Extending fusion rules with finite subgroups: A general construction of $Z_{N}$ extended conformal field theories and their orbifoldings

TL;DR

This paper develops a general construction for $Z_{N}$-extended fusion rings and modular partition functions, advancing the understanding of symmetry-graded topological and boundary conformal field theories.

Contribution

It introduces a new method to construct $Z_{N}$ symmetry extended fusion rings and their partition functions, applicable to multicomponent systems and domain wall configurations.

Findings

Constructed $Z_{N}$ extended fusion rings and modular partition functions.

Provided algebraic data for $Z_{N}$-graded topological and boundary conformal theories.

Linked partition functions to domain walls and RG flows via the folding trick.

Abstract

We construct the $Z_{N}$ symmetry extended fusion ring of bulk and chiral theories and the corresponding modular partition functions with nonanomalous subgroup $Z_{n}(\subset Z_{N})$. The chiral fusion ring provides fundamental data for $Z_{N}$- graded symmetry topological field theories and also provides algebraic data for smeared boundary conformal field theories, which describe the zero modes of the extended models. For more general multicomponent or coupled systems, we also obtain a new series of extended theories. By applying the folding trick, their partition functions correspond to charged or gapped domain walls or massless renormalization group flows preserving quotient group structures.