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 vectorstructure. We analyze some of these issues and show, in particular, howorientifold planes associated with $Sp(n)$ gauge groups can arise fromT-duality and how they can be interpreted in F-theory. We also, in an appendix,resolve a longstanding puzzle concerning the computation of $\Tr (-1)^F$ infour-dimensional supersymmetric Yang-Mills theory with gauge group SO(n).Comment: 50 pp.,harvma