Compactification of spacetime in SU($\mathrm{\infty }$) Yang-Mills theory

The compactification on a torus in $SU(\mathrm{\infty })$ Yang-Mills theory is considered. A special form of the configuration of a gauge field on a torus is examined. The vacuum energy and free energy in the presence of fermions coupled with this background in the theory are derived and possible symmetry breaking is investigated.