A classifying algebra for boundary conditions

TL;DR

This work addresses boundary conditions in rational conformal field theories with non-diagonal ${D_{\rm odd}}$-type modular invariants arising from a ${\mathbb Z}_2$ simple current. It constructs a finite-dimensional classifying algebra ${\tilde{\mathfrak A}}$ that governs boundary data, incorporating the charge-zero subalgebra ${\mathfrak A}_0$ and fixed-point (twisted) sectors via a fixed-point theory and the orbit Lie algebra via the fixed-point modular matrix ${\breve{S}}$. Boundary conditions correspond to irreps of ${\tilde{\mathfrak A}}$, with distinct handling for fixed points (two signs) and ordinary orbits, and annulus amplitudes are shown to be consistent with these structures. The results generalize known diagonal and integer-spin simple-current cases, providing a unified framework for open-closed RCFT with nontrivial modular invariants and suggesting connections to geometry and arithmetic symmetries.

Abstract

We introduce a finite-dimensional algebra that controls the possible boundary conditions of a conformal field theory. For theories that are obtained by modding out a Z_2 symmetry (corresponding to a so-called D_odd-type, or half-integer spin simple current, modular invariant), this classifying algebra contains the fusion algebra of the untwisted sector as a subalgebra. Proper treatment of fields in the twisted sector, so-called fixed points, leads to structures that are intriguingly close to the ones implied by modular invariance for conformal field theories on closed orientable surfaces.