Why gamma-matrices are associated with tetrads Lorentz rotation? In Zee's "QFT in nutshell" in a paragraph "Differential geometry of Riemann manifold" he states that Dirac gamma-matrices are associated with tetrads Lorentz rotation, so Dirac lagrangian in curved spacetime is written in a form
$$
\tag 1 L =\bar{\psi}\left(i\gamma^{a}\eta_{ab}e^{b}_{\mu}g^{\mu \nu}\partial_{\nu}-m \right)\psi. 
$$
Here $\eta_{ab}$ is Minkowski tensor, $e_{\mu}^{a}$ is tetrad. 
Hown to understand that statement? How to associate gamma-matrices with tetrades rotation? How from that statement follows the explicit form of lagrangian $(1)$?
 A: In order to deal with fermion fields in general relativity, that are representations of the Lorentz group, you want to work in a local inertial frame (LIF). To go back and forth between space-time indices $\mu=0,1,2,3$ and LIF indices $a=0,1,2,3$ you introduce the formalism of frame fields:
$$g_{\mu \nu}=e^a_\mu (x) \eta_{ab} e^b_\nu (x)$$
The generalization of gamma matrices to curved space is:
$$\gamma^{\mu}(x)=e_a^\mu (x)\gamma_{a} $$
where $\gamma^{a}$ are the usual constant matrices of quantum field theory in flat space. Dirac action is generalized to:
$$S=\int d^4x \sqrt{-g} (\bar{\psi}e_a^\mu\gamma^{a} D_{\mu} \psi-m\bar{\psi}\psi)$$
In which $D_{\mu}\psi_{\nu}=\partial_{\mu}\psi_{\nu}+\frac{1}{4} \omega_{\mu ab}\gamma^{ab}\psi_{\nu}-\Gamma_{\mu\nu}^{\rho}\psi_{\rho}$ for a spinor-vector field $\psi_{\nu}$. Actually for the spinor case $\psi$ of your question there isn't the $\Gamma $ piece in $D_{\mu}$. $\omega$ is the spin connection.
Reference: Freedman-Van Proeyen, Supergravity, p.178
