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This question was posted on mathoverflow (here) without too much success.

I'm hoping to read the famous Kapustin-Witten Paper "Electric-magnetic duality and the geometric Langlands program" and the related "The Yang-Mills equations over Riemann surfaces".

The following statement serves to explain the origin of my trouble: Let $G$ be a simple complex Lie group and $C$ a Riemann surface. Geometric Langlands is a set of mathematical ideas relating the category of coherent sheaves over the moduli stack of flat $G^{L}$-bundles ($G^{L}$ is the dual Langlands group of $G$) over $C$ with the category of $\mathcal{D}$-modules on the moduli stack of holomorphic $G$-bundles over $C$.

My problem: I have working knowledge of representation theory, but I'm completely ignorant about the theory of mathematical stacks and the possible strategies to begin to learn it. What specifically worries me is how much previous knowledge of $2$-categories is needed to begin.

My background: I've read Hartshorne's book on algebraic geometry in great detail, specifically the chapters on varieties, schemes, sheaf cohomology and curves. My category theory and homological algebra knowledge is exactly that needed to read and solve the problems of the aforementioned book. I'm also familiar with the identification between the topological string $B$-model branes and sheaves at the level of Sharpe's lectures.

Questions: I'm asking for your kind help to find references to initiate me on the theory of stacks given my physics orientation. What would be a good introductory reference on stacks for a string theory student? Is there any physics friendly roadmap to begin? Any familiar gauge/string theoretical analogies to start to develop intuition?

Any suggestion will be extremely helpful to me.

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  • $\begingroup$ sounds like a math question... $\endgroup$ Nov 8, 2020 at 1:46
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    $\begingroup$ I've had classes form Feynman, Thorne, Mandelstam, Gell-Mann, Zumino, Schwarz, Preskill...and I don't understand anything in this question. See ^^^zero^^^. $\endgroup$
    – JEB
    Nov 8, 2020 at 2:06
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    $\begingroup$ @ZeroTheHero I've admitted that definitely this question is about math. I just was wondering if a mathematical-physicists would have some recommendation to help a physics student to learn about this type of math that is relevant for papers written by physicists (Witten-Kapustin-Gukov) using physics (string theory and gauge theory). I'm sure my question is morally the same of that of a general relativity student hoping to learn more about differential geometry and is not out of place here. $\endgroup$ Nov 8, 2020 at 2:16
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    $\begingroup$ @JEB Sorry. I know this is specialized question; but I don't think is out of place here. Maybe someone with familiarity with the Witten's papers could help me to learn this is a physics friendly way. I really hope so. $\endgroup$ Nov 8, 2020 at 2:24
  • $\begingroup$ @RamiroHum-Sah If it's physics, someone might be able to help. I'm just saying, I don't know, and it's not for lack of exposure to physics. $\endgroup$
    – JEB
    Nov 8, 2020 at 2:34

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For the "physics part" of my question: I have found the papers String Orbifolds and Quotient Stacks , D-branes, orbifolds, and Ext groups and Stacks and D-Brane Bundles very useful, readable and explicit. Apparently the best strategy to deal with stacks in the context of a non-linear sigma model is to think on the target space as locally an orbifold (exactly the way mathematicians divulge the idea of a stack).

For the "math part" of my question: Angelo Vistoli textbook, Notes on Grothendieck topologies, fibered categories and descent theory, arXiv:math/0412512 is really pedagogical and self-contained. The best reference I've found so far.

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