Let us consider an orientable smooth 4-manifold $M$. Pick a vector bundle $T$ that's isomorphic to the tangent bundle $TM$. We then equip $T$ with a cotetrad (or coframe field) $e$ and (spin) connection $\omega$. This lets us describe a Riemannian metric on $M$ using a vector bundle map


$e: TM\rightarrow T$


We have that


$\omega\in so(4)$ (it's a $SO(4)$ connection) and $e\in \mathbb{R}^4$.


We can introduce now a $SO(5)$ connection $A$ such that


$A=\omega+e$ $\hspace{1.0cm}$(because $so(5)=so(4)\oplus \mathbb{R}^4$)


**My question is**: if I introduce a spin structure (spin bundle), can I define the following Dirac operator:


$D=\gamma^\mu(\partial\_\mu+A\_\mu)$ ?


Can I consider $A$ as a "pure" (gauge) connection as in the standard case?



Cotetrad, spin connection and Dirac operator
You have to be more precise. $A$ is an SO(5) connection but on what bundle? This is a formula for the Dirac operator which depends on local trivialization. I doubt that this defines a global operator in general (this works for a spin or $spin^c$ structures on TM and doesn't have to work on spin structure on every vetor bundle).


I will recommend you chapters "spin structures on vector bundles" and "dirac opetors" form book D.Salamon,Spin geometry and Seiberg-Witten invariants.


Hope it helped a little.


