  We study the phase structures of N=4 U(N) super Yang-Mills theories on R x
S^3/Z_k with large N. The theory has many vacua labelled by the holonomy matrix
along the non-trivial cycle on S^3/Z_k, and for the fermions the periodic and
the anti-periodic boundary conditions can be assigned along the cycle. We
compute the partition functions of the orbifold theories and observe that phase
transitions occur even in the zero 't Hooft coupling limit. With the periodic
boundary condition, the vacua of the gauge theory are dual to various
arrangements of k NS5-branes. With the anti-periodic boundary condition,
transitions between the vacua are dual to localized tachyon condensations. In
particular, the mass of a deformed geometry is compared with the Casimir energy
for the dual vacuum. We also obtain an index for the supersymmetric orbifold
theory.
