Note on Gauge Theories on $M/\G$ and the AdS/CFT Correspondence

It is well known that a weakly coupled $U(N)$ gauge theory on a torus with sides of length $L$ has extra light states with energies of order $1/(NL)$. We show that a similar result holds for gauge theories on $M/\G$ where $M$ is any compact Riemannian manifold and $\G$ is any freely acting discrete isometry group. As in the toroidal case, this is achieved by adding a suitable nontrivial flat connection. As one application, we consider the AdS/CFT correspondence on spacetimes asymptotic to $AdS_5/\G$. By considering finite size effects at nonzero temperature, we show that consistency requires these extra light states of the gauge theory on $S^3/\G$.