Supergravity

Einstein-Hilbert action , Einstein's equations , general relativity. D'Auria-Fre formulation of supergravity. M-theory super Lie algebra , type II super Lie algebra. W , Z , B-boson - electroweak field Yang-Mills field.

Specifically such a field. For instance there is, under mild assumptions, a unique maximally supersymmetric supergravity extension of the ordinary Einstein-Hilbert action on a 4-dimensional manifold.

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This in turn is obtained from the unique under mild assumptions maximally supersymmetric supergravity action functional on a 10,1 -dimensional spacetime by thinking of the 4-dimensional action function as being a dimensional reduction of the dimensional one. This uniqueness under mild conditions is one reason for interest in supergravity theories. Another important reason is that supergravity theories tend to remove some of the problems that are encountered when trying to realize gravity as a quantum field theory.

Originally there had been high hopes that the maximally supersymmetric supergravity theory in 4-dimensions is fully renormalizable.

Supergravity

In this first order formulation of gravity a field configuration is a Cartan connection with such coefficients. This perspective directly generalizes to supergeometry and yields the superspace formulation of theories of supegravity — super Cartan geometry. Then the direct sum. From this, a super-Cartan geometry is defined in direct analogy to the Cartan formulation of Riemannian geometry.

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Indeed, all the traditional literature on supergravity e. One subtlety to take care of is that this makes spacetime a super-spacetime locally modeled on super Minkowski spacetime.

Supergravity - Wikipedia

But the resulting theory is supposed to be a field theory on an ordinary spacetime locally modeled on ordinary Minkowski spacetime. This is enforced by a further constraint on the super-Cartan connection which forces it to be determined by the bosonic manifold underlying the given supermanifold. This constraint is variously known as the superspace constraints or as rheonomy. The other subtlety to take care of is that a key aspect of higher dimensional supergravity theories is that their field content necessarily includes, in addition to the graviton and the gravitino , higher differential n-form fields, notably the 2-form B-field of dimensional type II supergravity and heterotic supergravity as well as the 3-form C-field of dimensional supergravity.

This means that these higher dimensional supergravity theories are not in fact entirely described by super- Cartan geometry — but by super- higher Cartan geometry. The collection of all these exceptional classes constitutes what is known as the brane scan. But from a modern perspective this becomes crystal clear: So this collection of differential form data is no longer a Lie algebra valued differential form , it is an L-infinity algebra valued differential form , with values in the supergravity Lie 3-algebra. The Lie integration of this is a smooth 3-group G G which receives a map from the Lorentz group.

A solution to the bosonic Einstein equations of ordinary gravity — some Riemannian manifold — has a global symmetry if it has a Killing vector.

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Accordingly, a configuration that solves the supergravity Euler-Lagrange equations is a global supersymmetry if it has a Killing spinor: Here the notion of covariant derivative includes the usual Levi-Civita connection , but also in general torsion components and contributions from other background gauge fields such as a Kalb-Ramond field and the RR-field s in type II supergravity or heterotic supergravity.

This is where all the interest into these manifolds in string theory comes from. Notice though that nothing in the theory itself demands such a compactification. More generally, in the presence of other background gauge field s, the Calabi-Yau condition here is deformed. One also speaks of generalized Calabi-Yau space s. The equations of motion of those theories of supergravity which qualify as target spaces for Green-Schwarz action functional sigma models e.

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See at Green-Schwarz action — References — Supergravity equations of motion for pointers. Intranet Contact Sitemap Deutsch. Microscopic Quantum Structure and Dynamics of Spacetime. Research articles and preprints from Inspire. Textbook "Eisenstein series and automorphic representations - with applications in string theory". International Solvay Institutes, Brussels.

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Max Planck Yearbook Article. Neue Symmetriestrukturen in der Stringtheorie.