Communications in Mathematical Physics

, Volume 210, Issue 3, pp 733-784

First online:

Chiral Structure of Modular Invariants for Subfactors

  • Jens BöckenhauerAffiliated withSchool of Mathematics, University of Wales, Cardiff, PO Box 926, Senghennydd Road, Cardiff CF24 4YH, Wales, UK. E-mail: BockenhauerJM@cf.ac.uk; EvansDE@cf.ac.uk
  • , David E. EvansAffiliated withSchool of Mathematics, University of Wales, Cardiff, PO Box 926, Senghennydd Road, Cardiff CF24 4YH, Wales, UK. E-mail: BockenhauerJM@cf.ac.uk; EvansDE@cf.ac.uk
  • , Yasuyuki KawahigashiAffiliated withDepartment of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan.¶E-mail: yasuyuki@ms.u-tokyo.ac.jp

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Abstract:

In this paper we further analyze modular invariants for subfactors, in particular the structure of the chiral induced systems of M-M morphisms. The relative braiding between the chiral systems restricts to a proper braiding on their “ambichiral” intersection, and we show that the ambichiral braiding is non-degenerate if the original braiding of the N-N morphisms is. Moreover, in this case the dimensions of the irreducible representations of the chiral fusion rule algebras are given by the chiral branching coefficients which describe the ambichiral contribution in the irreducible decomposition of α-induced sectors. We show that modular invariants come along naturally with several non-negative integer valued matrix representations of the original N-N Verlinde fusion rule algebra, and we completely determine their decomposition into its characters. Finally the theory is illustrated by various examples, including the treatment of all SU(2) k modular invariants.