Gauge theories with non-trivial boundary conditions: Black holes

We study the partition function and entropy of U(1) gauge theories with multiple boundaries on the black holes background. The nontrivial boundary conditions allow residual zero longitudinal momentum modes and Wilson lines stretched between boundaries. Topological modes of the Wilson lines and other modes are also analyzed in this paper. We study the behavior of the partition function of the theory in different temperature limits, and find the transitions of dominances of different modes as we vary the temperature. Moreover, we find two different area contributions plus logarithm corrections in the entropy. One being part of the bulk fluctuation modes can be seen for finite-temperature black holes, and the other coming from vacuum degeneracy can only be seen in the superlow temperature limit. We have confirmed the mechanism and entropy found in the superlow temperature limit also persist for extremal black holes. The gauge fluctuation on the black hole background might help us understand some fundamental aspects of quantum gravity related to gauge symmetries.