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?