In one-dimensional quantum lattice systems, localized charges are naturally described by C*-correspondences rather than only by representations or endomorphisms. This research-oriented topics course develops Corey Jones’s DHR bimodule theory and its applications to fusion spin chains, quantum cellular automata, half-line charge categories, and categorical symmetry.

The first part establishes the required foundations: C*-algebras and quasi-local nets, unitary tensor categories and Drinfeld centers, Hilbert C*-modules and correspondences, C*-2-categories, Q-systems, finite-index conditional expectations, and Q-system realization. We then construct the C*-tensor category of localized and transportable DHR bimodules, its unitary braiding, and its functoriality under bounded-spread isomorphisms. For fusion spin chains, this leads to the braided equivalence between the DHR category and the Drinfeld center of the underlying fusion category.

The course next studies the Jones–Lim index for quantum cellular automata as a numerical measure of information flow complementary to the induced action on DHR sectors. We then construct intrinsic half-line charge categories and analyze the hypotheses under which their Drinfeld center recovers the DHR category. The final part develops the operator-algebraic SymTFT picture through physical-boundary inclusions, Lagrangian Q-systems, solitonic bimodules, and categorical symmetries realized by locality-preserving quantum channels. Applications include tensor-product and on-site realizability, symmetric states, anomaly-enforced gaplessness, and local topological order.

The course emphasizes detailed proofs of central results, concrete examples, and guided reading of recent research papers.