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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?Doing geometry using Feynman Path Integral?

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?

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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Alexander Chervov
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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?