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Explore state counting techniques in Kitaev quantum double models through this 88-minute conference lecture delivered by Benoît Doucot at the International Centre for Theoretical Sciences. Delve into the mathematical framework underlying these topological quantum models, which serve as fundamental examples of exactly solvable lattice gauge theories with anyonic excitations. Learn how to systematically count and classify the ground states and excited states in these models, understanding their relationship to quantum error correction and topological quantum computation. Examine the algebraic structures that govern state counting, including the role of quantum groups and their representation theory in determining the Hilbert space dimensions. Discover how these counting methods connect to broader concepts in topological order, including the relationship between ground state degeneracy and the topology of the underlying manifold. Gain insights into the physical interpretation of different state sectors and their relevance to quantum phase transitions and topological protection mechanisms. This presentation forms part of a comprehensive program on generalized symmetries and anomalies in quantum phases of matter, providing essential background for understanding how symmetry principles constrain and organize quantum many-body systems.
