ガロア第一論文及びその関連の資料スレ

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390: １３２人目の素数さん [] 2023/02/13(月) 15:48:09.11 ID:xsCTjZGt

>>389
つづき

In fact I now think the best way to understand a vertex algebra is to first really understand its topological analog, the structure of local operators in 2d topological field theory. If you check out any article about topological field theory it will explain that in a 2d TFT, we assign a vector space to the circle, it obtains a multiplication given by the pair of pants, and this multiplication is commutative and associative (and in fact a commutative Frobenius algebra, but I'll ignore that aspect here). It's very helpful to picture the pair of pants not traditionally but as a big disc with two small discs cut out -- that way you can see the commutativity easily, and also that if you think of those discs as small (after all everything is topologically invariant) you realize you're really describing operators labeled by points (local operators in physics, which we insert around a point) and the multiplication is given by their collision (ie zoom out the picture, the two small discs blend and look like one disc, so you've started with two operators and gotten a third).

Anyway this is getting long - to summarize, a vertex algebra is the holomorphic refinement of an E2
algebra, aka a "vector space with the algebraic structure inherent in a double loop space", where we allow holomorphic (rather than locally constant or up-to-homotopy) dependence on coordinates.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1615510393/390

391: １３２人目の素数さん [] 2023/02/13(月) 15:48:34.71 ID:xsCTjZGt

>>390
つづき

AND we get perhaps the most important example of a vertex algebra--- take X in the above story to be BG, the classifying space of a group G.
Then Ω^2X=ΩG is the "affine Grassmannian" for G, which we now realize "is" a vertex algebra.. by linearizing this space (taking delta functions supported at the identity) we recover the Kac-Moody vertex algebra (as is explained again in my book with Frenkel).

https://math.berkeley.edu/~frenkel/BOOK/
本(my book with Frenkel)"Vertex Algebras and Algebraic Curves" by Edward Frenkel and David Ben-Zvi 2001

https://web.ma.utexas.edu/users/benzvi/
David Ben-Zvi
https://en.wikipedia.org/wiki/David_Ben-Zvi
David Dror Ben-Zvi is an American mathematician, currently the Joe B. and Louise Cook Professor of Mathematics at University of Texas at Austin.[1]
Ben-Zvi earned his Ph.D. from Harvard University in 1999, with a dissertation entitled Spectral Curves, Opers And Integrable Systems supervised by Edward Frenkel.[2] In 2012, he became one of the inaugural Fellows of the American Mathematical Society.[3]
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1615510393/391

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