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?


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