Coset Graph Fusion Algebra

• Coset Graph Fusion Algebra is a commutative, associative algebra that encodes the fusion rules of sectors—such as boundaries and defects—in conformal field theories using coset graphs.
• It constructs fusion rules from coset graphs based on A–D–E Dynkin diagrams, where vertices represent physical sectors and the fusion product is defined via graph adjacency matrices.
• This framework enables explicit evaluation of modular invariants, quantum dimensions, and defect entropies across minimal and logarithmic conformal field theories.

A coset graph fusion algebra is a commutative, associative algebra whose structure constants encode the fusion rules of sectors—boundary conditions, defects, or representations—in rational, quasi-rational, or logarithmic conformal field theories (CFTs), typically constructed from coset vertex operator algebras or related lattice models. The algebraic structure is read directly from the adjacency matrices of a combinatorial object known as the coset graph, most notably of A–D–E Dynkin type and their various extensions. These coset graph fusion algebras unify the analysis of bulk and boundary CFT, quantum dimensions, topological defects, and partition function modularity, and arise as the Grothendieck or Verlinde rings of various categories of modules or defect lines.

1. Coset Graph Construction: Definitions and Symmetries

The central object in a coset graph fusion algebra is the coset graph $\widetilde G = (A_{m-1} \otimes G)/\mathbb{Z}_2$, where $A_{m-1}$ and $G$ are simply-laced Dynkin diagrams with coprime Coxeter numbers $m, m'$. The vertices are equivalence classes of pairs $(r,s)$ with $1 \leq r \leq m-1$, $1 \leq s \leq m'-1$, modulo the Kac conjugation $(r,s)\sim(m-r, m'-s)$ (for $G=A$) or $(r,a)\sim(m-r,a)$ (for $G\neq A$). This $\mathbb{Z}_2$ quotient enforces the symmetry structure of the associated minimal models and matches the identification of physical sectors under the fusion rules (Pearce et al., 26 Dec 2025, Pearce et al., 2010).

Edges of the coset graph connect vertices that are adjacent in the direct product graph $A_{m-1} \otimes G$; the result is a finite, typically regular, undirected graph. The Chebyshev recursion can be used to generate all adjacency (nimrep) matrices systematically from the fundamental one (Pearce et al., 26 Dec 2025).

2. Algebraic Structure: Fusion Rules and Graph Algebras

Each vertex of the coset graph corresponds to an elementary defect, boundary condition, or Grothendieck generator. The fusion product is defined by

$$(\mu)\times(\nu) = \sum_\rho \tilde N_{\mu\nu}{}^\rho\;(\rho),$$

where $\tilde N_{\mu\nu}{}^\rho$ is the number of admissible fusion channels compatible with the graph structure. The key facts:

• The set of fusion matrices $\{\tilde N_\mu\}$ defines a commutative, associative algebra over $\mathbb{Z}$.
• The fundamental fusion matrix (often labeled by the generator corresponding to the minimal adjacency or the "simple defect") suffices to generate all others recursively.

Diagonalization leads to a Verlinde-type formula for the structure constants:

$$\tilde N_{\mu\nu}{}^\rho = \sum_\ell \frac{\Psi_\mu^\ell \Psi_\nu^\ell}{\Psi_1^\ell} \overline{\Psi_\rho^\ell},$$

where $\Psi$ is the unitary matrix simultaneously diagonalizing the nimreps and the entry with label 1 is normalized to the identity (Pearce et al., 26 Dec 2025).

3. Coset and Diagonal Coset Vertex Operator Algebras

In the framework of vertex operator algebras (VOAs), the coset construction is realized as the commutant $C = \operatorname{Com}_{V_1}(W)$, where $V_1$ is a tensor product of affine VOAs $L_\mathfrak{g}(k,0) \otimes L_\mathfrak{g}(l,0)$ and $W \cong L_\mathfrak{g}(k+l,0)$ is the diagonal subalgebra (Lin, 2021, Feng et al., 18 Sep 2025).

The irreducible modules of $C$ correspond to orbits $[\lambda, \mu; \nu]$ (with appropriate charge-conservation constraints) in the weight lattice, and their fusion rules are tensor products of the individual affine fusion coefficients, subject to charge conservation $\lambda+\mu-\nu\in Q$. The result is:

$$[M^{[\lambda_1, \mu_1; \nu_1]}] \times [M^{[\lambda_2, \mu_2; \nu_2]}] = \sum_{[\lambda_3, \mu_3; \nu_3]} N^{(k)}_{\lambda_1, \lambda_2}{}^{\lambda_3} N^{(l)}_{\mu_1, \mu_2}{}^{\mu_3} N^{(k+l)}_{\nu_1, \nu_2}{}^{\nu_3} [M^{[\lambda_3, \mu_3; \nu_3]}],$$

a commutative, associative fusion ring (Lin, 2021). This structure is sometimes referred to as the "coset-graph fusion algebra."

