2 + 1 dimensional gravity as an exactly soluble system

Question

In Witten's article, 2+1-dimensional gravity as an exactly soluble system, I faced gauge algebras. Witten claims that for any scalar \(\lambda\), we have \(j_i\)s and \(P_i\)s related to each other as follows:

\([j_a ,j_b]=ε_{abc} j_c\)

\([j_a ,P_b]=ε_{abc} P_c\)

\([P_a ,P_b]=\lambdaε_{abc} j_c\)

He also claims, that in and only in $ISO(d-1,1)$ and $d=3$, do we have the following unique relations:

\([j_a ,j_b]=ε_{abc} j_c\)

\([P_a,P_b]=0, [j_a,P_b]=ε_{abc} P_c\)

How can I understand this? Any references?

This post imported from StackExchange Physics at 2015-03-29 12:41 (UTC), posted by SE-user Ali rezaie

Comments

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.

This post imported from StackExchange Physics at 2015-03-29 12:41 (UTC), posted by SE-user ACuriousMind

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 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).

This post imported from StackExchange Physics at 2015-03-29 12:41 (UTC), posted by SE-user Mitchell Porter

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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.

This post imported from StackExchange Physics at 2015-03-29 12:41 (UTC), posted by SE-user Mitchell Porter

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I have done a LaTeX-ing of this question from exactly what you wrote, but are you sure you meant to write "ISO" rather than "SO", and is the second set of equations actually supposed to have standard brackets, or should they be commutators (you might want to make it clear what they represent, if they are the former - anticommutators perhaps?)?

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It is ISO. I fixed the ISO commutation relations to match equations (2.9) of Witten.

Answer 1

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.

This post imported from StackExchange Physics at 2015-03-29 12:41 (UTC), posted by SE-user Mitchell Porter

Answer 2

I just fixed the ISO commutation relations to match equations (2.9) of Witten.

The first set of commutation relations represents for any nonzero value of $\lambda$ the Lorentz group $SO(3,1)$, the second those of $ISO(2,1)$. (That the value of $\lambda$ does not matter as long as it is nonzero can be seen by rescaling the $P_a$.)

$ISO(d-1,1)$ is a contraction of $SO(d,1)$ obtained by taking the limit $\lambda\to 0$. Both sets of formulas are only valid for $d=3$. (For other $d$, one cannot form the vector $j$, since the Levi-Civita symbol $\epsilon$ is defined only for $d=3$.) For other dimensions on must work with the tensor $J_{ab}$ instead, but then has analogous relations, though they look a bit different. $d=3$ is special as the antisymmetric matrices that make up the Lie algebra of $SO(3)$ are equivalent to axial 3-vectors through the formula for the corresponding basis vectors given directly above Witten's (2.8).

Contractions and $ISO(d-1,1)$ are, e.g., explained here; see also here.