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I'm a mathematician. I want to be able to read recent ArXiv postings on high energy physics theory (String theory) (and perhaps be able to do research). I want to understand compactifications, Dualities, D-branes, M-branes etc. What's the easiest way to do so provided I have the necessary knowledge in algebraic geometry, algebraic topology, analysis and differential geometry?

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Is this a question about Mathematics research? You might get better answers on a Physics Q&A website, such as physics.stackexchange.com –  Mark Grant Dec 13 '12 at 8:35
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I think it's more properly under the soft question category . I want to know the answer from a mathematician perspective. –  nabil Dec 13 '12 at 8:45
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I support this question being posted on MO. I think some of the users here might have a better feeling of what a mathematician needs to learn to have good intuitions in string theory-related mathematics. Also, I wanted to ask such a question myself ;) –  Piotr Achinger Dec 13 '12 at 8:46
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Community Wiki, please. –  Todd Trimble Dec 13 '12 at 11:54
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@David Roberts, you probably mean this - math.ias.edu/qft –  Asaf Dec 13 '12 at 13:34
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6 Answers 6

Many string theorists would like to know more algebraic geometry. There are a few of us who know algebraic geometry at a pretty high level (not me) but many more who would like to learn more and feel it would help with their research but find the literature very difficult. I think the optimal solution would be to find such a string theorist and agree that you will teach them algebraic geometry if they will teach you string theory.

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That's a really nice idea. On the covers of the IAS volumes mentioned by David Roberts, there are cartoons in four panels. The first two panels are set in the mid-60's; in one is a pair of physicists at a blackboard all excited about gauge theory, in the other is a pair of mathematicians at another blackboard all excited about Atiyah-Singer index theory. The second set of panels is set thirty years later; in the first the two physicists are staring with cartoon question marks at Atiyah-Singer index theory, and in the second the two mathematicians are likewise puzzling over gauge theory (cont.) –  Todd Trimble Dec 13 '12 at 19:56
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Clearly the foursome should have gotten together much earlier. I'm sure there are similar blackboards with mirror symmetry on one, enumerative algebraic geometry on the other, and the two groups should somehow get together. –  Todd Trimble Dec 13 '12 at 19:58
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It would be fun, but would require a lot of time and dedication from both parties. I'm tempted to ask the converse, how does a physicist who knows QM, QFT and string theory learn algebraic geometry, or at least the parts that are most relevant to string theory? The standard answer seems to be to read the first few chapters of Griffiths&Harris and lecture notes by Candelas and others but I wonder if there is a better answer that doesn't involve a willing algebraic geometer. –  Jeff Harvey Dec 14 '12 at 1:26
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Candelas once told me that, when he asked Atiyah how to learn algebraic geometry, Atiyah responded: "You can't". At first Candelas thought Atiyah was making a statement about him personally, but what he was saying is that algebraic geometry is such a large subject that understanding it is a full-time occupation. I'm sure string theory is the same. Therein lies the problem, and it doesn't help that when mathematicians and physicists talk about the same object they often do so in very different ways. –  anon Dec 14 '12 at 21:27
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I mean, if you are really trying to understand String Theory, then you're going to have to become fluent in Classical Mechanics, Quantum Mechanics, Quantum Field Theory, and General Relativity first... otherwise the papers are going to be unmotivated and you won't understand the linguistics and you won't know how the results connect to the universe (i.e. they're more than just a sequence of symbols which we call math).

That being said, assuming CM/QM/QFT/GR are under the belt, the best place to start is Green-Witten-Schwarz's (GWS) Superstring Theory, followed by skimming Polchinski's String Theory. This is supported by my string theory professor when I took it a while ago, Petr Horava (discoverer of D-branes). From here you can supplement other notes and papers.

In Vol.1 of GWS, chapter 2/3 will explain the bosonic string theory (i.e. ignoring fermions) and BRST quantization, which leads to a critical dimension $D=26$. Then chapter 4 will fix this with supersymmetry (i.e. putting back in the fermions), leading to the actual critical dimension $D=10$. After this, gauge anomalies and compactification and dualities and D-branes can start being assessed.

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From what I have gathered in physics , One writes an action and lagrangian invariant under some symmetry groups and derive the equation of motion and other consquences from the action. So,I guess one important thing is to understand symmetries and lagrangians –  nabil Dec 13 '12 at 18:09
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Understanding Lagrangians and symmetries is important, but some theories don't have Lagrangians. In fact such theories, like the (2,0) theory in six dimensions, are the focus of much recent research. Also, while Petr Horava has done much excellent work, including work that foreshadowed the discovery of D-branes, Joe Polchinski is the person who is generally credited with discovering D-branes in string theory. –  Jeff Harvey Dec 13 '12 at 18:21
    
But this credit isn't fully justified, it was an independent discovery. –  Chris Gerig Dec 13 '12 at 21:40
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Since I am a mathematician and also spent quite efforts on learning string theory, etc., let me add some comments.

