Christoffel symbols and Dirac matrices mathematical similarities? Maybe mine is a silly question, but are there mathematical similarities or common roots between the Christoffel symbols:
$ \nabla - \partial = \Gamma $
and the Dirac matrices $ ( \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2g^{\mu\nu}I )$ obtained through:
$ (\gamma^\nu \partial_\nu)(\gamma^\mu \partial_\mu) = \partial^\mu \partial_\mu $
EDIT: What I mean is that the Dirac matrices are obtained by trying to match two different derivatives, so I was wondering if that had some common ground with the Christoffel symbols that are defined as the difference between the connection  and the coordinate derivative.
 A: OK, I'm getting to this a little later than I originally thought I would but hopefully it's ok :)
So let's go back to the first equation you wrote down, but rearrange it a bit:
\begin{equation}
\nabla = \partial + \Gamma
\end{equation}
Now if I was teaching a first year GR course and someone showed that to me, I would yell at them because, for example, $\nabla \phi =\partial \phi$ with no Christoffel symbols. 
However, if we are willing to be a bit loose with our notation, then there's a sense in which that equation is ok. We just have to say that $\Gamma$ does not refer to the Christoffel symbols explicitly, but rather is a schematic statement saying that there is a connection piece of the covariant derivative that depends on the representation of the object that the covariant derivative acts on. If the object is a scalar, $\Gamma=0$. If the object is a vector, you get the normal christoffels. If the object is a tensor, $\Gamma$ stands for the specific combination of christoffels appropriate for that tensor. It's just like writing $D_\mu=\partial_\mu+ig A_\mu$ for a Yang Mills theory, strictly speaking you don't know what $A_\mu$ is until you know the representation of the object that the covariant derivative acts on.
So with that in mind, let's switch to the Einstein Cartan formalism. Instead of working with the metric $g_{\mu\nu}$ and associated connection $\Gamma^\mu_{\rho\sigma}$ as our variables, we work with the vielbein $e^a_\mu$ and the associated "spin connection" $\omega^{ab}_\mu$. The vielbein is defined by
\begin{equation}
g_{\mu\nu} = \eta_{ab} e^a_\mu e^b_\nu
\end{equation}
and it has a lot of nice properties you can read about elsewhere. The $\mu$ index is the standard tangent space index. Both $e$ and $\omega$ are spacetime one-forms, as you can see. The $a,b$ indices can be thought of as internal indices corresponding to an internal Lorentz group.
The main point is that there is an analogue of $\nabla$ for objects with local lorentz indices. We can call it $D$, and we can write a similarly schematic equation
\begin{equation}
D = d + \omega
\end{equation}
This formalism is most useful when dealing with forms (ie, with objects who can have any number of local lorentz $a,b$ indices, but all of whose spacetime indices $\mu,\nu$ are (1) downstairs and (2) totally antisymmetric), so the $\Gamma$ piece of the covariant derivative drops out. Hence I used the exterior derivative $d$ instead of the partial $\partial$.
This $D$ is a covariant derivative acting on the local lorentz indices. If I do a local lorentz transformation (which as it says on the tin is a local symmetry), this covariant derivative behaves like a local lorentz tensor.
Now acting on local lorentz scalars, $\omega=0$. Acting on local lorentz vectors you can work out the appropriate expression for $\omega$. Just for clarity, the covariant derivative acts on on lorentz vectors as
\begin{equation}
D_\mu V^a = \partial_\mu V^a + \omega^{ab}_\mu V^a
\end{equation}
where you have to work out the components of $\omega^{ab}_\mu$.
You should suspect that there should be some relation between the components of $\omega$ acting on a Lorentz vector, and the components of the usual Christoffel symbols which are the connection relevant for acting on a spacetime vector. Indeed there is such a relationship, it is
\begin{equation}
\omega^{ab}_\mu=e_\nu^a \partial_\mu e^{\nu,b} + e_\nu^a e^{\sigma b} \Gamma^\nu_{\sigma\mu}
\end{equation}
(the $e$ with an upper spacetime index $e^\mu$ is an inverse vielbein, ie the matrix inverse of the original vielbein).
Now the nice thing about this derivative $D$ is that unlike $\nabla$, it can act on spinors. The spinor carries no spacetime index but has 1 local lorentz index that lives in the spinor representation. In that representation,
\begin{equation}
D_\mu \psi = \partial_\mu \psi - \frac{i}{4}\omega^{ab}_\mu [\gamma^a,\gamma^b]\psi
\end{equation}
Here $\omega^{ab}_\mu$ is the same $\omega^{ab}_\mu$ used above. The factor of $-i/4$ depends on the specific gamma matrix conventions you used, I just stole this from wikipedia so I don't know the precise convention.
So in the schematic notation $D=d+\omega$, we see $\omega^{spinor} = -\frac{i}{4}\omega^{ab}_\mu[\gamma^a,\gamma^b]$. (It might be confusing that $\omega^{spinor}$ involves $\omega^{ab}_\mu$, but that's the cost of using schematic/sloppy notation. The point is that the connection piece of $D$ involves a $\Gamma$-like object when $D$ acts on local lorentz vectors, and involves the $\gamma$ matrices acting on local lorentz spinors).
So that's at least one sense in which your original statement is correct: there is a connection between the christoffels and the dirac matrices, due to working out properties of covariant derivatives. There is an object, $\omega$, which in different representations reduces to (1) essentially the chrisoffel symbols, or (2) the commutator of dirac matrices. 
