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Explore the representation theory of extended operators through the lens of Tannaka duality and Koszul duality in quantum field theory in this advanced lecture from the 2025 SCGCS School on Topological Symmetries and Defects in Quantum Field Theory. Delve into sophisticated mathematical frameworks that connect algebraic structures with physical theories, examining how Tannaka duality provides a bridge between categories and algebraic objects, while Koszul duality offers insights into the relationship between algebras and their derived categories. Learn how these powerful dualities apply to the study of extended operators in quantum field theory, providing new perspectives on topological symmetries and defect structures. Gain understanding of the mathematical foundations that underpin modern approaches to quantum field theory, particularly in the context of topological phases and symmetry-protected phenomena. This first part of the series establishes the theoretical groundwork for understanding how representation theory techniques can illuminate the structure of extended operators and their role in quantum field theoretic contexts.
