In string theory we study maps $X: \Sigma \to M$, where $\Sigma$ is the two dimensional worldsheet of the string and $M$ is the target manifold. When studying non-linear sigma models, for instance when we are looking at the Polyakov action for string theory, $\Sigma$ is often endowed with the two dimensional Minkowski metric.
From a topological perspective, however, it is well-known that a (compact) manifold admits a Lorentzian metric iff the Euler characteristic vanishes. This should mean that we can only take the metric on $\Sigma$ to be the Minkowski metric for genus 1 worldsheets. What do we do for other genus worldsheets?
(If the answer is something along the lines of "Wick rotate so you have a Riemannian signature", then my question is "Why is this a sensible thing to do?")