Orientation matters for NIMreps

Abstract

The problem of finding boundary states in CFT, often rephrased in terms of "NIMreps" of the fusion algebra, has a natural extension to CFT on non-orientable surfaces.This provides extra information that turns out to be quite useful to give the proper interpretation to a NIMrep. We illustrate this with several examples. This includes a rather detailed discussion of the interesting case of the simple current extension of A 2 level 9, which is already known to have a rich structure. This structure can be disentangled completely using orientation information. In particular we find here and in other cases examples of diagonal modular invariants that do not admit a NIMrep, suggesting that there does not exist a corresponding CFT. We obtain the complete set of NIMreps (plus Moebius and Klein bottle coefficients) for many exceptional modular invariants of WZW models, and find an explanation for the occurrence of more than one NIMrep in certain cases. We also (re)consider the underlying formalism, emphasizing the distinction between oriented and unoriented string annulus amplitudes, and the origin of orientation-dependent degeneracy matrices in the latter.

Modular Data: The Algebraic Combinatorics of Conformal Field Theory

“…This nim-rep arises as a natural action of M χ M on M χ N . As these partition functions of tori and cylinders appear so nicely here, it is tempting to ask about other surfaces, especially the Möbius band and Klein bottle, which also play a basic role in boundary RCFT [102,107].…”

“…Mathematically, these are nontrivial ring homomorphisms from the fusion ring into Z/mZ for some m. Partition functions associated to nonorientable surfaces (especially the Möbius strip and Klein bottle) are also important in boundary RCFT or open string theory-see e.g. [102,107]. We won't discuss these additional developments further in this paper.…”

Lattice realizations of the open descendants of twisted boundary conditions forsl(2) A–D–Emodels

“…The debate as to whether the constraint (3.50) is necessary for the Klein bottle coefficients is not settled. Refer to [3] for more detail.…”

“…= −χ 1,1 + 2χ 1,2 − 2χ 1,3 + 2χ 1,4 − χ 1,5 − 2χ 3,1 + 4χ 3,2 − 4χ 3,3 + 4χ 3,4 − 2χ 3,5 K(2)(3,3) = χ 1,1 − 2χ 1,2 + 3χ 1,3 − 2χ 1,4 + χ 1,5 + 2χ 3,1 − 4χ 3,2 + 6χ 3,3 − 4χ 3,4 + 2χ3,5 …”

Full Logarithmic Conformal Field theory — an Attempt at a Status Report

“…(Incidentally, even in the rational case the consistency of that full CFT was fully established only relatively recently . In this context it may also be of interest that there exist chiral rational CFTs to which there isn't associated a consistent full CFT with a torus partition function given by the ‘true diagonal’ modular invariant …”