Space-Time and Topological Orbifolds

I consider string propagation in a certain complexified version of space-time, in an attempt to gain some intuition about a possible unbroken phase of string theory. Topological \ensuremath{\sigma} models (related to Floer and Gromov theory) correspond from this point of view to the consideration of only strings with a selfdual pattern of momenta and windings. I also observe an amusing generalization of the high-energy saddle point studied by Gross and Mende; this generalization involves complex space-time and separate Riemann surface moduli for left-moving and right-moving modes.