# 2 + 1 dimensional gravity as an exactly soluble system [closed]

In Gomez's article (Higher spin part 2) http://arXiv.org/abs/1307.3200, I face to gauge algebras. Gomez says we have some algebras for $$ISO(2,1)$$ and $$SO(2,2)$$, $$SO(3,1)$$.

and he tells us only $$ISO(d-1,1)$$ and only for $$D=3$$ is a good and it is relevant and equivalent to 3D Chern-Simons.

Please give me a reference for understand it.

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## closed as unclear what you're asking by ACuriousMind♦, innisfree, Kyle Kanos, JamalS, user10851 Mar 29 '15 at 22:10

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• This is close to incomprehensible. Please try to rephrase your question such that it can be understood, explain your notation and which article you are talking about, and use MathJaX to typeset formulae. – ACuriousMind Mar 28 '15 at 20:42
• You can dissolve it in water? That's pretty cool... – Sean Mar 29 '15 at 0:12
• DON'T CLOSE THIS QUESTION! Rewrite it if necessary. There is a topic of real substance here, namely gauge theories corresponding to 2+1 gravity with various values of cosmological constant (that's the lambda). – Mitchell Porter Mar 29 '15 at 5:59
• I will try to answer it, and maybe I will suggest a rewrite of the question too, but I can't do it right away. – Mitchell Porter Mar 29 '15 at 6:00
• For the corrected formulas and another answer see physicsoverflow.org/29140 – Arnold Neumaier Mar 30 '15 at 8:51

Just a quick preliminary answer, I will fix it later.

The connection to general relativity is a change of variables in which the metric is replaced by a "spin connection" and a "frame field". These quantities can then be arranged in a new matrix, so the metric field has been rewritten as a different matrix-valued field, and the transformations (diffeomorphisms) allowed under the symmetry of general relativity (general covariance) map to gauge transformations of this new matrix-valued field. The commutation relations above, are for the group of these gauge transformations - J corresponds to translations, P to rotations and boosts. The actual group is different depending on whether we are in flat space, de Sitter space, or anti de Sitter space; the cosmological constant (which is respectively zero, positive, negative) shows up in the commutation relations as lambda. d=3 is special because only there is a gauge-invariant action for this rewrite of general relativity possible. ISO(2,1) is just the special case of lambda=0, flat space in 2+1 dimensions.

All this is scattered through section 2 of Witten's paper. Also see part 1.1 of the sequel.

Thanks to T.S. for a discussion of this and related papers a few years ago.

• Mitchell Porter thanks for your answer it was exactly my question please give me a reference for it. – Ali Mar 29 '15 at 14:41
• @Ali rezaie I don't have a single reference, that understanding came from many different sources. But you can do it yourself, step by step: – Mitchell Porter Mar 30 '15 at 11:18
• 1) Understand general relativity in 2+1 dimensions, in a vacuum spacetime, for the three cases. – Mitchell Porter Mar 30 '15 at 11:20
• 2) Understand the change of variables, first to spin connection and frame field, then their recombination into the new matrix. – Mitchell Porter Mar 30 '15 at 11:22
• 3) Understand how infinitesimal translations, rotations and boosts, act on the new matrix variable. – Mitchell Porter Mar 30 '15 at 11:24