Subgroups of $GL(4, \mathbf{R})$ as gauge groups?

In the present notes we always consider the symmetry group $\mathcal{F}^G$ acting on the tangent spaces of $\mathcal{S}$ as a global symmetry, namely the transformation acting on $T_s \mathcal{S}$ does not depend on $s$ and can be considered as a transformation of the linear space $\mathcal{T}$. Of course, it must preserve the cone $\mathcal{T}^+$, namely it must be a subgroup of $SL(4, \mathbf{R})$. Note that, while $SL(2, \mathbf{C})$ is considered as a gauge group on the spacetime $\mathcal{M}$ it represents a global symmetry when considered as a subgroup of $\mathcal{F}^G$.

One may ask if $\mathcal{F}^G$ can be considered as a gauge group, namely if a different group element can act on different tangent spaces. Of course one has to give up the absolute parallelism of $\mathcal{S}$ and therefore the operational interpretation of the vector fields described in Section 2.2. This interpetation can be criticized as a realistic description of the physical operations, but it provides a very useful guide for the theoretical intuition. We think that we cannot renounce to this guide in the present stage of the theoretical research.

We note that $\mathcal{S}$ is a generalization of a principal bundle, namely of the mathematical structure that describes a gauge theory defined on the spacetime manifold $\mathcal{M}$. A gauge theory defined on $\mathcal{S}$ would deserve the name of ``second gauge theory'' in analogy with the denomination ``second quantization'' used in the old times to indicate the construction of a quantum field theory. Then, why not to introduce a ``third gauge theory'' and so on?