# String worldsheets with periodic time?

If we try to work out the details of string theory over a worldsheet with genus 1, does that mean we have periodic time on the worldsheet? But if we have periodic time, and in some portions of the worldsheet, the target space time $X^0$ increases with $\tau$, won't it have to decrease with $\tau$ somewhere else on the worldsheet? But wouldn't such a decrease mean we have an anti-string mode?

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Dvořák, yes, unless the spacetime coordinate $X^0$ itself is a periodic variable, and it shouldn't be to avoid closed time-like curves, the $X^0$ has to decrease with $\tau$ as much as it increases.
And indeed, you may say that there are "antistrings": after all, one-loop diagrams such as torus correspond to the creation of particle-antiparticle pairs - which may be called "string-antistring" pairs. Of course, if we only talk about whole string and not their particular excited states, an antistring is nothing else than the original string - it's just oppositely oriented in space (it may have the opposite winding numbers, if there are any, too). Note that the reversed orientation of the time on the world sheet may be fully compensated by the reversed orientation of the spatial coordinate along the world sheet - only the overall $\epsilon_{\mu\nu}$ antisymmetric tensor on the world sheet knows about the orientation of the whole world sheet.
However, it's usually considered that the world sheet is Euclidean (signature) and such a world sheet should naturally be embedded into the Euclidean spacetime, too. In the Euclidean spacetime, there is no sharp difference between future-directed and past-directed vectors; after all, all vectors in the Euclidean spacetime are spacelike. So the sharp difference in the future/past sign disappears. The coordinate $\tau$ is just an auxiliary variable and one shouldn't be shocked that functions are not monotonic functions of $\tau$.