Extending fusion rules with finite subgroups: For a general understanding of quotient or gauging

TL;DR

The paper develops a systematic procedure to extend conformal field theories by $Z_{N}$ symmetry, constructing $Z_{N}$-graded extended fusion rings and corresponding modular partition functions, including nonanomalous subgroups $Z_{n}$. By leveraging symmetry-topological-field-theory (SymTFT) and the folding trick, it links these algebraic structures to domain walls and RG flows that preserve quotient group structures. It further provides general methods for extending tensored models, deriving partition functions for coupled systems via the least common multiple (lcm) and analyzing domain-wall preserving gcd structures, and uses operator counting with zero modes to demonstrate the utility of extended chiral fusion rings. The framework offers a pathway to classify symmetry-enriched topological orders and gauging phenomena in 2D CFTs/TQFTs, with potential applications to quantum spin liquids and related condensed-matter systems.

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 the $Z_{N}$ graded symmetry topological field theories and also provides algebraic data of smeared boundary conformal field theories describing 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.

Figures (2)

• Figure 1: The summary of the operations in this manuscript. Starting from a $Z_{N}$ symmetric chiral CFT or the corresponding symmetry topological field theory (SymTFT) $\mathbf{M}$, one can obtain the corresponding bulk CFT $\mathbf{B}$ or bulk fusion ring (black arrow). By extending this bosonic bulk CFT by chiral $Z_{N}$ operations or defects, one can obtain the $Z_{N}$ extended model (upper blue arrow). For the details of the chiral analog, $Z_{N}$ semionization, see the discussion in Fukusumi:2022xxeFukusumi_2022_cFukusumi:2023psx (lower blue arrow). It should be stressed that this operation is not a straightforward $Z_{N}$ extension of the bosonic chiral algebra or MTC in the categorical formalism in turaev2000homotopyfieldtheorydimensionkirillov2001modularcategoriesorbifoldmodelsFrohlich:2003hmmueger2004galoisextensionsbraidedtensoretingof2009weaklygrouptheoreticalsolvablefusion. By taking a subalgebra (or subring), one can obtain the SymTFT $\mathbf{S}$ (red arrows). By applying quotient or gauging operations to $\mathbf{F}$ or $\mathbf{S}$ by a subgroup $Z_{N'}\subset Z_{N}$, one will obtain the gauged model $\mathbf{F}/Z_{N'}$ or $\mathbf{S}/Z_{N'}$ respectively (purple arrows). The integer spin simple current orbifolding can be realized as the combinations of the lower two arrows under the condition $N=N'$. Roughly speaking, the extended model corresponds to a symmetry-enriched TO in condensed matter, and the quotient model corresponds to a quantum spin liquid in condensed matter.
• Figure 2: The relationship between theories. The quotient theory is defined by orbifolded theory $\mathbf{D}_{i}=\mathbf{B}_{i}/Z_{N_{i}/\text{gcd}(N_{1},N_{2})}$ (or parity zero sector of $\mathbf{F}_{i}/Z_{N_{i}/\text{gcd}(N_{1},N_{2})}$) corresponding to the red arrows in the figure. By applying the folding trick (black arrow) and quotient operation to $\mathbf{B}_{1} \otimes \overline{\mathbf{B}_{2}}$, one will obtain the (homo)morphism between $\mathbf{D}_{1}$ and $\mathbf{D}_{2}$. The new partition function and the corresponding extended theory $\mathbf{F}_{1\otimes 2}$ (blue arrow) will provide the algebraic data of the conformal interfaces. One can see the dual relationship between greatest common divisor and lowest common multiple as a dual relationship between quotient and extension. Hence, for further understanding of the quotient or gauging structure, it is necessary to study the extension of theories more thoroughly.