The study of supersymmetric gauge theories in physics has produced an incredibly rich collection of mathematical structures (for example the construction of Seiberg-Witten integrable systems and their quantization). I will explain (following work in progress with D. Nadler, A. Neitzke and T. Nevins) how to see such structures directly from the formalism of extended topological field theory. The main (and utterly conjectural) example is provided by a 6-dimensional field theory, Theory X (better known as the (2,0) 6d SCFT), whose structure unifies much of modern geometric representation theory. 

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