Toroidal Compactification Without Vector Structure

Many important ideas about string duality that appear in conventional $\T^2$ compactification have analogs for $\T^2$ compactification without vector structure. We analyze some of these issues and show, in particular, how orientifold planes associated with $Sp(n)$ gauge groups can arise from T-duality and how they can be interpreted in F-theory. We also, in an appendix, resolve a longstanding puzzle concerning the computation of $\Tr (-1)^F$ in four-dimensional supersymmetric Yang-Mills theory with gauge group SO(n).