A concrete example is the coset $$C = C(L_{\widehat{osp(1|2)}(2,0)}, L_{\widehat{osp(1|2)}(1,0)}^{\otimes 2})$$, where $C$ is realized as a simple current extension of the rational Virasoro VOA $L(c_{10,7},0)$, with $27$ inequivalent irreducible modules labeled by $(r,s)$, $r=1,3,5$, $1 \leq s \leq 9$, and fusion rules matching those of $\mathcal{M}(10,7)$ minimal model under a simple current identification (Feng et al., 18 Sep 2025).

4. Connections to Minimal Models, Defects, and Lattice Integrability

In minimal models $\mathcal{M}(m,m')$, the coset-graph fusion algebras encode both the bulk fusion rules and the combinatorics of topological defect lines and boundary conditions. The Perron–Frobenius eigenvector $\tilde d_\mu$ of the fundamental adjacency yields quantum dimensions (defect $g$-factors), satisfying:

$$\tilde d_\mu\,\tilde d_\nu = \sum_\rho \tilde N_{\mu\nu}{}^\rho\;\tilde d_\rho,$$

and giving, for example, the Affleck–Ludwig $g$-functions for boundary RG flows and defect entropy differences (Pearce et al., 26 Dec 2025).

The same fusion algebra structure emerges in the IR limit of RSOS lattice models, where transfer matrix fusion (T-system) and Y-system functional equations generate the algebraic recursion satisfied by the quantum dimensions and fusion graphs. Explicit examples—critical Ising $(A_2, A_3)$, tricritical Ising $(A_3, A_4)$, and the 3-state Potts $(A_4, D_4)$—demonstrate the generality (Pearce et al., 26 Dec 2025).

5. Role in Logarithmic and W-Extended Models

In logarithmic minimal models, which are not rational, a closed fusion algebra still arises at the level of the Grothendieck group of projective covers. Here, the twisted-affine coset graph $$A^{(2)}_{p,p'} = A^{(2)}_p \otimes A^{(2)}_{p'}/\mathbb{Z}_2$$ gives the fusion adjacency, with structure constants calculated directly from the graph adjacency, or via a Verlinde-like formula involving the projective modular $S$-matrix (Pearce et al., 2010). These data control both strip and torus partition functions for the $\mathcal{W}$-projective sectors and minimal sectors, including the distinctive features of logarithmic CFT such as multiplicities and indecomposables. The $\mathbb{Z}_2$ quotient is essential to enforce the correct identification of sectors and removal of twist fields (Pearce et al., 2010).

6. Applications: Modular Invariants, Entanglement, and Defect Combinatorics

Coset-graph fusion algebras provide explicit frameworks for evaluating:

• Modular invariant partition functions, both diagonal $A$-type invariants for projective or minimal sectors and their multiplicities.
• Defect line fusion and boundary condition classification, via nimreps of the algebra and corresponding adjacency matrices.
• Quantum dimensions, defect entropies, boundary $g$-functions, and symmetry-resolved entanglement entropies directly from Perron–Frobenius vectors.
• Dilogarithm identities for central charges and conformal weights in minimal models, with the coset quantum dimensions appearing as Y-system asymptotics (Pearce et al., 26 Dec 2025). These structures are universal across both unitary and nonunitary minimal models and their associated RSOS lattice realizations.

7. Tabular Overview: Fundamental Features of Coset-Graph Fusion Algebras

Aspect	Description	Reference
Graph type	$A_{m-1}\otimes G/\mathbb{Z}_2$ (A–D–E)	(Pearce et al., 26 Dec 2025)
Fusion algebra basis	Vertices of the coset graph, equivalence classes	(Lin, 2021)
Structure constants	Read from adjacency, or Verlinde-like formula	(Pearce et al., 26 Dec 2025)
Quantum dimensions	Perron–Frobenius eigenvector of adjacency	(Pearce et al., 26 Dec 2025)
Relation to CFT defects/boundary	Encodes fusion of defects/boundary in minimal models	(Pearce et al., 2010)

• (Feng et al., 18 Sep 2025) Structure and representations of the coset vertex operator algebra $C( L_{\widehat{osp(1|2)}(2,0), L_{\widehat{osp(1|2)}(1,0)^{\otimes 2})$
• (Lin, 2021) Trace functions and fusion rules of diagonal coset vertex operator algebras
• (Pearce et al., 26 Dec 2025) Unitary and Nonunitary A-D-E minimal models: Coset graph fusion algebras, defects, entropies, SREEs and dilogarithm identities
• (Pearce et al., 2010) Coset Graphs in Bulk and Boundary Logarithmic Minimal Models