Fixed points and fusion rings. Part 1

Abstract

In the first of this two-part series, we find `fixed point factorisation' formulas, towards an understanding of the fusion ring of WZW models. Fixed-point factorisation refers to the simplifications in the data of a CFT involving primary fields fixed by simple-currents. Until now, it has been worked out only for SU(n), where it has developed into a powerful tool for understanding the fusion rings of WZW models of CFT -- e.g. it has lead to closed formulas for NIM-reps and D-brane charges and charge-groups. In this paper, we generalise these formulas to the other classical algebras, laying the groundwork for future applications to fusion rings (Part 2). We also discuss connections with the twining characters of Fuchs-Schellekens-Schweigert.Comment: 22 page

Modular Invariants and Twisted EquivariantK-theory II: Dynkin diagram symmetries

“…Something similar to Conjecture 3 will hold for all other compact connected groups, and this should be worked out. As a first step, the analogue of fixed-point factorisation has been found for all G [3]. That there can possibly be subtleties is hinted by G = Spin(8) at level 2 (so κ = 8), for Z the full centre Z 2 × Z 2 .…”

“…Subfactor theory elegantly describes these extra structures but there still remains the problem of showing existence. For example, to establish that all modular invariants for SU (3) are realised by subfactors, Evans-Pugh [29,30] start with the nimrep graphs, construct for them the Boltzmann weights, construct from this the subfactors, compute the nimreps and recover the graphs they started with. What is needed is something more systematic, which for instance does not need to know what the nimrep graphs are expected to be.…”

“…The usual spectral sequence argument gives H 3 Gn/Z d (G n /Z d ; Z) Z d × Z, using H * Gn/Z d (pt; Z) Z⊕0⊕0⊕Z d ⊕Z⊕· · · and H 3 (G n /Z d ; Z) Z⊕0⊕0⊕Z⊕· · · ; likewise H 1 Gn/Z d (G n /Z d ; Z 2 ) Z 2 or 0, again depending on whether or not d is even. Finally,…”