nLab
super vielbein

Idea

A super vielbein is the analogue of a vielbein in supergeometry.

The canonical super vielbein

Let $N$ be a real spin representation and let $(x^a, \theta^\alpha)$ be the canonical coordinates on the supermanifold $\mathbb{R}^{d-1,1\vert N}$ underlying the super Minkowski spacetime super translation group. Then the canonical super vielbein is the $\mathbb{R}^{d-1,1\vert N}$-valued super differential form with components

• $\psi^\alpha \coloneqq \mathbf{d} \theta^\alpha$.

• $e^a \coloneqq \mathbf{d} x^a + \overline{\theta} \Gamma^a \mathbf{d} \theta$.

Related concepts

• super Cartan geometry

• generalized vielbein

• super-torsion

• geometry of physics – supersymmetry