From my basic understanding of very elementary topological string theory, we take the product of a Calabi-Yau threefold $X$ and our known and loved Minkowski space $M$, to get a ten-dimensional spacetime $M \times X$. Now, I know relativistically, the time "direction" depends on the observer in $M$, but regardless, the one time dimension definitely is one of the four dimensions of $M$.
It's this point which confuses me when it comes to topological strings studied as maps from algebraic curves into a fixed Calabi-Yau manifold. Certainly, identifying a string with an algebraic curve is taking into account its motion through time. And yet you're embedding it into a single fiber of $M \times X$ above some fixed point in $M$. In other words, the curve represents a string moving in time, yet since time is a direction in $M$, youre mapping the curve into something sitting at one point in time!
To take the simplest possible example, if you had a spacetime $\mathbb{R} \times S^{1}$, where the unbounded component was time, and the compact component was space, a string moving in time would be a cylinder living in this background. And yet the topological string/ Gromov-Witten slogan would be "let's map a cylinder into $S^{1}$."
Where exactly is my understanding abandoning me here?
Resolution: If I'm understanding correctly, my issue was arising from a confusion about the full, physical string theory vs. the topological sector of a string theory. I suppose topological string amplitudes compute highly interesting quantities (both mathematically and physically) despite this lack of an interpretation of moving in the time direction of the spacetime. As a mathematician, I'm more than happy with this resolution! I think in topological field theory, you have something similar: it may be very physically unrealistic to have something like the energy-momentum tensor having zero expectation value, but it nontheless, gives rise to an interesting topological sub-sector of a full, physical QFT.
