Knowlpedia

Definition

Super-Minkowski space

The affine superspace of spacetime and spinor coordinates equipped with the Lie supergroup structure integrating supertranslations.

Definition

Fix supertranslation data (V,S,Γ)(V,S,\Gamma), where Γ:Sym⁡2S→V\Gamma:\operatorname{Sym}^2S\to V. Super-Minkowski space is the affine supermanifold

MV∣S=(V, CV∞⊗ΛS∗)\mathbb M^{V|S}=(V,\ C^\infty_V\otimes\Lambda S^*)

equipped with the structure integrating the V0ˉ⊕S1ˉV_{\bar0}\oplus S_{\bar1}.

More precisely, let AA be a real Grassmann algebra. On AA-points, write ΓA(θ,θ′)=[θ,θ′]\Gamma_A(\theta,\theta')=[\theta,\theta'] for the bracket induced from Γ\Gamma on A1ˉ⊗SA_{\bar1}\otimes S. In exponential coordinates the exact two-step is

(x,θ)(x′,θ′)=(x+x′+12ΓA(θ,θ′),θ+θ′),(x,\theta)(x',\theta') =\left(x+x'+\tfrac12\Gamma_A(\theta,\theta'), \theta+\theta'\right),

where x,x′∈A0ˉ⊗Vx,x'\in A_{\bar0}\otimes V and θ,θ′∈A1ˉ⊗S\theta,\theta'\in A_{\bar1}\otimes S. Defining ΓA\Gamma_A as the induced bracket fixes the scalar sign convention in the displayed formula. The underlying reduced Lie group is the additive translation group of VV.

Its odd left-invariant directions form the , whose Levi bracket is induced by Γ\Gamma.

Distinctions

A general is merely a locally superringed space. Even an affine superdomain Vd∣sV^{d|s} is not super-Minkowski space until the spin representation, Γ\Gamma, and corresponding supertranslation structure have been chosen. Curved supergravity superspaces require further torsion and connection data and are not super-Minkowski space.

References
  1. P. Deligne and J. W. Morgan, “Notes on supersymmetry (following Joseph Bernstein),” in Quantum Fields and Strings: A Course for Mathematicians, Volume 1, American Mathematical Society, 1999, 41–97. Relevant: flat superspace and supertranslations.
  2. D. S. Freed, Five Lectures on Supersymmetry, American Mathematical Society, 1999. Publisher record. Relevant: superspace and super-Poincaré geometry.