Timeline for Roadmap to geometric Langlands for a mathematical physics student
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Jan 6, 2023 at 20:52 | comment | added | Daniel Waters | Okay great, I will keep all of this saved so that I know what to read when the time comes. I am interested in learning the GRT as well, which is part of why I was saying that about Kapustin-Witten, although I wasn't really aware of the self-contained nature of the paper. Thanks for all of your recommendations and advice! | |
| Jan 6, 2023 at 19:31 | comment | added | dhy | AFO on the other hand does require more mathematical background. This is not so close to what I do so take everything I say with a grain of salt, but my impression is that this is closely related to the general story of quantum enumerative geometry of symplectic resolutions, of which the prototypical case is the paper "Quantum Cohomology of the Springer Resolution" by Braverman-Maulik-Okounkov. After that I would suspect the paper to read is "Quantum Groups and Quantum Cohomology" by Maulik-Okounkov? | |
| Jan 6, 2023 at 19:24 | comment | added | dhy | @DanielWaters My point is that neither Hecke eigensheaves (which is just a definition) nor D-modules should be considered background for Kapustin-Witten, because they only really use the definitions (which they provide and explain.) There is of course independent value in learning about D-modules and why they appear in GRT (the book of Hotta-Takeuchi-Tanisaki is one place to learn this if you are interested) but I don't think knowing this theory will help for reading Kapustin-Witten. | |
| Jan 6, 2023 at 18:25 | comment | added | Daniel Waters | I have not read the paper as of yet, but I would like to be familiar with things like Hecke eigensheaves and $\mathcal D$-modules before reading the paper, as I like to have a solid background before diving into new things (as the conformal field theory and string theory part, I am still in the process of learning those things, and I don't need any recommendations for them at the moment) . But aside from Kapustin-Witten, I would need to know a bit about geometric representation theory for the paper by Aganagic, et. al. | |
| Jan 6, 2023 at 18:16 | comment | added | dhy | @DanielWaters In particular if Kapustin-Witten is your goal, $\infty$-categories, DAG, "classical" geometric representation theory, etc., are certainly not necessary prerequisite knowledge. | |
| Jan 6, 2023 at 18:14 | comment | added | dhy | @DanielWaters Is there a specific mathematical point you get stuck at when you try to read Kapustin-Witten? I would say those papers are written to require quite a bit of physics background and not very much math background (e.g. the introduction contains the sentence "No prior familiarity with the Langlands program is assumed; instead, we assume a familiarity with subjects such as supersymmetric gauge theories, electric-magnetic duality, sigma-models, mirror symmetry, branes, and topological field theory.") | |
| Jan 6, 2023 at 16:37 | comment | added | Daniel Waters | @dhy sure thing. I am interested in understanding the above linked papers by Witten, as well as the following paper of Aganagic, Frenkel, and Okounkov: arxiv.org/abs/1701.03146. In general, the connections between string/M-theory (in particular the $\mathcal N = (2,0)$ superconformal field theory) and GL. | |
| Jan 6, 2023 at 16:05 | comment | added | dhy | By the way, there are many different approaches to GL. If you have a concrete goal in mind (e.g., "I want to understand this paper _______ by ______") I can write a longer form answer giving a roadmap. | |
| Jan 6, 2023 at 16:04 | comment | added | dhy | I would actually actively recommend against reading any source on $\infty$-categories (unless you are intrinsically interested in them). The power of $\infty$-categories is that, once you understand the philosophy, they are very convenient to use, without having to delve into their foundational details. For the purposes of geometric Langlands, you will learn much more from seeing how they are used than actually reading about them... | |
| Jan 6, 2023 at 14:48 | comment | added | Daniel Waters | @TimothyChow Thanks, I'll add it to my reading list for $\infty$-categories. | |
| Jan 6, 2023 at 13:32 | comment | added | Timothy Chow | Markus Land's book Introduction to Infinity-Categories may be a friendlier introduction to $\infty$-categories than Lurie's book. | |
| Jan 6, 2023 at 6:04 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title, changed tag
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| Jan 6, 2023 at 3:07 | comment | added | Daniel Waters | Awesome! Thanks so much for your help! | |
| Jan 6, 2023 at 2:34 | comment | added | Exit path | Lurie’s book “Higher Topos Theory” is a classic reference for infinity categories, and the book “A Study in Derived Algebraic Geometry” by Gaitsgory-Rozenblyum is a pretty good introduction to DAG the way geometric Langlands theorists typically think about it. That said, there are probably numerous shorter/more user-friendly introductions to both subjects online | |
| Jan 6, 2023 at 1:32 | comment | added | Daniel Waters | Do you have any good references for those topics? | |
| Jan 6, 2023 at 1:30 | comment | added | Exit path | Also, some ideas which feature heavily in modern versions of the theory (and are absent from those notes) are higher category theory and derived algebraic geometry. The former is used everywhere (often in the guise of DG categories), and the latter is mostly used on the “spectral” side of things. For example, the derived geometric Satake equivalence of Bezrukavnikov and Finkelberg is an equivalence of two stable infinity categories, one of which is (Ind)-coherent sheaves on a derived scheme | |
| Jan 6, 2023 at 1:16 | comment | added | Exit path | That’s the one! | |
| Jan 6, 2023 at 1:10 | comment | added | Daniel Waters | Are you referring to arxiv.org/abs/hep-th/0512172 ? | |
| Jan 6, 2023 at 1:08 | comment | added | Exit path | Edward Frenkel has some introductory notes on conformal field theory and the Langlands program which are pretty good. You could start there and learn things as you need them. Since the field is pretty massive where you should go from there depends on your interests | |
| Jan 6, 2023 at 0:59 | history | asked | Daniel Waters | CC BY-SA 4.0 |