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Hello, I'm curious on whether say, a ph.d in mathematics with no experience in physics could pick up a book on String theory (say some intro for mathematicians) and learn it and then do research kinda "quickly"? From what I've understood, string theory is mostly mathematics so I would be interested in knowing.

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@Wadim, what does WUDU's vocabulary say to you? –  Mariano Suárez-Alvarez Jul 30 '10 at 1:09
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Ok, guys, let's forget vocabulary and background for a moment and let's assume the question is this: are there any sources that present string theory or other parts of physics relevant to mathematics in a way accessible to mathematicians? I know a couple of references (2 IAS volumes, Dolgachev's lectures etc.) but I'd be willing to know more, in particular since my impression of the above sources is that they all concentrate on concrete calculations rather than try to present a general picture. –  algori Jul 30 '10 at 1:20
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@algori: There are the two big mirror symmetry volumes published by the Clay Institute. The first is titled "Mirror Symmetry", is red/yellow, and has a bunch of authors (including Our Benefactor Ravi Vakil). The second is titled "Dirichlet Branes and Mirror Symmetry", is blue/orange, and also has a bunch of authors. –  Kevin H. Lin Jul 30 '10 at 2:03
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Kevin -- thanks, I'll take a look at those as soon as I get to the library. Andrew -- why would they ever want to critize your grammar? (Only joking; on a more serious note, this website is for discussing mathematics, not grammar, so you shouldn't pay too much attention to this kind of criticism.) –  algori Jul 30 '10 at 2:12
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Is there an area of mathematics that you can reads about and then do research "quickly"? Just looking at the author names of the Mirror Symmetry book suggested shows how many different areas of mathematics are involved. It seems to be all the motivation for constructions in string theory comes from physics, and i imagine it would be hard to do research without understanding the motivations and the goals. –  Sean Tilson Jul 30 '10 at 3:07

2 Answers 2

OK, I'll bite -- although I am not convinced that this is a suitable question for MO.

First of all, it depends what you mean by 'string theory'. There is mathematics which has been influenced (some deeply, some less so) by physics research billed as string theory:

  • Mirror symmetry comes to mind as one area where the influence has been crucial. To the two references in Wadim's comment, you could add the freely available Mirror symmetry (PDF) with contributions by Hori, Katz, Klemm, Pandharipande, Thomas, Vafa, Vakil and Zaslow.

  • Stability conditions in derived categories, particularly the work of Tom Bridgeland, taking inspiration from the work by string theorist Michael Douglas and collaborators.

  • Vertex operator algebras, conformal field theory,...

However, I would not say that working on mathematics which has been touched by string theory, necessarily implies that you are working on string theory.

If you really want to work on string theory, you have to learn some physics, if only because that is the language which the majority of practitioners still speak. A good resource (for mathematicians) to get you started are the lectures at the IAS activity Quantum Field Theory in the late 1990s.

One has to wonder, though, whether without any background in physics you really want to go down this path.

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It is definitely possible to start research in string theory without any background in physics. What is less certain is whether it is easily done by reading a book. The difficulty is to identify a good research problem that can be finished in a finite amount of time. A better strategy perhaps would be to scan the abstracts and articles in the hep-th arxiv, and to focus on papers that use the kind of mathematics that you're interested in. (A fairly recent book that touches on many of the current directions in string theory is "String Theory and M-Theory: A Modern Introduction", by Katrin and Melanie Becker and John Schwarz.)

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