how do string theorists study the mathematics required ?

Question

String theorists need to know algebraic geometry , algebraic topology , moduli spaces , characteristic classes etc. How do string theorists learn these topics ? I can't find an algebraic geometry textbook for physicists. If some one wants to develop new original ideas in string theory , He presumably need to know these topics in mathematics at a deep level. Is that right ?  Do one start learning algebraic geometry be reading a textbook such as Griffiths starting from the first chapter and so on ? Or just ignore the mathematics textbook and learn the required mathematics from physics sources ? I always find that knowing the general mathematical  theory makes many seemingly difficult constructions and ideas in physics look much easier. Is it worth it to spend a lot lot of time reading from textbooks on pure mathematics  if I'm eager to do the same kind of research that people such as Edward Witten do.  

Comments

Some other people might be able to give more on-topic advise, but hear this: Never, ever just read a book cover to cover. This will waste your precious time. Study with purpose. You first determine what it is that you actually need to understand and why you need to understand it. Then you learn just that and no more.

If you want to learn about string theory, first of all figure out if string theory builds on other physical theories. Find out why string theory uses the mathematics that you mention. There is always a good reason why physical theories use the mathematical structures and concepts that they do. Figure out what you then must learn and what you can ignore. Do not rely on one book to learn a topic and don't forget books aren't the only source of reliable information. A good teacher may be worth a dozen books, but you do have to know what to ask. Use many sources, physically and purely mathematically oriented. Compare how the sources differ in their treatment of your topic of choice. Form an opinion on the matter.

If you want to develop original ideas, you need to own your education. Just studying what other people tell you to study is not to your benefit, in my opinion.

---

One curious aspect of this is that existing string theory textbooks by and large remain at a rather superficial level that gives little impression of the mathematical substance. For instance the classical book "D-Branes" explains the all important gauge and gravitational anomalies (p. 161) essentially by saying "The beauty of the anomaly is that it is both a UV and an IR tool:".  On the one hand, it is what distinguishes physics students from mathematics students that they are used to be given such incomplete information, on the other hand there is a limit to how incomplete it may be while still being information. The textbook that presents string theory to the depth that it is actually understood maybe remains to be written. 

---

I've also been lately exploring some mathematics required for the more mathematical parts of string theory. It's my firm belief now that a full in-depth discussion of mathematics such as complex geometry, algebraic geometry in a way an advanced string theorist thinks about them does not exist. People praise Nakahara, but it's mostly just really a collection of definitions and theorems ripped off from proper math textbooks with no unifying theme whatsoever. For example the other day I was looking for an explicit coordinate formula for the Laplacian acting on p-forms and it was nowhere to be found in that book. The only good discussions I've found are very brief; chapters 12 and 15 in Volume 2 of Green Schwarz Witten, and the first few chapters of "Mirror Symmetry". Understanding geometry in the explicit and concrete way they present it should be your goal. And to me it seems like the only way of doing this is to pick up a certain result you think is important, pick up the machinery you need to know to derive/prove it from sources such as math textbooks, and try to "concretize" it.

---

I took a course on algebraic topology in a master's program in pure mathematics to fully grasp what algebraic topology is about. And to be honest I don't think a physicist would ever be able to understand algebraic topology if taught superficially. I suspect the same happens with algebraic geometry that I study now. Any string theorist would have the books of Hatcher, Griffiths and Harris or Hartshorne in their book selves. Indeed, the hard way is needed to learn those subjects. Even if you take the example of differential geometry. I studied differential geometry as physics undergrad and Master's student and found myself being in big trouble when doing it in the pure math course. The level of teaching is much deeper more detailed and only now I can say I do have some grasp in the fundamentals of differential geometry. The books I studied where by no means direct to physicists. String theorists know this and this is what makes them so special. In some cases they are as mathematicians as they are physicists and their knowledge and ability to use it is amazing.