[Paper Review] Irreps and Off-shell Invariant Actions of the N-extended Supersymmetric Quantum Mechanics

Francesco Toppan|ArXiv.org|Oct 16, 2006

Advanced Topics in Algebra4 references14 citations

TL;DR

This paper classifies irreducible representations of N-extended one-dimensional supersymmetry algebras using Clifford algebras and Cayley-Dickson constructions, demonstrating that division algebra structure constants—particularly octonionic ones—appear as coupling constants in off-shell invariant actions for N=8 supersymmetric quantum mechanics. It further links higher N supersymmetries (up to N=16) to sedenions and suggests their role in formulating 11D supergravity as a low-energy limit of M-theory.

ABSTRACT

The complete classification of the irreducible representations of the N-extended one-dimensional supersymmetry algebra linearly realized on a finite number of fields is presented. Off-shell invariant actions of one-dimensional supersymmetric sigma models are constructed. The role of both Clifford algebras and the Cayley-Dickson's doublings of algebras in association with the N-extended supersymmetries is discussed. We prove in specific examples that the octonionic structure constants enter the N=8 invariant actions as coupling constants. We further explain how to relate one-dimensional supersymmetric quantum mechanical systems to the dimensional reduction of higher-dimensional supersymmetric theories.

Motivation & Objective

• To classify irreducible representations (irreps) of the N-extended one-dimensional supersymmetry algebra for finite field multiplets.
• To construct off-shell invariant actions for N-extended supersymmetric sigma models beyond the superfield formalism.
• To explore the role of division algebras (real, complex, quaternions, octonions) and their Cayley-Dickson doublings in realizing higher N supersymmetries.
• To establish a connection between the structure constants of division algebras and coupling constants in off-shell actions, particularly for N=8.
• To investigate the potential of sedenions and higher algebras in formulating off-shell actions for 11D supergravity via dimensional reduction.

Proposed method

• Utilizes D-module representations to classify irreps of the N-extended D=1 supersymmetry algebra, mapping them to Clifford algebra representations.
• Applies the Cayley-Dickson doubling procedure to generate an infinite sequence of algebras: R → C → H → O → sedenions, linking each to increasing N supersymmetry.
• Constructs off-shell invariant actions for N=8 using octonionic structure constants as coupling constants, verified through explicit examples.
• Relates the N=16 supersymmetry algebra to the sedenions via Cl(0,15), showing its connection to the 11D supergravity multiplet (44 bosonic, 128 fermionic, 84 bosonic components).
• Demonstrates that the structure constants of division algebras enter both supersymmetry transformations and off-shell action Lagrangians, not just algebraic relations.
• Uses dimensional reduction to relate one-dimensional supersymmetric systems to higher-dimensional theories, especially 11D supergravity as the low-energy limit of M-theory.

Experimental results

Research questions

• RQ1 How can the irreducible representations of the N-extended one-dimensional supersymmetry algebra be completely classified for finite field multiplets?
• RQ2 In what way do the structure constants of division algebras (e.g., octonions) appear as coupling constants in off-shell invariant actions of N-extended quantum mechanics?
• RQ3 Can the Cayley-Dickson doubling procedure provide a systematic framework for constructing off-shell actions for N > 8, where superfield methods fail?
• RQ4 What is the role of sedenions and their non-division algebra structure in realizing N=16 supersymmetry and its connection to 11D supergravity?
• RQ5 How can one-dimensional supersymmetric quantum mechanics serve as a tool to explore off-shell formulations of 11D supergravity and M-theory?

Key findings

• The complete classification of irreducible representations of the N-extended D=1 supersymmetry algebra is achieved via D-module representations, with multiplets structured by Clifford algebra realizations.
• For N=8, the paper proves that the octonionic structure constants explicitly appear as coupling constants in off-shell invariant actions, a novel result not previously identified in the literature.
• The Cayley-Dickson doubling process generates an infinite sequence of algebras (R → C → H → O → sedenions), each associated with increasing N supersymmetry: N=1,2,4,8,16,…
• The N=16 supersymmetry algebra is realized on a 256-dimensional multiplet (128 bosonic and 128 fermionic fields), matching the on-shell content of 11D supergravity.
• The 11D supergravity multiplet (44 bosonic, 128 fermionic, 84 bosonic) carries a linear representation of the N=16 supersymmetry algebra, suggesting a deep algebraic link via sedenionic structure constants.
• The paper suggests that the octonionic covariantization prescription for N=8 actions can be extended to bi-octonions (N=9) and sedenions (N=16), offering a pathway toward an off-shell formulation of 11D supergravity.