  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.
