Duality Defects in $E_8$

TL;DR

This work classifies non-invertible Kramers-Wannier duality defects in the holomorphic $E_8$ lattice VOA ($V_{E_8}$, the chiral $(E_8)_1$ WZW model) arising from cyclic symmetries $\mathbb{Z}_m$ by realizing them as $\mathbb{Z}_2$ twists of invariant sub-VOAs and computing defect partition functions for small $m$. The authors develop a concrete, physically motivated framework that uses automorphisms of lattice VOAs, Kac's theorem on finite-order automorphisms, and a (2+1)d TFT perspective to construct and verify Tambara-Yamagami actions on holomorphic VOAs, including explicit results for $m=2,3,4,5$ and a computer-aided approach for higher $m$. They connect these 2d defect constructions to 2+1d anyon condensation and gapped boundaries, demonstrating how the axis-swapping $\mathbb{Z}_2$ action on $\mathrm{Irr}(V^{\mathbb{Z}_m})$ yields the TY defect lines and their defected partition functions, with cross-checks against fermionization and Potts-model realizations. The work provides a systematic procedure and a unifying TFT viewpoint that should extend to other holomorphic VOAs and higher-dimensional settings, and it outlines open questions about full associator data and higher-ality generalizations.

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.

Figures (7)

• Figure 1: On the left, an $X$ defect line (red) is inserted along a spatial slice, this gives a "twist" by $X$ in the time direction. On the right, the defect line is inserted along the time direction. Periodicitizing this setup on the right, we get the trace over the $X$-twisted Hilbert space $\mathcal{H}_X$.
• Figure 2: On the left, a topological defect line $X$ encircles a $\phi$ insertion in the plane. Since the line is topological, this is equivalent instead to the local operator $X\phi$. On the right, a topological defect line $X$ sweeps past the $\phi$ insertion leaving behind a local operator and a defect operator $\phi^\prime$. The defect operator is connected to $X$ by a topological tail (dotted).
• Figure 3: Left, the $X_\sigma$ defect sweeps past a $\sigma$ primary, turning it into the disorder operator $\mu$, plus a topological $\mathop{\mathrm{\mathbb{Z}}}\nolimits_2$ tail. Such a $\mu$ is traditionally called a "twist-field." Right, the $X_\sigma$ defect sweeps past the relevant operator $\epsilon$ and turns it into $-\epsilon$.
• Figure 4: Mathematically, the associator $\alpha$ is an element of $\mathop{\mathrm{Hom}}\nolimits((X_g\otimes X_h) \otimes X_k, X_g\otimes (X_h \otimes X_k))$ that defines in what way the two tensor products are equal. Physically, given a theory with non-trivializable anomaly $\alpha \in H^3(G,U(1))$, two networks of symmetry defect lines can be different up to a $U(1)$ phase.
• Figure 5: Given 4 defect lines $X_g$, $X_h$, $X_k$, and $X_{\ell}$, we only demand that the pentagon diagram commutes. This is familiar to those in RCFT as the pentagon identity for F-symbols.

Theorems & Definitions (7)

• Theorem 1: Theorem 8.6, kac_1990
• Proposition 1: DMNO, Proposition 4.8
• Proposition 2
• proof
• Proposition 3: carnahan2016regularity, Corollary 5.25
• Theorem 2
• proof