Topological operators, noninvertible symmetries and decomposition

Abstract

In this paper we discuss the relationship between noninvertible topological operators, one-form symmetries, and decomposition of two-dimensional quantum field theories, focusing on two-dimensional orbifolds with and without discrete torsion. As one component of our analysis, we study the ring of dimension-zero operators in two-dimensional theories exhibiting decomposition. From a commutative algebra perspective, the rings are naturally associated to a finite number of points, one point for each universe in the decomposition. Each universe is canonically associated to a representation, which defines a projector, an idempotent in the ring of dimension-zero operators. We discuss how Wilson lines on boundaries of two-dimensional theories decompose, and compute actions of projectors. We discuss one-form symmetries of the rings, and related properties. We also give general formulas for projection operators, which previously were computed on a case-by-case basis. Finally, we propose a characterization of noninvertible higher-form symmetries in this context in terms of representations. In that characterization, non-isomorphic universes appearing in decomposition are associated with noninvertible one-form symmetries. August 2021 D Miscellaneous group cohomology results 72 References 73

Generalized Symmetry Breaking Scales and Weak Gravity Conjectures

“…These are operators that, like more familiar symmetries, do not change under small deformations of their positions, but whose fusion algebra is more general than that of a group. These non-invertible symmetries have recently been explored in [53,54,[66][67][68][69][70][71]. Other arguments for conditions similar to (3.29) have appeared in [50][51][52].…”

Non-invertible topological defects in 4-dimensional $\mathbb{Z}_2$ pure lattice gauge theory

“…One of them is so-called "non-invertible symmetry." In 2 dimensions, such non-invertible symmetries are described by "fusion categories" and investigated actively in many papers including [2,3,4,5,6,7,8,9,10,11,12,13]. Non-invertible symmetries in higher dimensions are less understood than those in 2 dimensions.…”

Universal Deformations

“…Here we will study QFTs with local topological operators U α (x), where x is a point on the spacetime manifold, see e.g. [9,18,19,[24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. These operators generate a (d − 1)-form symmetry.…”