Verlinde lines, anyon permutations and commutant pairs inside $E_{8,1}$ CFT

We develop a defect-theoretic refinement of meromorphic 2d CFTs in which the ordinary torus partition function – often just the vacuum character – does not reveal how states organize under symmetry lines. Our central proposal is an \emph{equatorial projection} framework: from a commutant decomposition into commuting rational chiral algebras with categories $\mathcal{C}$ and $\widetilde{\mathcal{C}}$, we encode genus-one couplings by a non-negative integer matrix $M$ pairing characters and satisfying modular intertwiner relations. Invertible topological defect lines act directly on this gluing data (Verlinde lines diagonally via $S$-matrix eigenvalues, and anyon-permuting lines by braided-autoequivalence permutations), making modular covariance of defect amplitudes automatic and sharply distinguishing insertions that yield genuine modular invariants from those defining consistent non-holomorphic interfaces. We further show that the \emph{replacement rules} of \cite{Hegde:2021sdm, Lin:2019hks} arise as equatorial projections of defect actions, and we extend these constructions beyond two-character examples by systematically treating three-character commutant pairs in the $E_{8,1}$ theory. The unique $c=8$ meromorphic CFT $E_{8,1}$ serves as a universal testbed, producing new defect partition functions and clarifying the roles of $\mathrm{Pic}(\mathcal{C})$ and $\mathrm{Aut}^{\mathrm{br}}(\mathcal{C})$. Finally, we outline extensions to higher central charges (e.g.\ $c=32,40$), yielding modular-invariant non-meromorphic theories beyond the $c=24$ Schellekens landscape \cite{Schellekens:1992db} as defect/interface descendants of meromorphic parents.

💡 Research Summary

The paper introduces a novel “equatorial projection” framework to systematically organize and compute torus partition functions of meromorphic two‑dimensional conformal field theories (CFTs) when topological defect lines (TDLs) are inserted. In a meromorphic CFT the ordinary torus amplitude collapses to a single vacuum character, obscuring how states transform under non‑local symmetries. The authors remedy this by considering a pair of commuting rational chiral algebras, denoted 𝒞 and 𝒞̃, each equipped with a modular tensor category (MTC) of representations. The gluing of left‑ and right‑moving sectors along the “equator” of the torus is encoded by a non‑negative integer matrix M whose entries M_{ij} count how many times the i‑th simple object of 𝒞 is paired with the j‑th simple object of 𝒞̃. Consistency demands that M intertwines the modular S‑ and T‑matrices of the two categories: S_𝒞 M = M S_𝒞̃ and T_𝒞 M = M T_𝒞̃.

The central insight is that any invertible TDL—whether a Verlinde line (associated with a simple object of the MTC) or an anyon‑permuting line (associated with a braided auto‑equivalence)—acts on the gluing data by a simple matrix transformation. A Verlinde line labelled by a simple object a contributes a diagonal matrix Λ_a whose entries are the S‑matrix eigenvalues for a; inserting this line along the temporal cycle transforms M → Λ_L M Λ_R, where Λ_L and Λ_R are the diagonal matrices for the left‑ and right‑moving copies of the line. An anyon‑permuting defect corresponds to a permutation matrix P implementing the braided auto‑equivalence; its insertion yields M → P M P^{-1}. In the most general case of a product of such defects, the transformation can be written as M → R_L M R_R, with R_L,R_R belonging to the group generated by diagonal S‑eigenvalue matrices and permutation matrices. Because the modular intertwining conditions are covariant under conjugation, the defect‑deformed matrices automatically satisfy the same modular constraints, guaranteeing modular covariance of all defect amplitudes.

The authors show that the “replacement rules” previously introduced in the literature (e.g., Hegde–Lin) are precisely special instances of these defect actions: replacing one side of a commutant pair while preserving modular invariance corresponds to acting with a permutation matrix on the right (or left) factor of M. Thus the replacement rules acquire a clear physical interpretation as the effect of specific invertible defects.

The framework is illustrated in depth using the unique c = 8 meromorphic CFT, E₈₁, which serves as a clean laboratory because its vacuum character is the only torus amplitude in the absence of defects. The authors enumerate all three‑character commutant pairs (𝒞, 𝒞̃) that embed into E₈₁, including (B₁,₁, B₆,₁), (D₂,₁, D₆,₁), (II₂, G⊗₂₂,₁), (A₄,₁, A₄,₁), and (A⊗₂₂,₁, A⊗₂₂,₁). For each pair they compute the explicit gluing matrix M, identify the Picard group Pic(𝒞) (generated by invertible simple currents) and the braided auto‑equivalence group Aut^{br}(𝒞), and demonstrate how these groups act on M. The resulting defect partition functions fall into two classes: (i) those that reorganize into genuine modular‑invariant, non‑holomorphic partition functions, interpretable as new rational CFTs; and (ii) those that remain consistent genus‑one observables but cannot be interpreted as full CFT partition functions, instead describing conformal interfaces (defect boundaries) between distinct theories.

Beyond the c = 8 case, the authors argue that the same construction applies to higher‑central‑charge meromorphic theories, such as c = 32 and c = 40. By selecting appropriate commutant pairs inside these larger theories, one can generate a plethora of non‑holomorphic modular invariants that lie outside the classic Schellekens c = 24 classification. In this way, the equatorial projection framework provides a systematic method to produce new RCFTs and conformal interfaces from known meromorphic parents, extending the landscape of admissible two‑dimensional CFTs.

In conclusion, the paper unifies defect actions, modular intertwining, and the replacement‑rule paradigm into a single algebraic language based on the gluing matrix M. This not only clarifies the role of Verlinde and anyon‑permuting lines in shaping torus amplitudes but also offers a practical computational toolkit for constructing and classifying defect‑decorated meromorphic CFTs and their non‑holomorphic descendants. Future directions include applying the framework to four‑dimensional supersymmetric constructions, exploring connections with higher‑form symmetries, and systematically classifying all admissible commutant pairs for higher‑c meromorphic theories.