GR as a gauge theory: there's a Lorentz-valued spin connection, but what about a translation-valued connection?

Question

Given an internal symmetry group, we gauge it by promoting the exterior derivative to its covariant version:

$$ D = d+A, $$

where $A=A^a T_a$ is a Lie algebra valued one-form known as the connection (or gauge field) and $T$ the algebra generators.

For GR, we would like to do the same thing with the Poincaré group. But the Poincaré group isn't simple, but rather splits into translations $P$ and Lorentz transformations $J$. I would thus expect two species of connections:

$$ D = d + B^a P_a + A^{ab} J_{ab}. $$

But the covariant derivative of GR as usually found in textbooks is:

$$ \nabla_\mu = \partial_\mu +\frac{1}{2}(\omega^{\alpha \beta})_\mu J_{\alpha \beta}, $$

where $\omega$ is the spin connection. It is defined for any object that has a defined transformation under $J$, i.e., under Lorentz transformations, like spinors or tensors. But it makes no mention of the translation generator $P$. What happened? Shouldn't I have this extra gauge field?

Answer 1

One can make a Poincare-Lie-Algebra-valued Cartan Connection by setting $$ \eta = \tau_a {\bf e}^{*a} + \frac 12 \sigma^{ab} \omega_{ab} $$ where $\tau_a$ and $\sigma^{ab}$ are the Lie algebra generators of translation and Lorentz transformations, and ${\bf e}^{*a}= e_\mu^a \,dx^\mu$, $\omega_{ab}= \omega_{ab\mu}\,dx^\mu$ are the co-frame and connection one-forms. Then some use of the Poincare Lie algebra shows that the curvature decomposes as
$$ F\equiv d\eta+ \eta\wedge \eta= \tau_a T^a +\frac 12 \sigma^{ab} R_{ab}, $$ where $T^a = d{\bf e}^{*a}+ {\omega^a}_b \wedge {\bf e}^{*b}$ is the torsion and ${R^a}_b= d{\omega^a}_b+ {\omega^a}_c \wedge{\omega^c}_b $ the usual Riemann curvature. Thus the torsion is seen as the translation part of the curvature.

People go on to make some sort of gauge theory out of this, but it is not a conventional principle-bundle gauge theory, and I have never understood how the physical fields become sections of an associated bundle. A standard reference is Reviews of Modern Physics Vol 48, no 3 (1976) General Relativity with Spin and Torsion, Foundation and Prospects, by F W Hehl, P von de Heyde and G D Kerlick. Personally I find their notation impenetrable, but that is my failing, I suppose.

Answer 2

In 2+1 dimensions general relativity with Einstein–Hilbert action with or without
cosmological constant is equivalent to a gauge theory with a gauge group one of $\mathrm{ISO}(2,1)$, $\mathrm{SO}(3,1)$ or $\mathrm{SO}(2,2)$ (depending on the presence of cosmological constant and its sign) and a pure Chern–Simons action.

The gauge field is a Lie-algebra-valued one form $$ A_i= e^a_i P_a + \omega^a_i J_a,$$ where $P_a$ are translation generators and $J_a=\frac 12 \epsilon_{abc}J^{bc}$ are generators of Lorentz transformations.

A good reference for this is a paper:

• Witten, E. (1988). $2+1$ dimensional gravity as an exactly soluble system. Nuclear Physics B, 311(1), 46-78, doi, online pdf.

This paper also contains the following passage on the four dimensional case:

In the last twenty years, many physicists have wished to combine together the vierbein $e_i{}^a$ and the spin connection $\omega_i{}^a{}_b$ into a gauge field of the group $\mathrm{ISO}(d - 1,1)$. The idea is that the spin connection would be the gauge field for Lorentz transformations, and the vierbein would be the gauge field for translations. One then tries to claim that “general relativity is a gauge theory of $\mathrm{ISO}(d - 1,1)$”. However, there has always been something contrived about attempts to interpret general relativity as a gauge theory in that narrow sense. One aspect to the problem is that in four dimenisons, for instance, the Einstein action (2.2) is of the general form $\int e \wedge e \wedge (d \omega + \omega^2)$. If we interpret $e$ and $\omega$ as gauge fields, we should compare this to a gauge action $\int A \wedge A \wedge (dA + A^2)$. But there is no such action in gauge theory. So we cannot hope that four-dimensional gravity would be a gauge theory in that sense.