Tannakian QFT's 1 - Tudor Dimofte - Universität Hamburg
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- SFB 1624 : Lectures on Tannakian QFT's
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18.10.2024
Tannakian QFT's 1
Lecture 1: From spark algebras to quantum groups.
I'll begin with an intro/overview of the lecture series. Then I'll propose a definition of “Tannakian QFT” in the context of 3d topological theories that allows line operators to be represented by generalized quantum groups (aka ribbon Hopf algebras), explicitly realized as algebras of operators (“spark algebras”) on topological boundaries. (This is closely related to upcoming work of Theo Johnson-Freyd and David Reutter.) I’ll explain how to implement the procedure for finite group gauge theory (Dijkgraaf-Witten theory).
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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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