Supergravity

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A simplified explanation of the topics of this article are given in the article Why 10 dimensions?

In theoretical physics, a supergravity theory is a field theory combining supersymmetry and general relativity.

Like any field theory of gravity a supergravity theory contains a spin-2 field whose quantum is the graviton. Supersymmetry requires the graviton field to have a superpartner. This field has spin 3/2 and its quantum is the gravitino. The number of gravitino fields is equal to the number of supersymmetries. Supergravity theories are believed to be the only consistent theories of interacting massless spin 3/2 fields.

History

Supergravity was initially proposed as a four-dimensional theory in 1976, but was quickly expanded to Kaluza-Klein Supergravity by merging it with Kaluza-Klein theory. In this form, the 11-dimensional theory generated considerable excitement as the first potential candidate for the Theory of Everything. This excitement was built on four pillars:

• Nahm showed that 11 dimensions was the largest number of dimensions consistent with a single graviton, and that a theory with more dimensions would also have particles with spins greater than 2.

• Shortly afterwards, Ed Witten showed that 11 was the smallest number of dimensions that was big enough to contain the the gauge groups of the Standard Model, namely SU(3) for the strong interactions and SU(2) times U(1) for the electroweak interactions.

• In 1978, Cremmer, et al. showed that in 11 dimensions, there is only a single choice for the matter fields that gives an equal numbers of fermions and bosons in the theory; that is, there is only a single choice of matter fields that is consistent with supersymmetry.

• In 1980, P. G. O. Freund and Rubin showed that compactification from 11 dimensions could occur in only one of two ways, leaving only 4 or 7 macroscopic dimensions (the other 7 or 4 being compact).

Thus, the first two results established 11 dimensions uniquely, the third result eliminated most choices in how to construct the matter fields, and the last result explained why the observed universe appears to be four-dimensional. Of course, the second result is only necessary if we want supergravity to be a theory of everything.

Failings of the 11-dimensional model, support for 10 dimensions

The initial excitement over 11-dimensional supergravity soon waned, as various failings were discovered, and attempts to repair the model failing as well. Problems included:

• The compact manifolds containing the standard model were not compatible with supersymmetry, and could not hold quarks or leptons. One suggestion was to replace the compact dimensions with the 7-sphere, with the symmetry group SO(8), or the squashed 7-sphere, with symmetry group SO(5) times SU(2). These aren't big enough to contain the standard model, and so additional composite fields must be added.

• Until recently, the physical neutrinos seen in the real world were believed to be massless, and appeared to be left-handed, a phenomenon referred to as the chirality of the Standard Model. It was very difficult to construct a mode from 11 dimensions that posses such chirality. The recent phenomenon of neutrino oscillations lessen this concern a bit.

• Supergravity results in an unrealistically large cosmological constant in four dimensions, and that constant is difficult to remove, even with fine-tuning.

• Quantization of the theory leads to quantum field theory gauge anomalies rendering the theory inconsistent.

Many of these difficulties can be avoided by moving to a 10-dimensional theory involving superstrings. However, moving to 10 dimensions looses all of the strengths of the 11-dimensional theory, including the fact that its a Kaluza-Klein theory; a variety of ad-hoc particle and gauge fields must be added.

The core breakthrough for the 10-dimensional theory was a demonstration by Green, Schwarz and David Gross that there are only two supergravity models in 10 dimensions where all of the anomalies cancel. These were theories built on the groups SO(32) and , the direct product of two copies of E₈.

Relation to superstrings

A supergravity theory is generally the zero-length limit of a superstring theory (i.e., the limit in which the string is approximated as having zero length, and treated as a dimensionless point-particle), with the exception of "maximal" 11-dimensional supergravity, which is a limit of M-theory (most likely the limit in which the membranes are treated as having zero volume). However, this doesn't necessarily mean that string theory/M-theory is the only possible UV completion of supergravity and supergravity can be studied in its own right.

The underlying spacetime is a supermanifold and its symmetries are superdiffeomorphisms.

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