[Paper Review] Aspects of on-shell supersymmetry

Jos Gheerardyn|ArXiv.org|Nov 12, 2004

Particle physics theoretical and experimental studies1 references3 citations

TL;DR

This paper investigates on-shell supersymmetry in classical field theories that lack a Lagrangian formulation, demonstrating that such theories admit broader classes of potentials and target space geometries than standard on-shell actions allow. Using superconformal tensor calculus and Scherk-Schwarz dimensional reduction, the author constructs a massive, non-chiral (type IIA) supergravity in ten dimensions from eleven-dimensional supergravity without an action, showing that supersymmetry can organize dynamics even without a Lagrangian, and deriving explicit supersymmetric solutions via Killing spinor equations.

ABSTRACT

We study classical field theories that do not admit an action principle, implying that their formulation is entirely in terms of equations of motion. In the theories that we consider, supersymmetry is realized on the mass-shell. We will show that the supersymmetry algebras previously studied in the literature, allow for a broader class of potentials and for new target space manifolds. We discuss extensively theories with eight supercharges as well as a massive ten dimensional maximal supergravity.

Motivation & Objective

• To explore classical field theories that are supersymmetric and Poincaré-covariant but do not admit a Lagrangian formulation, focusing on on-shell supersymmetry.
• To extend the class of allowed target space geometries in nonlinear sigma models beyond those constrained by action-based supersymmetry.
• To demonstrate that consistent dimensional reduction of an action-based theory (11D supergravity) can yield a non-action-based theory (massive IIA supergravity).
• To develop a method for constructing supersymmetric solutions in non-Lagrangian theories using Killing spinor equations and cohomogeneity-one geometries.
• To show that on-shell supersymmetry can serve as a powerful organizing principle for dynamics even in the absence of an action.

Proposed method

• Utilizes superconformal tensor calculus to derive the structure of on-shell N=2, d=5 supergravity and its multiplets (vector, hypermultiplets, Weyl multiplet).
• Applies a generalized Scherk-Schwarz reduction to 11D supergravity, introducing a mass parameter via a non-isometric U(1) shift in the compactified dimension.
• Imposes a specific z-dependence on higher-dimensional fields (e.g., metric ∝ e^{2mz}) to preserve Poincaré invariance in the lower-dimensional theory.
• Derives the effective 10D massive IIA supergravity by requiring invariance under the reduced supersymmetry algebra and solving the resulting equations of motion.
• Constructs supersymmetric solutions by solving the Killing spinor equation in the form (∂μkν − ∂νkμ)ΓμΓνϵ = mϵ, where k is a homothetic Killing vector.
• Maps hypercomplex structures to quaternionic-like geometries via a frame formalism and identifies SU(2) invariance as a key symmetry in the reduction.

Experimental results

Research questions

• RQ1 Can supersymmetry be used to construct consistent field theories without a Lagrangian action?
• RQ2 What class of target space geometries is allowed in on-shell supersymmetric nonlinear sigma models when no action exists?
• RQ3 How can dimensional reduction of an action-based theory like 11D supergravity lead to a non-action-based theory such as massive IIA supergravity?
• RQ4 What are the conditions under which a higher-dimensional theory with a homothetic Killing vector reduces to a lower-dimensional supersymmetric theory with mass terms?
• RQ5 How can supersymmetric solutions be systematically constructed in non-Lagrangian supergravity theories?

Key findings

• The paper constructs a massive, non-chiral (type IIA) supergravity in ten dimensions from 11D supergravity via Scherk-Schwarz reduction, which does not admit a conventional action principle.
• The resulting 10D theory is distinct from Romans' massive IIA supergravity and features a mass term generated by a non-isometric compactification with a U(1) shift symmetry.
• Supersymmetric solutions are constructed by reducing 11D plane-wave and Ricci-flat cone backgrounds, preserving 16 supersymmetries in the 10D theory.
• The reduction procedure relies on a metric ansatz ∝ e^{2mz} and a z-dependent spinor ansatz ∝ e^{mz/2}, which preserve supersymmetry when the background admits a homothetic Killing vector.
• The paper establishes a mapping between hypercomplex and quaternionic-like geometries via a frame formalism, showing that the reduction preserves SU(2) invariance and curvature constraints.
• It is shown that on-shell supersymmetry allows for more general target spaces and potentials than those constrained by an action, extending the scope of supersymmetric field theories.