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

Naveen Balaji Umasankar, Arpit Das

Abstract

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.

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

Abstract

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 and , we encode genus-one couplings by a non-negative integer matrix pairing characters and satisfying modular intertwiner relations. Invertible topological defect lines act directly on this gluing data (Verlinde lines diagonally via -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 theory. The unique meromorphic CFT serves as a universal testbed, producing new defect partition functions and clarifying the roles of and . Finally, we outline extensions to higher central charges (e.g.\ ), yielding modular-invariant non-meromorphic theories beyond the Schellekens landscape \cite{Schellekens:1992db} as defect/interface descendants of meromorphic parents.
Paper Structure (33 sections, 424 equations, 1 figure, 1 table)

This paper contains 33 sections, 424 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Equatorial projection framework
  4. Topological defect lines
  5. Half-gauging
  6. Equatorial projection principle
  7. Double-sided equatorial projection
  8. Example I: $(B_{r,1}, B_{7-r,1})$
  9. Example II: $(D_{r,1}, D_{r,1})$
  10. Towards a classification
  11. Orbifold Construction
  12. $B_{r,1}$ orbifolds
  13. $A_{r,1}$ orbifolds
  14. $D_{r,1}$ orbifolds
  15. $G_{r,1}$ orbifolds
  16. ...and 18 more sections

Figures (1)

  • Figure 1: Gluing chiral hemispheres: a VOA $V$ with $\text{Rep}(\mathcal{C})$ on the northern hemisphere and $\widetilde{V}$ with $\text{Rep}(\widetilde{\mathcal{C}})$ are sewn along the boundaries at points $i\in I_{\mathcal{C}}$ and $j\in I_{\widetilde{\mathcal{C}}}$ by gluing matrix $M$, yielding the glued sphere with equator $S_{\text{eq}}^{1}$.