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

Upon compactification on a circle, SU(N) gauge theory with all fields in th e 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(N) topological symmetry arises in the context of the AdS/CFT corres pondence, and why the symmetry group is Z(N) rather than U(1). This provide s a test of the AdS/CFT correspondence for finite N. If the theory is formu lated on R-3 x S-1 with anti-periodic boundary conditions for fermions arou nd the S-1, the topological symmetry is spontaneously broken; we show that the domain walls are D-strings, and hence that flux tubes associated with m agnetic confinement can end on the domain walls associated with the topolog ical symmetry. For the (0; 2) A(N-1) superconformal field theory in six dim ensions, we demonstrate an analogous phenomenon: a Z(N) global symmetry gro up arises if this theory is compactified on a Riemann surface. In this case , the domain walls are M-theory membranes.