- #1
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Hi,
I'm rather confused about the procedure in which people obtain gravity from gauged (super)-Poincare algebras. Let me outline what this procedure is.
*First you gauge the Poincare algebra with generators P and M
*You obtain two gauge fields: the vielbein (associated with P) and the spin connection (associated with M)
*After that you put the curvature of P, called R(P), to zero: R(P)=0
Now, this R(P)=0 constraint has two effects:
1) The spin connection becomes a dependent field; the number of constraints equals the number of components of the spin connection, and thus it can be solved
2) One can "exchange" the P-transformations for general coordinate transformations (gct's), which is what you want: in a theory of gravity there are no P-transformations, but just gct's and local Lorentz transformations (GR can as such be defined on a tangent bundle with these two transformations as right- and left transformations)
However, this second step is not clear to me. Gauging the Poincare algebra is done a la Yang-Mills, so the algebra is realized on the gauge fields. So the parameter of the P-transformations is NOT a vector lying in the tangent space, right? It's just an internal parameter. However, in the end you want the Local Lorentz transformations to act in the tangent space, so where do you make this identification?
Also, this exchanging of the P-transformations, which is often described in texts about supergravity, is not clear to me. The relation in the Poincare case says that
<br /> \xi^{\lambda}\partial_{\lambda}e_{\mu}^a + \partial_{\mu}\xi^{\lambda}e_{\lambda}^a = \xi^{\lambda}R_{\lambda\mu}^a (P) + \delta_{P}(\xi^{\lambda}e_{\lambda}^b)e_{\mu}^a + \delta_M(\xi^{\lambda}\omega_{\lambda}^{ef})e_{\mu}^a<br />
The LHS is a gct, and so we see that putting R(P)=0 we get a relation between gct's, P-transformations with a gauge parameter involving the vielbein, and a local Lorentz transformation involving the spin connection. However, these are NOT the usual gauge transformations, so how is this "exchange" precisely done?
Also, obviously you change the gauge algebra, and is it guaranteed that it will still close on the fields?
Thanks in forward!
I'm rather confused about the procedure in which people obtain gravity from gauged (super)-Poincare algebras. Let me outline what this procedure is.
*First you gauge the Poincare algebra with generators P and M
*You obtain two gauge fields: the vielbein (associated with P) and the spin connection (associated with M)
*After that you put the curvature of P, called R(P), to zero: R(P)=0
Now, this R(P)=0 constraint has two effects:
1) The spin connection becomes a dependent field; the number of constraints equals the number of components of the spin connection, and thus it can be solved
2) One can "exchange" the P-transformations for general coordinate transformations (gct's), which is what you want: in a theory of gravity there are no P-transformations, but just gct's and local Lorentz transformations (GR can as such be defined on a tangent bundle with these two transformations as right- and left transformations)
However, this second step is not clear to me. Gauging the Poincare algebra is done a la Yang-Mills, so the algebra is realized on the gauge fields. So the parameter of the P-transformations is NOT a vector lying in the tangent space, right? It's just an internal parameter. However, in the end you want the Local Lorentz transformations to act in the tangent space, so where do you make this identification?
Also, this exchanging of the P-transformations, which is often described in texts about supergravity, is not clear to me. The relation in the Poincare case says that
<br /> \xi^{\lambda}\partial_{\lambda}e_{\mu}^a + \partial_{\mu}\xi^{\lambda}e_{\lambda}^a = \xi^{\lambda}R_{\lambda\mu}^a (P) + \delta_{P}(\xi^{\lambda}e_{\lambda}^b)e_{\mu}^a + \delta_M(\xi^{\lambda}\omega_{\lambda}^{ef})e_{\mu}^a<br />
The LHS is a gct, and so we see that putting R(P)=0 we get a relation between gct's, P-transformations with a gauge parameter involving the vielbein, and a local Lorentz transformation involving the spin connection. However, these are NOT the usual gauge transformations, so how is this "exchange" precisely done?
Also, obviously you change the gauge algebra, and is it guaranteed that it will still close on the fields?
Thanks in forward!