Anti-de Sitter space and the center of the gauge group

Upon compactification on a circle, SU(N) gauge theory with all fields in the adjoint representation acquires a Z_N global symmetry because the center of the gauge group is Z_N. For N = 4 super Yang-Mills theory, we show how this Z^ topological symmetry arises in the context of the AdS/CFT correspondence, and why the symmetry group is Z_N rather than U(1). This provides a test of the AdS/CFT correspondence for finite N. If the theory is formulated on R³ x S¹ with anti-periodic boundary conditions for fermions around the S¹, the topological symmetry is spontaneously broken; we show that the domain walls are D-strings, and hence that flux tubes associated with magnetic confinement can end on the domain walls associated with the topological symmetry. For the (0, 2) A_N-1 sup er conformai field theory in six dimensions, we demonstrate an analogous phenomenon: a ZN global symmetry group arises if this theory is compactified on a Riemann surface. In this case, the domain walls are M-theory membranes.