[Paper Review] Four dimensional cubic supersymmetry

Michel Rausch de Traubenberg|ArXiv.org|Dec 5, 2003

Algebraic structures and combinatorial models22 references3 citations

TL;DR

This paper proposes a four-dimensional non-supersymmetric extension of the Poincaré algebra based on cubic (third-order) fermionic generators, termed cubic supersymmetry (3SUSY), which closes via a symmetric three-fold product rather than anticommutation. It constructs a consistent field-theoretic realization using bosonic multiplets containing scalars, vectors, and two-forms, and derives an invariant Lagrangian and Noether currents, demonstrating that the algebra closes via triple commutators and is compatible with gauge symmetry only under standard gauge fixing.

ABSTRACT

A four dimensional non-trivial extension of the Poincaré algebra different from supersymmetry is explicitly studied. Representation theory is investigated and an invariant Lagrangian is exhibited. Some discussion on the Noether theorem is also given.

Motivation & Objective

• To explore non-supersymmetric extensions of the Poincaré algebra that evade the Coleman-Mandula and Haag-Lopuszanski-Sohnius theorems.
• To construct a field-theoretic realization of a cubic supersymmetry algebra (3SUSY) in four dimensions, distinct from standard supersymmetry.
• To investigate the representation theory of the 3SUSY algebra, identifying multiplets of fermions and bosons with specific spin content.
• To derive an invariant Lagrangian for the bosonic multiplet and analyze its gauge structure and energy boundedness.
• To examine the applicability of the Noether theorem in the context of cubic algebras, where symmetry charges close via triple commutators.

Proposed method

• The paper employs F-Lie algebras with Z3-gradation, where the zero-graded part is a Lie algebra (sp(4,R)) and the non-zero parts are represented by adjoint generators.
• The algebra is defined by cubic relations: {A_a, A_b, A_c} = g_{ab}J_c + g_{ac}J_b + g_{bc}J_a, with J_a generating translations and Lorentz transformations.
• A field-theoretic realization is constructed using scalar, vector, and two-form fields, with a Lagrangian invariant under 3SUSY transformations.
• The Lagrangian is expressed in a Fermi-like form, with kinetic terms for φ, A, and B fields, and includes gauge-fixing terms for vector and two-form fields.
• Noether currents are derived from the Lagrangian using the standard procedure, and the associated charges are shown to close via triple commutators.
• The algebraic structure is realized both abstractly via generators and concretely via commutators acting on fields in Hilbert space.

Experimental results

Research questions

• RQ1 Can a non-trivial extension of the Poincaré algebra exist that is not supersymmetry, and if so, what algebraic structure supports it?
• RQ2 How can a cubic (third-order) fermionic symmetry be consistently realized in four-dimensional relativistic field theory?
• RQ3 What are the physical multiplets (fermionic and bosonic) that arise from the 3SUSY algebra, and what are their field content and mass degeneracy?
• RQ4 Is it possible to construct a gauge-invariant Lagrangian for 3SUSY, and if so, under what conditions does gauge symmetry remain compatible?
• RQ5 How does the Noether theorem apply to algebras that close via cubic (rather than quadratic) relations, and what are the implications for conserved currents and charges?

Key findings

• The 3SUSY algebra is realized via an F-Lie algebra with Z3-gradation, where the three-fold symmetric product closes the algebra and generates translations.
• Fermionic multiplets consist of three degenerate chiral fermions, while bosonic multiplets contain one scalar, one vector, and one antisymmetric two-form field.
• The constructed Lagrangian for the bosonic multiplet is invariant under 3SUSY transformations and takes a Fermi-like form with kinetic terms for all fields.
• Gauge symmetry is compatible with 3SUSY only when standard gauge-fixing terms (e.g., 't Hooft-Feynman) are included for vector and two-form fields.
• The Noether current is conserved and leads to charges that realize the algebra via triple commutators, with the triple product of charges yielding the Poincaré generators.
• The energy density is not obviously bounded from below due to negative signs in vector field kinetic terms, indicating a need for further analysis of the field manifold upon inclusion of interactions.