Anomalous function of a Lorentz-violating QED effective action and the relation between compact bulk scalar propagator and path integral duality

In this paper, we consider a compact five dimensional spacetime with the structure $\mathcal{M}^{1,3}\times S^{1}$. Generally speaking, motion on such a structure will break Lorentz invariance, allowing for causal bulk signals to propagate superluminally. Based on recent articles, we calculate the anomalous function of a gauge invariant but Lorentz-violating term in the $4D$ QED effective action by using path integral. Finally, we find that the compact bulk scalar propagator and path integral duality are consistent, this result brings a new perspective: the behavior of breaking Lorentz invariance caused by dimensional compactness can be seen as path integral duality.