Modular Invariants from Subfactors:¶Type I Coupling Matrices and Intermediate Subfactors |
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Authors: | Jens Böckenhauer David E Evans |
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Institution: | (1) School of Mathematics, University of Wales Cardiff, PO Box 926, Senghennydd Road, Cardiff CF24 4YH, Wales, UK, GB |
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Abstract: | A braided subfactor determines a coupling matrix Z which commutes with the S- and T-matrices arising from the braiding. Such a coupling matrix is not necessarily of “type I”,
i.e. in general it does not have a block-diagonal structure which can be reinterpreted as the diagonal coupling matrix with
respect to a suitable extension. We show that there are always two intermediate subfactors which correspond to left and right
maximal extensions and which determine “parent” coupling matrices Z
± of type I. Moreover it is shown that if the intermediate subfactors coincide, so that Z
+=Z
−, then Z is related to Z
+ by an automorphism of the extended fusion rules. The intertwining relations of chiral branching coefficients between original
and extended S- and T-matrices are also clarified. None of our results depends on non-degeneracy of the braiding, i.e. the
S- and T-matrices need not be modular. Examples from SO(n) current algebra models illustrate that the parents can be different, Z
+≠Z
−, and that Z need not be related to a type I invariant by such an automorphism.
Received: 8 December 1999 / Accepted: 15 February 2000 |
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