# Derivation of the supergravity action in 11D

The Einstein-Hilbert action of general relativity is uniquely determined by general covariance and the requirement that only second derivatives in the metric appear. Yang-Mills theory can be motivated in a similar way. In the original paper of Scherk, Julia, Cremer there are some arguments given from which they deduced the form of the action. They are only sketched however. Is there a more complete exposition of the derivation in the literature, or possibly even a uniqueness result as in the case of general relativity or Yang-Mills theory?

• The arguments relied heavily on computer algebra to figure out how to close the terms, and I don't think people have replicated the calculations since, but we have consistency checks from string theory relating to M-theory, so the computer program didn't have a bug. The action must be unique, since the multiplet structure is very simple--- it's just the graviton plus the three-form field, and the gravitino that's it. The argument from counting the number of components is pretty persuasive. What do you find unconvincing here? – Ron Maimon May 22 '12 at 3:34
• A counting argument can only give a neccessary condition not a sufficient one. I would be satisfied if there were a theorem that stated, that there is a (unique?) supersymmetric action, without exhibiting it explicitly. Ideally the theorem should give an algorithm to compute the action. This would be the situation one is in the case of General Relativity or Yang-Mills. – orbifold May 22 '12 at 17:43
• When there is a further restriction on spins >2 counting should be enough to establish uniqueness. You can only have one spin-2 object, some spin-1 objects (various dimension forms), and exactly one spin-3/2 object, you can't have 2 gravitinos (because then you would need N=2 SUSY's and you would get spins >2). So you need 1 graviton plus N spin-1/2 and an assorted collection of spin-1's for k<11, and it should be possible to exclude all these by counting. It is remarkable that a SUSY action exists for the graviton+3-form and gravitino all by themselves. But I agree, a proof would be nice. – Ron Maimon May 22 '12 at 19:43
• It's exactly the same condition. One looks for the most general 11D field content and action that has exactly 2 derivatives in each term - the presence of a fermion is counted as 1/2 of a derivative (square root as a derivative) and a fundamental boson such as the metric tensor or a gauge potential is dimensionless. If you require 11D supersymmetry of such a theory, the 11D SUGRA action is the unique solution up to field redefinitions. One may prove those things starting from the free spectrum and adding interactions, motls.blogspot.com/2012/04/royal-status-of-11-dimensional.html – Luboš Motl May 23 '12 at 8:16
• @LubošMotl : Is the procedure of beginning from the free spectrum and adding interactions trial and error, or does it follow some algorithm? I imagine that you begin by postulating some supersymmetry transformation between the fields, compute and find what terms do not cancel, add those and so on. At some point you find that the remainder can be written as a total derivative and you are done. I think that is more or less what was done in the original paper. Given that there are many of those miraculous actions I feel that there should be a systematic way. – orbifold May 27 '12 at 19:07

## 1 Answer

A beautifully conceptual derivation of 11d SuGra was given in

• Riccardo D'Auria, Pietro Fré, Geometric Supergravity in D=11 and its hidden supergroup, Nuclear Physics B201 (1982) 101-140 (nLab)

using the excellent supergeometric methods later laid out in their textbook

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Supergravity and Superstrings - A Geometric Perspective World Scientific, 1991 (nLab)