I agree with David Roberts suggesting this (published book is a little more complete, but not essentially), I partly agree with Chris Gerig "This is more just QFT, and would be a good mathematical perspective after you understand the physics of QFT / String Theory..." I would say that this more concerns lectures by mathematicians: Deligne, Kazhdan, Bernstein, which I would suggest to skip at first reading. And just look at physicits lectures: Faddeev, Gawedzki (fall semester) and Witten, D'Hoker (spring semester)

"... , in particular I don't think it help for the papers that get posted on arXiv hep-ph and hep-th." Well, yes, this volume does not cover most interesting developments made in 90-ies, but nevertheless as some basics sources, it should be Okay.

Let me also agree with Chris Gerig "I mean, if you are really trying to understand String Theory, then you're going to have to become fluent in Classical Mechanics, Quantum Mechanics, Quantum Field Theory, and General Relativity first... "

The way which many people in Russia are doing this - is via volumes by Landua Lifshitz. Let me say that volumes 1-3 (Classical mechanics, Field theory(Classical electrodynamics and General relativity), Quantum mechanics) are quite accessible for mathematicians, even for last years undergrads. But this does not contain Feynman path integral. You may look at Feynman, R.P. and Hibbs, A.R. Quantum Mechanics and Path Integrals. Also LL does not include quantum field theory. You may look at Ramond's short book. And IAS volume discussed above.

You may also look at Igor Dolgachev's (mathematician) "Introduction to physics" http://www.math.lsa.umich.edu/~idolga/lecturenotes.html


Another question it might be worth to figure out - what aim you are setting for yourself. To become a physicist? Let me tell the story - a friend of mine started as a physicist, but later turned to mathematics, I asked him why ? (cause he is really smart and surely had good perspectives). He answered: "you know in physics 1+2+3+4+.... = -1/12, can you live with this ? Me not." Another story about I. Gelfand who being at Rutgers decided to learn some physics, it is started Okay, but at some point, physicist said "here we divide by the volume of the diffeomorphism group" (you always do it in Faddeev's-Popov approach), after that Gelfand stopped this. (The story from my friend who was Gelfand's student and was personally there). I mean for a mathematician absence of proof/(clear understanding) is like a teeth pain, but true physicists will not even observe a problem :) So there is certain cultural and mental difference, and should choose what is more suitable for you.

However, my strong feeling is that mathematical community MUST somehow "learn/absorb/rework/rethink" ideas of QFT and string theory. There are certain important tools and ideas which are now hidden in some physical language and sometimes looks as trick, heuristics, etc..., but should be cleared out, polished, placed in the right position of our mathematical knowledge. We are at certain point where the part of math. community and hep-th community are quite close to each other, this will not be forever. So it is important not to loose a chance of gaining physical "intuition" and making from it mathematical theory.

Let me speculate a little on the possible place of physical ideas in math. It seems to me they are to certain extent "differential geometry of specific infinite-dimensional manifolds". I mean typically in physics we consider the space of all smooth maps from one manifold to another. We write a kind of differential form on this space and integrate it. The problem is that such integration is ill-defined business, however it somehow works. I think that the manifolds we work are not some abstract infinite-dimensional manifolds, but we should take into account that we consider the space of maps from one finite-dim manifold to another finite-dim - an this will lead to certain "semi-infinite" structures. Like vertex operator algebras more or less are loop algebras of finite-dimensional Lie algebras. To certain extent these ideas kind be made quite precise in topological quantum fields theories see e.g. this discussion: Doing geometry using Feynman Path Integral?

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You could do worse than starting with this book: http://books.google.com.au/books/about/Quantum_Fields_and_Strings.html?id=ecruIiTk05EC&redir_esc=y.

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String theory is HUGE! –  David Roberts Dec 13 '12 at 10:23
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An electronic version of this book is available in DjVu format: libgen.org/…, libgen.org/… –  Dmitri Pavlov Dec 13 '12 at 14:49
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This is more just QFT, and would be a good mathematical perspective after you understand the physics of QFT / String Theory, in particular I don't think it help for the papers that get posted on arXiv hep-ph and hep-th. –  Chris Gerig Dec 13 '12 at 16:28
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Yeah, you could do worse. Hopefully you could do better, too, because those two volumes are really quite high-level and demanding. It's almost as if it could be titled "Quantum Fields for Fields Medalists" (as in, Fields Medalist Witten assigns "superhomework" to Fields Medalist Deligne and other high-powered friends, and their submitted solutions essentially form a book within a book). –  Todd Trimble Dec 13 '12 at 16:41
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Isn't everything Witten writes superhomework for mathematicians anyway? :-P –  David Roberts Dec 13 '12 at 21:46
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the most basic book I know of is Enumerative Geometry and String Theory by Sheldon Katz.

http://books.google.co.uk/books?id=cvNMmma0pQMC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

but of course it doesn't even begin to scratch the surface of the topics you (and not only you) want to understand.

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I want to understand compactifications , Dualities , D-branes , M-branes etc. What's the easiest way to do so provided I have the necessary knowledge in algebraic geometry , Algebraic topology, analysis and differential geometry

I have the following book and deals with all the topics you mentioned in addition to Conformal Field Theory for a layman's perspective.

http://books.google.com/books/about/String_Theory_Demystified.html?id=S4JyPgw4ZlAC

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