Tannakian QFT's 4 - Tudor Dimofte - Universität Hamburg
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29.10.2024
Tannakian QFT's 4
Lecture 4: Holomorphic QFT's
In this final lecture I'll move outside the fully topological world, and consider representation theory for line operators in partially holomorphic QFT's. I'll explain how in 3d holomorphic-topological theories this leads to "chiral categories" controlled by dg-shifted versions of Yangians; and in 4d leads to generalized Yangians (related to work of Costello, Costello-Yamazaki-Witten).
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Quantum field theories often contain rich collections of extended operators, or "defects," intimately connected to symmetries and their dynamics. Given some degree of topological invariance, extended operators are expected to organize themselves into mathematical categories, with various additional structures, such as tensor products coming from collisions of operators. However, it can be quite difficult physically to fully identify/compute these categories — encoding the complete set of extended operators and their correlation functions.
In this series of lectures, I’ll discuss systematic approaches to tackling this problem, fundamentally inspired by representation theory (of categories) in mathematics, and the ideas of Tannaka and Koszul duality. I’ll present some recent/ongoing work of mine (joint with Wenjun Niu, and parts with Victor Py, Thomas Creutzig, and Chris Beem), noting also that Tannaka/Koszul duality have a long and rich history in physics, spanning several decades.
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