Algebraic Geometry in String Theory?

Solution 1:

I only know about string theory from a (rather great) distance, and with the perspective of a pure mathematician who has colleagues in mathematical physics who think about the theory (some of whom were trained as physicists).

With this warning given, let me say that it seems to me that it would be near impossible to understand string theory without some understanding of algebraic geometry. I would adopt an analytic point of view, such as in the book by Griffiths and Harris (Principles of algebraic geometry), since this is going to be closer to the language that physicists use than a more algebraic treatment. You could also look at the books Quantum fields and strings: a course for mathematicians, by Deligne, Witten, et. al., which is based on a year long series of courses given at the IAS in 96-97, by Witten among others. I don't know how comprehensible these will be (since they are written from the point of view of leading those with rather strong mathematical training into some kind of understanding of the physics), but they may give an idea of what kind of geometry you should learn, and what kind of perspectives on that geometry would be useful.