nLab super Minkowski space

Context

Riemannian geometry

Riemannian geometry

superalgebra

and

supergeometry

Applications

Gravity

gravity, supergravity

Contents

Idea

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

Definition

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

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

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

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

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

Revised on May 22, 2013 13:14:10 by Urs Schreiber (84.153.217.239)