Research abstract of Y. Kawahigashi for 2001-02

We made the following study on algebraic quantum foeld theory.

Consider a net of subfactors {N(I) \subset M(I)}_{I \subset S^1} on S^1, split S^1 into four intervals I_1,I_2,I_3,I_4 in the counterclockwise order, and construct subfactors M(I_1) \vee M(I_3) \subset (N(I_2) \vee M(I_4))' and M(I_1)\vee M(I_3) \subset (N(I_2) \vee N(I_4))'. These are the "relative versions" of the subfactors considered by Longo, Mueger and the author [30]. In the second half of paper [35], we have shown that these subfactors are isomorphic to the Longo-Rehren subfacotrs arising from the systems of irreducible endomorphisms of M(I) resulting from \alpha^+-induction and \alpha^\pm-induction, respectively. By our previous results, they are also duals of the generalized Longo-Rehren subfactors arising from the modular invariants produced by \alpha^\pm-induction.

Next, we have naturally defined a real-valued invariant, "central charge" c, for diffeomorphism covariant nets of factors on the circle S^1, and gave a complete classification of such nets for the case c < 1 with Longo in [36]. They are labelled with certain pairs of the A-D-E Dynkin diagrams. We have three infinite series and four exceptionals. Our main tools are \alpha-induction for nets of subfactors and modular invariants.

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