Let $M$ be a Lorentzian spin $4$ manifold, i.e. admits a spin structure $Spin^+(M)\rightarrow M$, which is just a principal $Spin^+(1,3)$ bundle over $M$, which is compatible with the bundle of oriented, and time oriented, orthonormal frames over $M$, denoted $SO^+(M)$. Furthermore, let $P\rightarrow M$ be a principal $G$ bundle over $M$, where $G$ is a compact Lie group. We can then write the Yang-Mills-Dirac Lagrangian on $M$ as:
\begin{align*} \mathscr{L}_{YMD}[A,\Psi]=-\frac{1}{2}\langle F^A_M,F^A_M \rangle_{Ad(P)}-m^2\langle \Psi,\Psi\rangle_{S\otimes E}+\Re\left(\langle\Psi,D_A\Psi \rangle_{S\otimes E}\right). \end{align*} Here $F^A_M$ is the curvature two form of a gauge potential $A$, $Ad(P)$ is the adjoint bundle $Ad(P)=P\times _{Ad}\mathfrak{g}$, $S=Spin^+(M)\times _{\kappa}\mathbb{C}^4$ is the spinor bundle (where $\kappa$ is the spin representation), $E=P\times_\rho V$ is a vector bundle associated to $P$ (where $\rho$ is a representation of $G$ on $V$), making $S\otimes E$ a twisted spinor bundle, and $\Psi$ a gauge multiplet spinor field. Finally, $D_A$ is the twisted Dirac operator, which is given in a local gauge and orthonormal frame $e_i$ by \begin{align*} D_A\psi=\gamma^i\left(d\psi(e_i)+\frac{1}{4}\xi_{ab}(e_i)\gamma^{ab}\psi+\rho_*(A(e_i))\psi\right) \end{align*} where $\gamma^i$ is a gamma matrix, and $\gamma^{ab}=\frac{1}{2}[\gamma^a,\gamma^b]$. The one forms $\xi_{ab}$, are the one forms defining the Levi-Civita connection in this orthonormal frame, i.e. $$\nabla e_a=\xi_{ab}\eta^{ab}\otimes e_b.$$
I know that this is only a "classical" description of this picture, as I am not doing anything with path integrals, but I have noticed that two of the objects in $\mathscr{L}_{YMD}$ depend on the metric $g$, namely the Yang-Mills term, and the term which includes the twisted Dirac operator. Furthermore, when I pass to the action: \begin{align} S_{YMD}=\int_M \mathscr{L}_{YMD}[A,\Psi]\text{dvol}_g \end{align} where in coordinates $$\text{dvol}_g=\sqrt{-\det g}dx^1\wedge \cdots \wedge dx^4$$ each term will at least mildly depend on the metric if I vary the action with respect to $g$.
My question is then this: If I add the Einstein Hilbert action to the Yang-Mills-Dirac action, will this then describe a theory of gravity where the matter Lagrangian is $\mathscr{L}_{YMD}$? I know as a quantum theory, GR ends up being non-renormalizable, but does this set up describe the Lagrangian of that non renormalizable theory?
