Topological BF Sectors and Operator–Algebraic Chern–Simons Theories

We formulate an operator–algebraic construction of three-dimensional Chern–Simons type defect field theories from Buchholz–Fredenhagen sectors of Haag–Kastler nets. The basic input is an admissible datum [Formula: see text]), where M is a von Neumann algebraic vacuum phase, [Formula: see text] is a unitary modular topological BF sector category, and [Formula: see text] is a finiteindex Morita realization of [Formula: see text] on M. Thus the datum records not only the modular tensor category of topological charges, but also its realization by finite-index correspondences and Connes fusion in the von Neumann Morita bicategory. The category [Formula: see text] arises from a bicategorical topological scaling-limit hypothesis, together with finiteness and nondegeneracy assumptions. We define the finite-index Morita target, prove the closure properties of finite-index correspondences under Connes fusion and spatial tensor products, and formulate the admissibility conditions under which BF sectors act as Morita endomorphisms of M. Assuming the framed defect Cobordism Hypothesis for the chosen admissible target, each datum [Formula: see text] determines a framed extended defect TQFT. We prove that this assignment is invariant under Morita equivalence and therefore descends to equivalence classes of admissible data. Its image is the class of framed defect theories produced from admissible BF-sector Morita data; under a reconstruction hypothesis the descended map is injective, hence bijective onto this constructed class. The result is a structural classification of the operator–algebraic Chern–Simons defect theories determined by topological BF sectors and their finite-index Morita realizations, rather than of Haag–Kastler nets themselves.