How does the first-order analysis extend to the (A)dS algebra in supergravity?

[haushofer]

Hi,

I have a question concerning gravity obtained by gauging symmetry algebras, a method used often in the context of supergravity.

One can gauge the Poincaré algebra with generators {P,M}. P generates translations, M generators Lorentz transformations, schematically given by

$$[P,P]=0, \ \ \ \ [M,P]=P, \ \ \ \ \ [M,M]=M$$
This gives two gauge fields, one for P and one for M:
$$e_{\mu}{}^a, \ \ \ \omega_{\mu}{}^{ab} \ \ \ (1)$$
In the first order formalism one then imposes a curvature constraint. The curvature R of translations, R(P), is put to zero: R(P)=0. This amounts to putting the torsion to zero. The effect is that the $\omega_{\mu}{}^{ab}$ can be solved for if one introduces inverse fields $e^{\mu}{}_a$, and the local P-translations on the remaining field $e_{\mu}{}^a$ can be regarded as a combination of a general coordinate transformation and a Lorentz transformation (this is not trivial, but nevertheless true). This makes one to identify $e_{\mu}{}^a$ as vielbein.

One can then impose the Vielbein postulate to express the Levi-Civita connection $\Gamma^{\rho}_{\mu\nu}$ in terms of the gauge fields (1). Then the Riemann tensor can be expressed in terms of the curvature of Lorentz transformations R(M),

$$R^{\mu}{}_{\nu\rho\sigma} = -e^{\mu}{}_a R_{\rho\sigma}{}^{ab}(M) e_{\nu \ b}$$

In this way one can "obtain general relativity by a gauging procedure on the Poincare algebra".

My question is: how does this first-order analysis extend to the (A)dS algebra? The problem is that the (A)dS algebra is semi-simple (in constrast with the Poincare algebra), so effectively one has 1 generator. Splitting things up becomes messy. If one applies the above analysis naively to the (A)dS algebra written as

$$[P,P]=M, \ \ \ \ [M,P]=P, \ \ \ \ \ [M,M]=M$$

then one doesn't get the deformation [P,P]=M into the Riemann tensor via the Vielbein postulate; because the curvature R(P) doesn't change, the spin connection doesn't change, and if the Vielbein postulate doesn't change, the Riemann tensor doesn't change, and so effectively one is gauging the Poincaré algebra again.

I looked up some literature about this (Mansouri, Ortin's "gravity and strings", Wilczek) but I would like to understand the precise reason why the 1-order formalism a la Poincaré goes wrong. And how to improve it, of course.

So if anyone has an idea: I would be happy with some comments :)

 

[haushofer]

I'm kicking it in the hope that somebody recognizes the problem. :)

 

[mitchell porter]

For reference, I will link to: some earlier threads https://www.physicsforums.com/showthread.php?p=2781243" of "gauge" versus "string" approaches to quantum gravity in 3 dimensions.

In the string versus loop debate, I am very busy just trying to understand string. But of course I have seen the numerous debates in which string theorists say that treating gravity as a gauge theory is an inherently doomed approach. I can list some of the assertions - LQG is unpredictive because it has infinitely many coupling constants, it can't recover the classical limit, it only has kinematics but not dynamics, the Hilbert space is too big (continuum-many dimensions) and the method of quantization is http://golem.ph.utexas.edu/string/archives/000299.html" (note the statement: "these are the same methods used in LQG for quantizing the gravitational field in 3+1 dimensions") - but I couldn't reproduce the arguments unaided. That would require a level of sophistication somewhat beyond where I'm at, and I continue to feel that the best way to reach that level is just to study the string until it makes sense, rather than to study the anti-loop arguments until they make sense. I figure that by the time you really understand string theory, you should certainly understand enough about gravity and field theory to form an independent judgment about these matters.

Nonetheless, recently I'm reading papers by http://arxiv.org/abs/1105.5632" state, with the intuition that it's relevant.)

haushofer, I don't mean to hijack your thread, but I'm hoping that if anyone decides to answer your question, it might shed some light on the broader context.