Duality defects in E8

Abstract

We classify all non-invertible Kramers-Wannier duality defects in the E8 lattice Vertex Operator Algebra (i.e. the chiral (E8)1 WZW model) coming from ℤm symmetries. We illustrate how these defects are systematically obtainable as ℤ2 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.

The Branes Behind Generalized Symmetry Operators

“…We find that when the $$SL(2,\mathbb {Z})$$ bundle of the F‐theory model is non‐trivial, these models generically have a non‐invertible symmetry simply because the fusion algebra for the generalized symmetry operators contains multiple summands. This is quite analogous to what has been observed in the context of various field theoretic constructions [ 14,38,43,44,55,56,62,64,65,74,80–82,84,85,87,96,98,99,102,128,131–143 ] as well as some recent holographic models. [ 100,101 ]…”

Exploring non-invertible symmetries in free theories

“…Non-invertible symmetries are long known to exist in two-dimensional rational CFTs since the work of Verlinde [2] (see also [3][4][5][6][7][8][9][10][11][12][13][14]). It was recently realized that non-invertible symmetries also exist in non-conformal theories in two dimensions [15], as well as in QFTs in higher dimensions [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30], and have turned out to be useful in various contexts [31][32][33][34][35][36][37][38][39][40].…”

Non-invertible defects in 5d, boundaries and holography

“…A Wilson Loop deep in the bulk may then move towards the boundary and open up. 14 Notice though that Wilson lines cannot open anywhere else in the bulk, so that the 1-form symmetry is not explicitly broken outside of the boundary. Instead, points at which Wilson lines end define objects naturally charged under the boundary 0-form symmetry U(1) (0) ∂ (see the discussion above).…”

“…Symmetries obeying a non-invertible fusion law, usually called non-invertible symmetries, have been studied in 2d QFT's [2][3][4][5][6][7][8][9][10][11][12][13][14]. They have also been recently realized in higher dimensions by introducing duality defects, a generalization of the usual Krammers-Wannier duality defect [15][16][17][18][19][20], and condensation defects from gauging p-form symmetries in submanifolds [21].…”