SUSY Algebras and Unitary Representations

Supersymmetry extends spacetime symmetry with fermionic generators whose anticommutators constrain energy and particle multiplets. This chapter connects the algebra to unitary states, central-charge bounds, and shortening, while separating representation theory from the dynamical question of which states a theory actually contains.

Enter this chapter

Begin with Graded Spacetime Symmetry and the Supersymmetry Theorems to state the assumptions under which spacetime symmetry admits a graded extension.

Helpful background. Spinors, Conjugations, Bilinears, Chirality, and Fierz Identities; Lorentz Field Representations and Poincaré Particle Representations; Multiplets, Invariants, and Selection Rules provide the spinor reality, Poincaré representations, and symmetry multiplets used here.

Topics

1. Graded Spacetime Symmetry and the Supersymmetry Theorems — States the hypotheses behind the supersymmetry extension theorems.
2. Supersymmetry Across Dimensions, Signatures, and Reality Conditions — Counts real supercharges using dimension-dependent spinors.
3. The Four-Dimensional N=1 Super-Poincaré Algebra — Builds the minimal four-dimensional algebra.
4. Extended Supersymmetry, R-Symmetry, and Central Charges — Introduces R-symmetry and central extensions.
5. Massive and Massless Unitary Supermultiplets — Constructs physical multiplets at fixed momentum.
6. BPS Bounds, Shortening, and Multiplet Recombination — Derives positivity bounds and shortened representations.
7. On-Shell Supermultiplets and Supersymmetric Ward Identities — Packages external states and amplitude identities.
8. Superconformal Algebras, Shortening Data, and CFT Dynamics — Separates superconformal representation data from CFT dynamics.

How the ideas fit together

At fixed momentum, positive supercharge anticommutators become fermionic creation and annihilation operators, generating a multiplet from a chosen state. A sharp positivity bound can make some combinations null, reducing the multiplet’s size. That shortening is a kinematic statement: it does not establish that a charge sector contains a state saturating the bound. Nor should physical-state counting be confused with counting covariant field components, which may include gauge redundancy and auxiliary fields. Weinberg 2000, §§ 25.4–25.5.

Where to go next

Continue to Supersymmetric Quantum Mechanics, Cohomology, and the Witten Index to construct the Hamiltonian from a supercharge and its Hilbert-space adjoint.

References

• Weinberg, Steven. The Quantum Theory of Fields. Volume III: Supersymmetry. Cambridge University Press, 2000. DOI.