Toroidal compactification without vector structure

Many important ideas about string duality that appear in conventional T2 compactification have analogs for T2 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). (author)