)=(2,0) corresponds to a deformed Jacobi group and describes Lichnerowicz's original algebra of constant curvature, differential geometric operators acting on symmetric tensors. The case (2p,Q)=(0,1) is simply the {\cal N}=1 superparticle whose supercharge amounts to the Dirac operator acting on spinors. The (2p,Q)=(0,2) model is the {\cal N}=2 supersymmetric quantum mechanics corresponding to differential forms. (This latter pair of models are supersymmetric on any Riemannian background.) When Q is odd, the models apply to spinor-tensors. The (2p,Q)=(2,1) model is distinguished by admitting a central Lichnerowicz-Dirac operator when the background is constant curvature. The new supersymmetric models are novel in that the Hamiltonian is not just a square of super charges, but rather a sum of commutators of supercharges and commutators of bosonic charges. These models and superalgebras are a very useful tool for any study involving high rank tensors and spinors on manifolds.