A black hole toy model with non-local and boundary modes from non-trivial boundary conditions

We study gauge theories between two parallel boundaries with non-trivial boundary conditions, which serve as a toy model for black hole background with two boundaries near the horizon and infinite, aiming for a better understanding of the Bekenstein-Hawking entropy. The new set of boundary conditions allows boundary modes and non-local modes that interplay between the two boundaries. Those boundary modes and Wilson lines stretched between the two boundaries are carefully analyzed and are confirmed as physical variables in the phase space. Along with bulk fluctuation modes and topological modes, the partition function and entropy of all physical modes are evaluated via Euclidean path integral. It is shown that there are transitions between the dominance of different modes as we vary the temperature. The boundary fluctuation modes whose entropy is proportional to the volume dominate at high temperatures, and the boundary-area scaled boundary modes and Wilson lines are the more important at low temperatures. At super-low temperatures, when all the fluctuation modes die off, we see the topological modes whose entropy is the logarithm of the length scales of the system. The boundary modes and non-local modes should have their counterparts in a black hole system with similar boundary conditions, which might provide important hints for black hole physics.