nLab
super Minkowski space

Idea

The spacetime in supergeometry/supergravity which is the super-analog of ordinary Minkowski spacetime.

Definition

Ordinary $(d+1)$-dimensional Minkowski space can be understood as the quotient $\mathrm{ISO}(d,1)/(\mathrm{SO}(d,1))$ of the Poincare group by the Lorentz group – the translation group.

Analogously, the for each $N\in \mathbb{N}$ the $N$-extended supermanifold Minkowski superspace or super Minkowski space is the quotient of supergroups

$$\overline{\mathrm{Osp}}(d+1\mid N)/(\mathrm{SO}(d,1)\times \mathrm{SO}(N))$$\bar Osp(d+1|N)/ (SO(d,1)\times SO(N))

where $\overline{\mathrm{Osp}}(d+1\mid N)$ is…

The super-translation group.

Properties

Cohomology and super $p$-branes

As opposed to ordinary Minkowski space, the de Rham cohomology of super-Minkowski space contains nontrivial exceptional cocycles. These serve as the WZW terms for the Green-Schwarz action functional (see there for more) of super-$p$-branes propagating on super-Minkowski space.

References

for instance page 370, part II section II.3.3

• Leonardo Castellani, Riccardo D'Auria, Pietro Fre, Supergravity and Superstrings - A Geometric Perspective