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at string theory there used to be a stub-section “Fields medal work induced by string theory. I have now expanded that to the following keyword list:
Pure mathematics work which came out of string theory and was awared with a Fields medal includes the following.
Richard Borcherds, 1998
Maxim Kontsevich, 1998
formality theorem and formal deformation quantization via holography of Poisson sigma-model string.
Edward Witten, 1990
knot invariants via WZW model-string/Chern-Simons theory holography;
elliptic genus, Witten genus and rigidity via superstring partition functions;
Grigori Perelman, 2006
I didn’t know that Perelman’s work was related to string theory…
Ricci flow is the renormalization group flow of the bosonic string sigma model in a pure gravity background. But the bosonc string admits two more massless background fields: the dilaton and the B-field. Perelman observed that allowing also the dilaton doesn’t change the qualitative properties of the flow as needed to complete Hamilton’s proof strategy, but does change it quantitatively such as to finally make the pinching behaviour controllable. That allowed to compete the proof.
One might wonder: why not switch on the B-fiel, too, then. But I suppose the flow would then actually find its fixed points where the sigma model becomes conformal. That’s what you’d be after for string physics, but it would no longer serve Hamilton’s proof (unless one could show that the flow necessarily finds the SU(2) WZW model I suppose.)
We now interpret this as a dilaton. Are you sure this is the interpretation which lead Perelman to use this term ? Have in mind that dilaton is about gravity (in many variants) and not necessarily in the variants called string theory.
Borcherds work is about CFT, and CFT main advances just before his work by Belavin, Polyakov, Zamalodhcikov, Knizhnik were mainly motivated by condensed matter physics in 2d. Some of his awarded work has also being largely motivated by the study of monster group as well as automorphic forms, subjects closer to number theory, group theory, lattices and codes.
Besides, I am sorry to tell, but I have bad feeling about introducing this title. It legitimizes propaganda and force, we should be more humble about things. The subject of the medal work mentioned is RELATED to string theory, it is extremely hard and unfair to say that the work is INDUCED by string theory. Of course all work preceding some work is in so many areas, hence logically it is “induced” by work in most areas in mathematics, as the theorems from their are used substantially. And in the way toward mirror symmetry algebraic geometry, and algebraic number theory insights were so much deeper than the outline which suggested it in derivation of importance of topological sigma models in string theory. The rest was independent work. Gromov-Witten invariants were devised by Gromov in a setup of symplectic geometry. D-branes of type II in terms of derived category of coherent sheaves in work of Kontsevich in 1994 were continuation of Beilinson-Bondal-Kapranov on enhanced derived categories of coherent sheaves in 1980s, and the Polchinski introduction of D-branes under that name in 1995 was chronologically after the algebraic geometry work of Kontsevich. Of course, we are lucky that a couple of people like Witten and Kontsevich indeed have such a grasp both of deep physics and deep mathematics. But we should not make a REDUCTION of this to “string theory” (and in particular not the reduction of CFT to string theory).
Conclusion: I suggest related replacing induced.
Sure, I changed “induced” to “closely related”, for political correctness. But personally I think this is quite unfortunate.
Regarding the dilaton: I have no idea if Perelman picked up the dilaton from the string literature or if he redscovered it independently. Luckily in mathematics concepts exists independently of who discovers them and when. The dilaton is always the dilaton, just as the Monster group is always the Monster group, independently of who rediscovers it.
I have added more references at Ricci flow:
The identification of Ricci flow with the renormalization group flow of the bosonic string sigma-model is reviewed for instance in
Kasper Olsen, From Polyakov to Perelman
E. Woolgar, Some Applications of Ricci Flow in Physics, Can.J.Phys.86:645,2008 (arXiv:0708.2144)
Mauro Carfora, Renormalization Group and the Ricci flow (arXiv:1001.3595)
and discussed in more detail for instance in
T Oliynyk, V Suneeta, E Woolgar, Irreversibility of World-sheet Renormalization Group Flow, Phys.Lett. B610 (2005) 115-121 (arXiv:hep-th/0410001)
T Oliynyk, V Suneeta, E Woolgar, A Gradient Flow for Worldsheet Nonlinear Sigma Models, Nucl.Phys. B739 (2006) 441-458 (arXiv:hep-th/0510239)
Arkady Tseytlin, On sigma model RG flow, “central charge” action and Perelman’s entropy, Phys.Rev.D75:064024,2007 (arXiv:hep-th/0612296)
Would it be fair to add Maryam Mirzakhani to the list in view of her work on moduli spaces, e.g., here? Bridging the symplectic/complex divide brings the work close to mirror symmetry.
Ah this overview of topological recursion agrees:
…the mathematical meaning of mirror symmetry is unclear. Despite this, quantum topology has received a number of Fields medals for work in and around mirror symmetry, including Jones (1990), Witten (1990), Kontsevich (1998), and Mirzakhani (2014).
Myself, I don’t know how Mirzakhani’s work really relates to mirror symmetry. But if you find (or since you found) trustworthy citations that make the connection, then it makes sense to go ahead and add that information to the entry. (This goes without saying, right?)
It’s more lack of knowledge. So it seems she provided a new proof of the Witten conjecture, which ought to be enough to include her, so I’ve done that.
Thanks for reminding me of that entry. I have touched wording, formatting and hyperlinking at Witten conjecture.
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