Vortices on the world sheet and string's critical dynamics

Non-perturbative effects in the first quantized string propagating in non-simply connected space are considered. These effects are due to vortices on the string world sheet, which have to appear in properly regularized two-dimensional field theory describing the world sheet's dynamics. The strings on the lattices are discussed and the dynamical reasons for self dual lattices are suggested. It is shown that non-positively definite lattices, such as a hyperbolic, can be described as an antiferromagnetics in the formalism of functional integral. The critical dynamics on non-simply connected orbifolds and Calabi-Yau manifolds are shortly considered. The two new issues are: the field condensate mechanism different from the Hosotani one, and the open question about the real dimension of compact manifold.