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This book provides a self-contained presentation of supergravity theories from its fundamentals to its most recent union with string and superstring theories. (3) Supergravity and superstrings: A Geometric perspective. Vol. 1: Mathematical foundations - Castellani, L. et al. Singapore, Singapore: World Scientific ().
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Even better: while constraining just the bosonic component of the super-torsion tensor to vanish implies the 11d super-gravity equations of motion, constraining the entire super-torsion tensor to vanish furthermore constrains the gravitino field strength to vanish and hence implies the purely bosonic Einstein equations in 11d. People like to say that gravity is the result of locally gauging Lorentz symmetry.
A physics account of this that does it right and still sticks to the component notation that you are used to from the physics textbooks is.
This canonical Minkowski vielbein field has the obvious but important property that it is translation invariant, expressed by the equation. This condition is essentially the principle of equivalence. This should be intuitively clear. The precise mathematical statement and its generalization to higher order torsions is due to the remarkable article.
In terms of any such choice, then the corresponding torsion is. So far this defines a torsion-free spacetime geometry. In general however, this of course need not yet satisfy the equations of motion of gravity. In order to phrase these, one defines the Riemann curvature as the 2-form. See chapter I.
A super-Cartan geometry now is a super-spacetime locally modeled on this superspace, and an H -structure on this is a choice of super-vielbein, encoding a field configuration of supergravity graviton and gravitino. A super-vielbein is a superform with local bosonic components. We might hence be tempted to define. The correct expression is. Precisely the same holds true also for super-Lie groups, just with a sign thrown in whener two odd-graded elements change position.
This basic fact of super-symmetry has the following profound implication: as opposed to ordinary bosonic Minkowski spacetime, its extension to super-Minkowski spacetime has a built-in non-vanishing torsion, with components. While there is hence non-trivial torsion in super-Minkowski spacetime if we decompose it into bosonic and fermionic components, it may be more natural to regard it as a unified structure and regard the combination. Therefore we may now generalize all of the previous story of gravity encoded in Cartan geometry to super-Cartan geometry.
For a convenient listing of these equations and all the torsion and Bianchi identities that go into them, see pages of Supergravity and Superstrings — A Geometric Perspective. No further super-Einstein equations have to be imposed by hand.
They are already implied and do imply the bosonic super-torsion constraint. Hence setting also this to zero, hence constraining the entire super-torsion to vanish, further restricts from general solutions to dimensional supergravity to purely bosonic solutions of dimensional gravity. What remains is an ordinary field of gravity in 11 d subject to the ordinary bosonic vacuum Einstein equations in 11d.
You mention the gauging of the Lorentz symmetries.
In order to remove the local translations from your algebra you need to set the torsion which is the field strength of translations to zero. This is called a conventional constraint, because it enables one to solve for the spin connection. It starts appearing with equation 3.