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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

632: １３２人目の素数さん [] 2023/02/19(日) 11:17:26.11 ID:ynjTT/Eh

>>631
つづき

https://en.wikipedia.org/wiki/Vorlesungen_%C3%BCber_Zahlentheorie
Vorlesungen uber Zahlentheorie (German for Lectures on Number Theory) is the name of several different textbooks of number theory. The best known was written by Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863.
Dirichlet and Dedekind's book
This translation does not include Dedekind's Supplements X and XI in which he begins to develop the theory of ideals.
The German titles of supplements X and XI are:
Supplement X: Uber die Composition der binaren quadratische Formen (On the composition of binary quadratic forms)
Supplement XI: Uber die Theorie der ganzen algebraischen Zahlen (On the theory of algebraic integers)

https://archive.org/details/vorlesungenber00lejeuoft
textsVorlesungen uber Zahlentheorie. Hrsg. und mit Zusatzen versehen
by Lejeune-Dirichlet, Peter Gustav, 1805-1859; Dedekind, Richard, 1831-1916
Publication date 1894
Topics Number theory
Publisher Braunschweig F. Vieweg
Collection gerstein; toronto
Digitizing sponsor University of Toronto
Contributor Gerstein - University of Toronto
Language German
(DOWNLOAD OPTIONS PDFより)

つづく

http://rio2016.5ch.net/test/read.cgi/math/1615510393/632

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