Supersymmetric Quantum Mechanics and Super-Lichnerowicz Algebras

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.

39 pages, LaTeX, fixed typos, added refs, final version to appear in CMP

References in corpus (18)

• Partial Masslessness of Higher Spins in (A)dS
• Gauge Invariances and Phases of Massive Higher Spins in (A)dS
• SUSY in the sky
• On geometric equations and duality for free higher spins
• Exotic tensor gauge theory and duality
• Gauge invariant formulation of massive totally symmetric fermionic fields in (A)dS space
• Tensor gauge fields in arbitrary representations of GL(D,R): II. Quadratic actions
• Tensor fields of mixed Young symmetry type and N-complexes
• Worldline approach to vector and antisymmetric tensor fields
• Constant Curvature Algebras and Higher Spin Action Generating Functions
• Geometric Second Order Field Equations for General Tensor Gauge Fields
• Worldline approach to vector and antisymmetric tensor fields II
• Higher spin fields from a worldline perspective
• Conformal Invariance, N-extended Supersymmetry and Massless Spinning Particles in Anti-de Sitter Space
• Supersymmetry and the cohomology of (hyper)Kaehler manifolds
• Kahler spinning particles
• Particles, Superparticles and Super Yang--Mills
• Classical description of spinning degrees of freedom of relativistic particles by means of commuting spinors

Cited by in corpus (17)

• Massless Poincare modules and gauge invariant equations
• Tractors, Mass and Weyl Invariance
• Weyl Invariance and the Origins of Mass
• Consistent Non-Minimal Couplings of Massive Higher-Spin Particles
• Detours and Paths: BRST Complexes and Worldline Formalism
• U(N) spinning particles and higher spin equations on complex manifolds
• Half-integer Higher Spin Fields in (A)dS from Spinning Particle Models
• Extended SUSY quantum mechanics: transition amplitudes and path integrals
• BRST Detour Quantization
• Generalized Einstein Operator Generating Functions
• Higher-Spin Modes in a Domain-Wall Universe
• The Graded Differential Geometry of Mixed Symmetry Tensors
• Superconformal quantum mechanics and the exterior algebra
• U(N|M) quantum mechanics on Kaehler manifolds
• (p,q)-form Kaehler Electromagnetism
• The Involutive System of Higher-Spin Equations
• Conformally equivariant quantization for spinning particles