We review the framework subfactors provide for understanding modular invariants. We discuss the structure of a generalized Longo-Rehren subfactor and the relationship between the coupling matrices of such subfactors, modular invariance and local extensions. We relate results of Kostant, in the context of the McKay correspondence for finite subgroups of SU(2), to subfactors. A direct proof of how alpha-induction produces modular invariants is presented.

arXiv:math/0204281Apr 23, 200217 citations

Physics Letters B

Modular invariants from simple currents. An explicit proof

A.N. Schellekens, Shimon Yankielowicz B

Published Aug 1, 1989• 80 citations

Publisher

DOI: 10.1016/0370-2693(89)90948-9

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The twisted convolution identity and ghost

r

-SICs from finite quantum dilogarithms

Radchenko and Wheeler (RW) recently proved a finite pentagon relation for real quadratic special values of the modular quantum dilogarithm and used it to establish the rank-

1

twisted convolution identity conjectured by the current authors. RW gave an explicit argument in the principal case and remarked that their proof holds for all rank-

1

admissible tuples. We extend their proof to all rank-

r

admissible tuples and provide an explicit dictionary between the modular quantum dilogarithm and the Shintani-Faddeev modular cocycle conventions in the respective papers. Thus, we establish that, if

d,r

are positive integers such that

r<\frac{d-1}{2}

and

\frac{d^2-1}{r(d-r)} \in \mathbb{Z}

, then there exist ghost

r

-SICs: i.e., configurations of

d^2

rank-

r

subspaces in

\mathbb{C}^d

that satisfy a non-Hermitian equichordal condition. Under the Stark conjecture, these configurations are Galois conjugate to Hermitian equichordal configurations called

r

-SICs (or rank-

r

SIC-POVMs).