Modular invariants from subfactors: type I coupling matrices and intermediate subfactors
A type III factor \(N\) with a finite braided system of endomorphisms \(_N {\mathcal X}_N\) produces ``statistics'' unitary matrices \(S\) and \(T\) that define a finite dimensional representation of the modular group \(\text{SL}(2,{\mathbb Z})\) [cf. \textit{K.-H. Rehren}, ``Braid group statistics and their superselection rules'', in \textit{Kastler D.} (ed.): ``The algebraic theory of superselection sectors'' (Palermo, 1989), World Sci. Publishing, River Edge, NJ, 333-355 (1990)]. As shown in [\textit{J. Böckenhauer}, \textit{D. E. Evans} and \textit{Y. Kawahigashi}, Commun. Math. Phys. 208, No. 2, 429-487 (1999; Zbl 0948.46048)], the matrix \(Z=(Z_{\lambda ,\mu})_{\lambda,\mu \in _N {\mathcal X}_N}\) with entries \[ Z_{\lambda ,\mu} =\langle \alpha_\lambda^+ ,\alpha_\mu^- \rangle , \] where \(\alpha_\lambda^\pm\) denotes the \(M-M\) morphism obtained from \(\lambda\) by \(\alpha\)-induction, using braiding and respectively opposite braiding, \(\langle \;,\;\rangle\) denotes the dimension of the interwiner space, and \(N\subset M\) is a finite index inclusion of type III factors compatible with the system \(_N {\mathcal X} _N\). The matrix \(Z\) is not necessarily of type I. The authors show within this framework the existence of two intermediate subfactors \(N\subset M_\pm \subset M\), naturally associated to the vacuum column, respectively to the vacuum row of \(Z\), such that the inclusions \(N\subset M_\pm\) give rise to coupling matrices \(Z^\pm\) of type I with \(Z^+_{\lambda,0}=Z_{\lambda,0}\) and \(Z^-_{0,\lambda}=Z_{0,\lambda}\). Moreover, the entries of the ``type I parents'' \(Z^\pm\) of \(Z\) are expressed as \[ Z^\pm_{\lambda,\mu} =\sum\limits_{\tau \in _M {\mathcal X}^0_M} \langle \tau ,\alpha_\lambda^\pm \rangle \langle \tau, \alpha_\mu^\pm \rangle ,\qquad \lambda,\mu \in _N {\mathcal X} _N , \] where \(_M {\mathcal X}^0_M\) is the ambichiral system associated to the system \(_N {\mathcal X}_N\) by \(\alpha\)-induction. In the process, the Galois type correspondence from [\textit{M. Izumi}, \textit{R. Longo} and \textit{S. Popa}, J. Funct. Anal. 155, No. 1, 25-63, (1998; Zbl 0915.46051)] is essentially used. The relationship between the new statistics matrices associated to the braided ambichiral systems \(_{M_\pm} {\mathcal X}^0 _{M_\pm}\) and the original \(S\) and \(T\) matrices is also established. Finally, using the \(\text{SO}(16l)\) current algebra models at level \(1\), one shows the existence of occurences where \(M_+ \neq M_-\) and \(Z^+ \neq Z^-\).
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