Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography

TL;DR

The paper tackles the fusion-rule data of noninvertible and categorical symmetries in topological orders and CFTs by developing a bulk semionization framework for $Z_N$-symmetric models. It constructs a bulk semion algebra via extension of a bosonic SFC to a $Z_N$-graded FC and extracts a subalgebra that encodes CCFT/TO fusion data, establishing a bulk-edge (SymTFT) correspondence. A key result is a ring isomorphism linking the bulk semion algebra to CCFT fusion data, formalizing topological holography and allowing CCFT fusion to be obtained from bulk CFT data; the Ising/Majorana example illustrates a double-semion structure and dualities, including a vanishing fusion scenario for different bulk representations. The framework is then related to established category-theoretic structures, suggesting a route to universal CCFT/TO descriptions across dimensions and highlighting potential connections to RG flows and generalized symmetry concepts.

Abstract

The algebraic or ring structure of anyons, called the fusion rule, is one of the most fundamental research interests in contemporary studies on topological orders (TOs) and the corresponding conformal field theories (CFTs). Recently, the algebraic structure realized as generalized symmetry, including non-invertible and categorical symmetry, captured attention in the fields. Such non-abelian anyonic objects appear in a bulk CFT or chiral CFT (CCFT), but it has been known that the construction of a CCFT contains theoretical difficulties in general. In this work, we study the structure of the fusion rules in $Z_{N}$ symmetric chiral and bulk conformal field theories and the corresponding TOs. We propose a nontrivial expression of subalgebra structure in the fusion rule of a bulk CFT. We name this subalgebra ``bulk semion". This corresponds to the fusion rule of the CCFT and categorical symmetry of the TO or symmetry topological field theory (SymTFT). This gives a bulk-edge correspondence based on the symmetry analysis and corresponds to an anyon algebraic expression of topological holography in the recent literature. The recent topological holography is expected to apply to systems in general space-time dimensions. Moreover, we present a concise way of unifying duality (or fractional supersymmetry), generalized or categorical symmetry, and Lagrangian subalgebra. Our method is potentially useful to formulate and study general TOs, fundamentally only from the data of bulk CFTs or vice versa, and gives a clue in understanding CCFT (or ancillary CFT more generally).

Figures (2)

• Figure 1: Relationship between category theories and our formalism. We note the symbols $\phi, \theta, \Psi, \Phi$ in the corresponding theories to clarify the relations. The blue arrow corresponds to the boson condensation discussed in Bais:2008niKong:2013ayaKawahigashi:2021hds. The yellow arrows correspond to the simple current extension. The respective red arrow or combination of red arrows is called topological holography or sandwich construction in modern literature. The resultant $Z_{N}$ graded CCFT seems to correspond to a premodular fusion category (PMFC)Wang:2023uojSohal:2024qvqEllison:2024svgKikuchi:2024ibt, but further investigations are necessaryFukusumi:2023psx. The $Z_{N}$ semionization $\{\theta_{a,0}\}\rightarrow \{\phi_{a,p}\}$ in the figure corresponds to the procedure in the author's worksFukusumi:2022xxeFukusumi_2022_c but this is different from $Z_{N}$ extension of MTC in etingof2009weaklygrouptheoreticalsolvablefusionBarkeshli:2014cna. Similar interesting figures can be seen in Lu:2025gptEtingof:2009yvg, but they are also different from the $Z_{N}$ extension in this manuscript (We thank Zhengdi Sun for the related discussions clarifying this point).
• Figure 2: A proposal constructing $D$-dimensional Ancillary CFT (ACFT) and $D+1$-dimensional TO from $D$-dimensional CFT. The ACFT proposed in Nishioka:2022ook is the generalization of a chiral CFT in general space-time dimensions. The doubling trick PhysRevLett.54.1091Cardy:1986gw, a method to relate a bulk and chiral CFT, has also been revisited in Nishioka:2022ook. This is a modification of the similar figure in Fukusumi:2023psx by the author. We also note a related proposal by using category theoryKong:2024ykr.