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Upon compactification on a circle, SU(N) gauge theory with all fields in theadjoint representation acquires a $Z_N$ global symmetry because the center ofthe 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/CFTcorrespondence, and why the symmetry group is $Z_N$ rather than U(1). Thisprovides a test of the AdS/CFT correspondence for finite N. If the theory isformulated on $R^3 \times S^1$ with anti-periodic boundary conditions forfermions around the $S^1$, the topological symmetry is spontaneously broken; weshow that the domain walls are D-strings, and hence that flux tubes associatedwith magnetic confinement can end on the domain walls associated with thetopological symmetry. For the (0,2) $A_{N-1}$ superconformal field theory insix dimensions, we demonstrate an analogous phenomenon: a $Z_N$ global symmetrygroup arises if this theory is compactified on a Riemann surface. In this case,the domain walls are M-theory membranes.Comment: 14 pages, harvma

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