(I wrote the introduction section for the sake of completeness, notation and study. The question per se, is written in the section "My Question")
Introduction
On the one hand of nature, we have gravity; after a course on general relativity we realize two basic things:
$1)$ The mathematical structure of spacetime (manifolds and its differential geometry)
$2)$ General Relativity (GR) deals with a fundamental interaction of nature called gravity; its dynamics, from quantum field theory $[1]$ to the large scale structure of universe $[2]$, is given by Einstein Field Equations:
$$\delta S_{g_{\mu\nu}} = \frac{-1}{16\pi}\int_{(\mathcal{M},g)} R\sqrt{-g}dx^{4}=0 \implies R_{\mu\nu} -\frac{1}{2}R g_{\mu\nu} = 8\pi T_{\mu\nu}. \tag{1}$$
On the other hand of nature, we have the standard model; after some quantum field theory we realize that the free fields of the fundamental interactions of nature are expressed in a flat spacetime background approximation, given by $[3]$:
$$\delta S_{\mathrm{SM}} = \int_{(\mathbb{R}^{4},\eta)} \Big\{\frac{-1}{4\pi} B_{\mu\nu}B^{\mu\nu} - \frac{1}{4\pi} W^{a}_{\mu\nu}W_{a}^{\mu\nu} - \frac{1}{4\pi} G^{a}_{\mu\nu}G_{a}^{\mu\nu} \Big\}dx^{4} . \tag{2}$$
Fibre Bundles
Given a manifold $\mathcal{M}$, we construct another manifold $\mathcal{B}_{\mathcal{M}}$, called fibre bundle. A fibre bundle is a mathematcal structure with the following algebraic elements $[4]$:
$1)$ A differentiable manifold $\mathcal{E}$ called the total space.
$2)$ A differentiable manifold $\mathcal{M}$ called the base space.
$3)$ A differentiable manifold $\mathcal{F}$ called the typical fibre (or just fibre).
$4)$ A surjection $\pi: \mathcal{E} \to \mathcal{M}$ called the projection.
$5)$ The inverse image $\pi^{−1}(p)$ called the fibre at $p$.
$6)$ A Lie group $G$ called the structure group, which acts on $\mathcal{F}$ on the left.
Therefore, the fibre bundle is the whole structure: $\mathcal{B}_{\mathcal{M}} = \big(\mathcal{E}, \mathcal{M},\pi,\mathcal{F},G\big)$ and a intuitive picture is given in Figure $1$.
Figure $1$: A fibre bundle
Important examples can be given:
Tangent Bundle: $\mathcal{TM} = \big(\mathcal{E}, \mathcal{M},\pi,T_{p}\mathcal{M},GL(m,\mathbb{R})\big)$
Principal Bundle: $\mathcal{P} = \big(\mathcal{E}, \mathcal{M},\pi,G,G\big)$
Fibre Bundles in Standard Model
The mathematical setting of the spacetime introduces the tangent bundles (which in basic GR seems to be almost irrelevant at a first glance). But, for the other interactions, the principal bundles plays a paramount role. The stage here is the minkowski spacetime $(\mathbb{R}^4,\eta)$. If you introduce just the electromagnetic interaction, we should expect the tangent and principal fibre bundles to be:
$\mathcal{T}\mathbb{R}^{4}_{\mathrm{Electromagnetism}} = \big(\mathcal{E}, (\mathbb{R}^{4},\eta),\pi,\mathbb{R}^{4},\mathcal{Poin}\big)$
Where $\mathcal{Poin}$ is the poincaré group.
$\mathcal{P}_{\mathrm{Electromagnetism}} = \big(\mathcal{E}, (\mathbb{R}^{4},\eta),\pi,U(1),U(1)\big)$
For the whole standard model:
$\mathcal{T}\mathbb{R}^{4}_{SM} = \big(\mathcal{E}, (\mathbb{R}^{4},\eta),\pi,\mathbb{R}^{4},\mathcal{Poin}\big)$
$\mathcal{P}_{SM} = \big(\mathcal{E}, (\mathbb{R}^{4},\eta),\pi,U(1)\otimes SU(2)\otimes SU(3),U(1)\otimes SU(2)\otimes SU(3)\big)$
My Question
Given a curved spacetime, we should expect a general tangent bundle as $\mathcal{TM} = \big(\mathcal{E}, \mathcal{M},\pi,T_{p}\mathcal{M},GL(m,\mathbb{R})\big)$ . But, I don't know for sure if the "principal bundle for gravity" is exactly:
Principal Bundle: $\mathcal{P} = \big(\mathcal{E}, \mathcal{M},\pi,GL(m,\mathbb{R}),GL(m,\mathbb{R})\big)$
My question is: what is the gauge group of gravity?.
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$[1]$ Birrell.N.D. & P.C.W.Davis Quantum Field Theory in Curved Space
$[2]$ Weinberg. S. Cosmology
$[3]$ https://en.wikipedia.org/wiki/Mathematical_formulation_of_the_Standard_Model
$[4]$ Nakahara.M. Geometry, Topology and Physics
