# Yang-Mills connections for Palatini $f(R)$ Theory

It is known that for some modified theories of gravity such as Palatini $$f(R)$$ Theory, we have two kinds of the connections: Affine connection and the Christoffel symbol, which are related to each other by variating the action with respect to the connection. For example for the Palatini $$f(R)$$ we have

$$\Gamma^{a}_{bc}= \{^{a}_{bc} \} +\Delta^{a}_{bc}$$

where $$\{^{a}_{bc} \}$$ is the Christoffel symbol and $$\Delta^{a}_{bc}$$ is a function of the derivative of the $$f(R)$$.

Now, I want to know which of these connections should be considered for the su(2) Yang-Mills algebra? Since the matter section of the Palatini theory is not a function of connection, contrary to metric-affine theory, does it make any difference in our chosen connection for the related Yang-Mills theory?