mathematical prerequisites for string theory

Question

I don't have a degree in physics or math and I'm trying to understand physics on my own. I just love playing with mathematical structures and find a way to use them in physics. What are the mathematical topics that one should master to do research in string theory ? To be more specific , I have been reading lee's smooth manifolds and Hatcher's algebraic topology but I think that it takes too long to finish them and solve most problems because I attempt to find concrete examples for everything I read . Is there a shortcut to be able to reach interesting applications of these things quickly and fully understand them at the same time ? What are the required topics for string theory ? What's the best way to read math books for someone who study physics ? Another unrelated question ,Is it possible to get into masters / PhD program without having studied physics at undergraduate level ?

Comments

 There seems to be too many and too soft questions mixed into a single thread, I suggest to move it to chat once it receives enough attention, say 30 views? Now that it has received a lovely answer, I retract my suggestion.

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Even people having followed all academic steps (university, PhD, etc.) have difficulties in finding a job for doing the researches of their interest.

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@VladimirKalivianski, true, but OP doesn't seem to be asking about job perspective.

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Do you know about physics stack exchange and physicsforums?

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@physicsnewbie I think this question is suitable here, although it may be better in chat, or broken into separate questions, but it's fine.

Answer 1

To learn the basics you need to understand in a reasonable level classical mechanics, quantum mechanics and special relativity. The conformal field theory you need will be presented on the string theory textbooks. Actually even the basics of special relativity and quantum mechanics is presented in Zweibach's "A first course in string theory".  Once you have understood the quantized string you will want to understand the spectrum. For this you will need a lot, but a lot, of group theory (including how to decompose spinor representations for different dimensions). Then, you will want to understand current algebras and thus a lot of CFT. Later you will need to understand BRST quantization (thus some elements of cohomology), D-branes etc. To understand string amplitudes you might as well need some alg. geometry and the story gets more complicated...

The best thing to do is to follow Susskind's "Theoretical Minimum" which should prepare you well to understand some basics of string theory, but it is definitely hard to understand well string theory without solid fundations in classical and quantum mechanics, special and general relativity and of course quantum field theory. If you want to professionally work on the area maybe it would be a good idea to do a Master's in theoretical or mathematical physics first. It is possible. When I was doing a masters one of my classmates was a chemical engineer and he excelled in the master (after many many hours of work though). As for the Hatcher and algebraic topology it wont really help you for a while. I cannot remember from the top of my head now if homotopy is really needed in any of the classic textbooks, and the only thing usually needed is some elements of cohomology (de Rham, BRST etc).

Good luck.

Comments

Thanks. This is a good answer. How is it possible to get into master/PhD program in theoretical physics without academic credentials , letters of recommendations and research experience? I do have a degree and my GPA is high but the degree subject is irrelevant (in biology )

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Look, this depends a lot in the country you will study at. If you have done a degree in biology then you must have taken some calculous and some physics courses and surely some professors might be able to give you a recommendation letter. If you say you have been studying algebraic topology and similar stuff maybe it won't be as hard to convince a committee to get in a physics program. It depends therefore on the universities you gonna apply to and the material you are going to present. Some people might put you forward to take some undergrad courses before you get admitted.