I am wondering what currently our mathematicians do related to conformal field theory, (I know currently it is a central topic, but I have only a vague idea what mathematicians do in there), or more generally topological field theory, field theory… I am extremely appreicated if there is any survey paper.
[Math] What do mathematicians currently do in conformal field theory (or more general field theory)
conformal-field-theorymp.mathematical-physicsquantum-field-theory
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One must distinguish between quantum/classical on the string world-sheet and in spacetime. Both of your statements are basically correct, but should read something like "CFT theory is the space of classical solutions to the spacetime equations of string theory" and "Quantization of the the world-sheet sigma model of a string theory gives rise to a CFT."
In a little more detail, the sigma-model describing string theory propagation on some manifold M is a 2-dimensional quantum field theory which in order to describe a consistent string theory must be a conformal field theory. The "classical limit" of this 2-dimensional field theory is a limit in which some measure of the curvature of M is small in units of the string tension. To construct a CFT one must solve the sigma-model exactly, including world-sheet quantum effects.
The coupling constants of the sigma-model are fields in spacetime such as the metric $g_{\mu \nu}(X(\sigma))$ on $M$ where $X: \Sigma \rightarrow M$ define the embedding of the string world-sheet $\Sigma$ into $M$. Now there is also a spacetime theory of these fields. You can think of it as a ``string field theory". At low-energies it can sometimes be usefully approximated by a theory of gravity coupled to some finite number of quantum fields, but in full generality it is a theory of an infinite number of quantum fields. Roughly speaking, each operator in the CFT gives rise to a field in spacetime. The spacetime string field theory lives in 10 dimensions for the superstring or 26 dimensions for the bosonic string and it also has a classical limit. The classical limit is $g_s \rightarrow 0$ where $g_s$ is a dimensionless coupling constant. It appears in perturbative string theory as a factor which weights the contribution of a Riemann surface by the Euler number of the surface. It can also be thought of as the constant (in spacetime) mode of a scalar spacetime field known as the dilaton.
The main point is that there are two notions of classical/quantum in string theory, one involving the world-sheet theory, the other the spacetime theory. In order to avoid confusion one must be clear which is being discussed. Unfortunately string theorists often assume it is clear from the context.
In response to the further question about the space of string fields, I would suggest that you have a look at the introductory material in http://arXiv.org/pdf/hep-th/9305026. You may also find http://arXiv.org/pdf/hep-th/0509129 useful. I should add that while string field theory has had some success recently in the description of D-brane states, it is not widely thought to be a completely satisfactory definition of non-perturbative string theory.
As Robert Israel indicates, if you really wanted to define a QFT, it would certainly be very analytic, say, to define the path integral. So analytic, in fact, that no one can do it for even semicomplicated theories.
However, one of the big math/physics developments in the last few decades is a class of QFTs where the observables are topological in nature. These are the topological QFTs or TQFTs. In these theories, you can ignore all or most of the (too) hard analysis and deal with much more well-defined spaces. Of course, from the mathematics point of view, these theories still involve a path integral that isn't defined (due to all that hard analysis we're ignoring), but enough structure can be found and has been developed to lead to all sorts of cool mathematics (see the mathematical definition of TQFTs, most recently axiomatized by Lurie building on lots of prior work). And even without this structure, physical intuition about these not mathematically well-defined theories has led to countless conjectures, theorems and the like, for example in mirror symmetry and various invariants like Donaldson and Seiberg-Witten.
Best Answer
CFT/QFT/TFT/etc. is a huge subject...
Here are some random references off the top of my head...
Segal, "The definition of conformal field theory".
Costello, "Topological conformal field theories and Calabi-Yau categories" -- This is (essentially) the 2d version of the (Hopkins-)Lurie/Baez-Dolan cobordism hypothesis that Lennart mentions. See also Kontsevich-Soibelman, "Notes on A-infinity...". This stuff is closely related to mirror symmetry, which is - in physics terms - a duality between certain field theories (or sigma models). Mirror symmetry by itself is already a huge enterprise...
See papers by Yi-Zhi Huang for stuff about vertex operator algebras and CFTs.
One can consider string topology from a field theory viewpoint... see for example Sullivan, "String Topology: Background and Present State" and Blumberg-Cohen-Teleman, "Open-closed field theories, string topology, and Hochschild homology". This is actually related to the work of Costello, Lurie, Kontsevich mentioned above -- see e.g. section 2.1 of Costello's paper.
An important problem is that of making rigorous some of the things that physicists do in QFT, such as path integrals. See Costello, "Renormalization and effective field theory" and also Borcherds, "Renormalization and quantum field theory".
There's also Chern-Simons theory... Gromov-Witten theory... Kapustin-Witten theory... Rozansky-Witten theory...
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