Twisted boundary states and representation of generalized fusion algebra

TL;DR

This work extends the Cardy/NIM-rep framework to twisted boundary states by a finite automorphism group $G$, defining the generalized fusion algebra $\mathcal{F}(\mathcal{A};G)$ that combines ordinary and twisted representations. It proves a Verlinde-like formula for generalized fusion coefficients and constructs irreducible representations that may have dimension greater than one when $G$ is non-abelian. The central result is that any consistent set of twisted boundary states furnishes a NIM-rep of $\mathcal{F}(\mathcal{A};G)$, with explicit checks in $so(8)_1$, $u(1)_k$, $su(3)_k$, and $su(3)_1^{\oplus 3}$ demonstrating non-commutative structure and packing of ordinary NIM-reps into generalized ones. The paper also connects graph fusion algebras to generalized fusion algebras in simple current extensions and discusses broader implications for automorphism lifts and invariant classification, outlining avenues for future NIM-rep classification and algebraic understanding of exceptional invariants.

Abstract

The mutual consistency of boundary conditions twisted by an automorphism group G of the chiral algebra is studied for general modular invariants of rational conformal field theories. We show that a consistent set of twisted boundary states associated with any modular invariant realizes a non-negative integer matrix representation (NIM-rep) of the generalized fusion algebra, an extension of the fusion algebra by representations of the twisted chiral algebra associated with the automorphism group G. We check this result for several concrete cases. In particular, we find that two NIM-reps of the fusion algebra for $su(3)_k (k=3,5)$ are organized into a NIM-rep of the generalized fusion algebra for the charge-conjugation automorphism of $su(3)_k$. We point out that the generalized fusion algebra is non-commutative if G is non-abelian and provide some examples for $G = S_3$. Finally, we give an argument that the graph fusion algebra associated with simple current extensions coincides with the generalized fusion algebra for the extended chiral algebra, and thereby explain that the graph fusion algebra contains the fusion algebra of the extended theory as a subalgebra.

Figures (4)

• Figure 1: Joining two open strings with boundary condition $[\lambda^*, 0]$ and $[0, \mu]$ yields a string with $[\lambda^*, \mu]$.
• Figure 2: Joining two open strings with boundary condition $[\lambda^*, 0]$ and $[0, \tilde{\mu}]$ yields a string with $[\lambda^*, \tilde{\mu}]$. The thick line stands for a twist by the automorphism $\omega \in G$.
• Figure 3: Extended Dynkin diagram of $E_6$. $\theta$ stands for the highest root of $E_6$. Each box expresses the simple roots of $su(3) \subset E_6$.
• Figure 4: The automorphism group $S_3$ of $su(3)^{\oplus 3}$ has a lift to $E_6$.