Study of a gauge theory with a pure $Z(N)$ lattice in 4D
This is a solution attempt at the last problem from the 2025 Rudolf Ortvay competition proposed by Dávid Pesznyák. I want to develop my skills in this particular subject, so I paraphrase the problem here as a discussion. Firstly, there is many interest in describing our world with gauge theories for technical reasons such as renormalization of observables, but consider first that such theories involve many symmetries that are natural to rationalize. Question the fact that electrons all have the same charge or that systems with conservation of energy are far simpler to describe. Those can be viewed as symmetries (assumptions) that in many cases describe our world very well up to a given generalization. A common way to achieve a gauge theory is with a path integral formulation, which can be viewed through the lens of the lattice field theory method, making sense of infinite-dimensional integrals with a small spacetime lattice of length $\Lambda$. The gauge field $U_{\mu}\left(x\righ...