Suppose I have a local gauge group $g$ that is a $C^0$-function over Minkowski spacetime. This implies that the corresponding gauge connection $A_\mu$ is not continuous, i.e. it can have value infinity. More concrete, an interesting case would be the case where $A_\mu$ can have the following values:

Either zero or $\infty$.

If $A_\mu$ is equal to $\infty$, the Yang-Mills action $S$ vanishes. One can define a boundary of the spacetime manifold (assumed as flat) $\partial M$ that separates the singularity of the connection. Would these assumptions lead to the theory

$\sum_{k_1, \dots, k_n=0}^1 \prod_{i=1}^n (\int \mathcal{D}[All. other. fields]e^{iS})_{k_i}$?

Here, the indices $k_i$ indicate from which points the expression $(\int \mathcal{D}[All. other. fields]e^{iS})_{k_i}$ will be separated. If for point number $i$ the value of $k_i$ is zero than the action is not defined on this region, otherwise it is defined. This would induce a sum over all manifold topologies.

Can such a theory be formulated?