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Dualities on QFT–also called Quantum Field Theory dynamics–is a huge and fundamental research area. However, despite underpinning major mathematical breakthroughs such as the work of Kapustin and Witten on the geometric Langlands conjectures, it is still an area of difficult access for both physicists and mathematicians.

In particular, while many mathematicians would benefit from being able to work with dualities, those who try to familiarise themselves with it often encounter two problems: either the sources for learning about dualities lack rigour, or the available mathematical references on QFT dynamics approach the subject as it was done 10~20 years ago.

Also, just as in elementary number theory, without clear organisation, one may naively get the impression that research on dualities is just a haphazard compilation of unrelated results. Of course, the expert knows better.

But how can a mathematician or a mathematics student become proficient in using dualities as tools? In particular, what surveys, lecture notes, or even original papers on QFT dynamics are most accessible to mathematicians?

(While modern mathematical references would be ideal, feel free to recommend (possibly old) sources aimed at physicists only)


A (temporary) side question: would including String Theory dualities make the question too broad?

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General References for Physicists

  1. Lectures on 4d $\mathcal{N}=1$ dynamics and related topics (Yuji Tachikawa);
  2. $\mathcal{N}=2$ supersymmetric dynamics for pedestrians (Yuji Tachikawa).

General References for Mathematicians

  1. A pseudo-mathematical pseudo-review on 4d $\mathcal{N}=2$ supersymmetric quantum field theories, by Yuji Tachikawa;

  2. Part 4 of Quantum Fields and Strings: A Course for Mathematicians.

AGT Correspondence

  1. Introduction to AGT duality (for physicists, by Jose Miguel Zapata Rolon);
  2. A Pedagogical Introduction to the AGT Conjecture (for physicists, by Robert J. Roger);
  3. 4d/2d Correspondence: Instantons and $\mathcal{W}$-algebras (for physicists, PhD thesis of Jaewon Song. While it is a research thesis, it also contains a few introductory sections).
  4. AGT correspondence seminar notes (for mathematicians, by various authors; notes by Pavel Safronov)

Kapustin-Witten on S-duality and Geometric Langlands

  1. Chris Elliott's thesis (for mathematicians, which aims to make parts of KW mathematically rigorous);
  2. Langlands Program, Field Theory, and Mirror Symmetry (survey for physicists, by Kazuki Ikeda);
  3. Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4 (for mathematicians, by Kevin Costello, which gives a derived geometry perspective to some aspects of KW);
  4. Geometric Langlands Twists of N = 4 Gauge Theory from Derived Algebraic Geometry (for mathematicians, research paper by Chris Elliott);
  5. An Algebraic Introduction to Kapustin-Witten Theory (Lecture notes for mathematicians by Chris Elliott).

Background

  1. Volumes 1 and 2 of Quantum Fields and Strings: A Course for Mathematicians.
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    $\begingroup$ The majority of these are aimed at physicists, however. Other references, especially mathematical, would be greatly appreciated. $\endgroup$ – Théo de Oliveira Santos Dec 27 '18 at 4:08

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