On the Categorical Spectrum of Topological Quantum Field Theories

Explore the mathematical foundations of topological quantum field theories through this conference talk delivered at Harvard CMSA's Workshop on Quantum Field Theory and Topological Phases via Homotopy Theory and Operator Algebras. Delve into Kitaev's original suggestion that invertible topological quantum field theories of varying dimensions should assemble into a spectrum or generalized homology theory, and examine the candidate spectrum proposed by Freed and Hopkins. Learn how not-necessarily-invertible TQFTs should form a 'categorical spectrum' - an analogue of a spectrum with non-invertible cells at each level. Discover the existence of a unique categorical spectrum that satisfies reasonable assumptions about collections of compact, very finite, and discrete TQFTs, including the requirement that its invertibles align with Freed and Hopkins' framework. Understand the key assumptions underlying this construction, examine how the categorical spectrum appears in low dimensions, and explore its potential applications for studying gapped boundaries of anomaly theories in higher dimensions. The presentation draws from ongoing collaborative research with Theo Johnson-Freyd and provides insights into advanced topics in mathematical physics and category theory.