Multi-interval Subfactors and Modularity of Representations in Conformal Field Theory
Yasuyuki Kawahigashi, Roberto Longo, Michael Mueger
TL;DR
The paper addresses how multi-interval inclusions in rational conformal nets encode the full superselection (sector) structure. It shows that, under split property and Haag duality (or modular PCT), the two-interval inclusion is the Longo–Rehren LR inclusion generated by the complete sector system, yielding a global index $\mu_A$ equal to the global index $I_{global}$ and a non-degenerate braiding (modularity). Through canonical endomorphisms, $\alpha$-induction, and LR nets, it also establishes that the quantum double description governs the multi-interval case and extends to $n$ intervals with an iterated LR structure. This work provides a model-independent link between local observables and the global topological data (Verlinde data) of the theory, with implications for 1+1D conformal QFT and topological quantum field theory.
Abstract
We describe the structure of the inclusions of factors A(E) contained in A(E')' associated with multi-intervals E of R for a local irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo-Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of A. As a consequence, the index of A(E) in A(E')' coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of A form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry.