Non periodic Ishibashi states: the su(2) and su(3) affine theories

TL;DR

This paper extends the understanding of boundary conditions and symmetry actions in su(2) and su(3) affine theories on a cylinder by incorporating non-periodic Ishibashi states and analyzing their impact on twisted partition functions.

Contribution

It introduces a formalism for including all non-periodic bulk sectors in the analysis of boundary conditions and symmetry actions in affine theories, with explicit results for su(2) and su(3).

Findings

Determines the action of symmetry groups on boundary conditions.

Computes twisted partition functions for su(2) and su(3).

Provides a graph-based summary of symmetry properties.

Abstract

We consider the su(2) and su(3) affine theories on a cylinder, from the point of view of their discrete internal symmetries. To this end, we adapt the usual treatment of boundary conditions leading to the Cardy equation to take the symmetry group into account. In this context, the role of the Ishibashi states from all (non periodic) bulk sectors is emphasized. This formalism is then applied to the su(2) and su(3) models, for which we determine the action of the symmetry group on the boundary conditions, and we compute the twisted partition functions. Most if not all data relevant to the symmetry properties of a specific model are hidden in the graphs associated with its partition function, and their subgraphs. A synoptic table is provided that summarizes the many connections between the graphs and the symmetry data that are to be expected in general.