Duality Defects in $E_8$

TL;DR

This paper classifies and constructs non-invertible duality defects in the $E_8$ lattice VOA, connecting algebraic, topological, and physical perspectives to advance understanding of symmetries in conformal field theories.

Contribution

It systematically classifies duality defects in the $E_8$ VOA arising from $bZ_m$ symmetries and provides explicit constructions and proofs using VOA and topological field theory techniques.

Findings

Classification of duality defects from $bZ_m$ symmetries.

Explicit defect partition functions for small $m$.

Rigorous proof linking VOA automorphisms to Tambara-Yamagami actions.

Abstract

We classify all non-invertible Kramers-Wannier duality defects in the $E_8$ lattice Vertex Operator Algebra (i.e. the chiral $(E_8)_1$ WZW model) coming from $\mathbb{Z}_m$ symmetries. We illustrate how these defects are systematically obtainable as $\mathbb{Z}_2$ twists of invariant sub-VOAs, compute defect partition functions for small $m$, and verify our results against other techniques. Throughout, we focus on taking a physical perspective and highlight the important moving pieces involved in the calculations. Kac's theorem for finite automorphisms of Lie algebras and contemporary results on holomorphic VOAs play a role. We also provide a perspective from the point of view of (2+1)d Topological Field Theory and provide a rigorous proof that all corresponding Tambara-Yamagami actions on holomorphic VOAs can be obtained in this manner. We include a list of directions for future studies.