nLab The Cauchy Problem in Classical Supergravity

This entry is about the article

• Yvonne Choquet-Bruhat,

  The Cauchy Problem in Classical Supergravity Letters in Mathematical Physics 7 (1983) 459-467. 0377

Summarry

Here is the quintessence of the article.

Terminology

The article considers super fields valued in a “single, arbitrary, large enough” Grassmann algebra in the sense exposed in the book

• Bryce de Witt, Supermanifolds, Cambridge Monographs on Mathematical Physics, 1984, 1992

In the following we shall not quite follow this, but make use of the observation that this really means that we are working over the sheaf topos on the site of superpoints, as described at super ∞-groupoid. This changes nothing about the actual computations and formulas, but somewhat strenghtens the conceptual background.

Field configurations

Let $\mathfrak{siso}(d,1)$ be the super Poincare Lie algebra in some dimension. The field configuration space of supergravity is the groupoid of Lie-algebra valued forms $\Omega^1(X, \mathfrak{siso})$ on a given spacetime manifold $X$. This is the super-groupoid that assigns to a given superpoint $\mathbb{R}^{0|q}$ with function algebra the Grassmann algebra $\Lambda_q$ the ordinary groupoid of Lie-algebra valued forms of the ordinary Lie algebra $(\mathfrak{g} \otimes \Lambda_q)_{even}$:

$$\Omega^1(X, \mathfrak{g}) : \mathbb{R}^{0|q} \mapsto \Omega^1(X, (\mathfrak{g} \otimes \Lambda_q)_{even}) \,.$$

Its objects are the field configurations, its morphisms the gauge transformations. The collection of objects over $\mathbb{R}^{0|q}$ is the collection of dg-algebras homomorphims

$$Hom_{dgAlg}(W(\mathfrak{g} \otimes \Lambda_q)_{even}), \Omega^\bullet(X)) \,,$$

where on the left we have the Weil algebra of $(\mathfrak{g} \otimes \Lambda_q)_{even}$. The Weil algebra is a free dg-algebra on the generators

• $\{e^a \otimes (\Lamba_q^*)_{even} \}$

• $\{\omega^{a b} \otimes (\Lamba_q^*)_{even} \}$

• $\{\psi^{\alpha} \otimes (\Lamba_q^*)_{odd} \}$ .

Accordingly a field configuration over $\mathbb{R}^{0|q}$ is

1. the graviton given by

   1. the vielbein $\{E^a_\mu : X \to (\Lambda_q)_{even}\}$;

   2. the spin connection $\{\Omega_\mu{}^{a b} : X \to (\Lambda_q)_{even}\}$;

2. the gravitino given by $\Psi_\mu^\alpha : X \to (\Lambda_q)_{odd}$.

The Cauchy problem

The Euler-Lagrange equations for these fields are schematically of the form

$$R_{\mu \nu}^{a b } + F = 0$$

$$D \Psi = 0 \,.$$

These are over each $\mathbb{R}^{0|q}$ equations in $C^\infty(X, \Lambda_q)$. In degree 0 in the Grassmann generators this is just the ordinary Einstein equaitons of gravity for the 0-Grassmann degree component of the fields. This is a well defined and causal Cauchy problem.

The observation now is: the equations can be solved by induction over the total number of Grassmann generators. In each step, the problem is a well-defined causal Cauchy problem in a version of ordinary gravity coupled to a bunch of extra fields.

category: reference