Tannakian QFT's 2 - Tudor Dimofte - Universität Hamburg
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- SFB 1624 : Lectures on Tannakian QFT's
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22.10.2024
Tannakian QFT's 2
Lecture 2: Tannaka and Koszul
I’ll delve into the world of cohomological TQFT’s, and explain how Tannaka duality induces Koszul duality in a perturbative, cohomological theory. This will allow us to connect the ribbon Hopf algebras of Lecture 1 with Koszul duality for E2 and E3 algebras (and work of Tamarkin, Lurie, Costello, and others). I’ll also give more examples of quantum groups controlling line operators in 3d gauge theories with continuous groups, including Chern-Simons theory and twists of 3d N=4 gauge 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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