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arXiv:2408.16811 · Journal of High Energy Physics

Notes on gauging noninvertible symmetries. Part II. Higher multiplicity cases

Alonso Perez-Lona, Daniel G. Robbins, Eric Sharpe, Thomas Vandermeulen, Xingyang Yu

Published May 7, 2025• 8 citations
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Abstract

A bstract In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions, extending our previous work. Specifically, in this work we discuss more general gauged noninvertible symmetries in which the noninvertible symmetry is not multiplicity free, and discuss the case of Rep( A 4 ) in detail. We realize Rep( A 4 ) gaugings for the c = 1 CFT at the exceptional point in the moduli space and find new self-duality under gauging a certain non-group algebra object, leading to a larger noninvertible symmetry Rep(SL(2, ℤ 3 )). We also discuss more general examples of decomposition in two-dimensional gauge theories with trivially-acting gauged noninvertible symmetries.

DOI
10.1007/jhep05(2025)066

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A bstract In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions, extending our previous work. Specifically, in this work we discuss more general gauged noninvertible symmetries in which the noninvertible symmetry is not multiplicity free, and discuss the case of Rep( A 4 ) in detail. We realize Rep( A 4 ) gaugings for the c = 1 CFT at the exceptional point in the moduli space and find new self-duality under gauging a certain non-group algebra object, leading to a larger noninvertible symmetry Rep(SL(2, ℤ 3 )). We also discuss more general examples of decomposition in two-dimensional gauge theories with trivially-acting gauged noninvertible symmetries.
A bstract In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions. We specialize to certain gaugeable cases, specifically, fusion categories of the form Rep(H) \textrm{Rep}\left(\mathcal{H}\right) Rep(H) Rep H for H \mathcal{H} H H a suitable Hopf algebra (which includes the special case Rep( G ) for G a finite group). We also specialize to the case that the fusion category is multiplicity-free. We discuss how to construct a modular-invariant partition function from a choice of Frobenius algebra structure on H∗ {\mathcal{H}}^{\ast } H∗ H ∗ . We discuss how ordinary G orbifolds for finite groups G are a special case of the construction, corresponding to the fusion category Vec( G ) = Rep(ℂ[ G ] * ). For the cases Rep( S 3 ), Rep( D 4 ), and Rep( Q 8 ), we construct the crossing kernels for general intertwiner maps. We explicitly compute partition functions in the examples of Rep( S 3 ), Rep( D 4 ), Rep( Q 8 ), and Rep(H8) \textrm{Rep}\left({\mathcal{H}}_8\right) Rep(H8​) Rep H 8 , and discuss applications in c = 1 CFTs. We also discuss decomposition in the special case that the entire noninvertible symmetry group acts trivially.
We use the language of von Neumann subfactors to investigate non-invertible symmetries in two dimensions. A fusion categorical symmetry \mathcal{C} 𝒞 , its module category \mathcal{M} ℳ , and a gauging labeled by an algebra object \mathcal{A} 𝒜 are encoded in the bipartite principal graph of a subfactor. The dual principal graph captures the quantum symmetry \mathcal{C}' 𝒞 ′ obtained by gauging \mathcal{A} 𝒜 in \mathcal{C} 𝒞 , as well as a reverse gauging back to \mathcal{C} 𝒞 . From a given subfactor N \subset M N ⊂ M , we derive a quiver diagram that encodes the representations of the associated non-invertible symmetry. We show how this framework provides necessary conditions for admissible gaugings, enabling the construction of generalized orbifold groupoids. To illustrate this strategy, we present three examples: Rep (D_4) ( D 4 ) as a warm-up, the higher-multiplicity case Rep (A_4) ( A 4 ) with its associated generalized orbifold groupoid and triality symmetry, and Rep (A_5) ( A 5 ) , where A_5 A 5 is the smallest non-solvable finite group. For applications to gapless systems, we embed these generalized gaugings as global manipulations on the conformal manifolds of c=1 c = 1 CFTs and uncover new self-dualities in the exceptional SU(2)_1/A_5 S U ( 2 ) 1 / A
We extend the decomposition conjecture to 2d quantum field theories with a gauged Rep(H) symmetry category for H a finite-dimensional semisimple Hopf algebra with Rep(G) trivially-acting and Vec(Γ) the remaining symmetry, for G, Γ finite groups. We check our extension by explicitly computing partition functions, and by verifying that previous results arise as special cases. Furthermore, we compute the topological operators responsible for enforcing the decomposition. Then, drawing from these results, we formulate a plausible decomposition conjecture for the even more general case of Rep(H″) trivially-acting and Rep(H′) the remaining symmetry, for H′, H″ Hopf algebras, not necessarily associated with groups.
A bstract We discuss two-dimensional conformal field theories (CFTs) which are invariant under gauging a non-invertible global symmetry. At every point on the orbifold branch of c = 1 CFTs, it is known that the theory is self-dual under gauging a ℤ 2 × ℤ 2 symmetry, and has Rep( H 8 ) and Rep( D 8 ) fusion category symmetries as a result. We find that gauging the entire Rep( H 8 ) fusion category symmetry maps the orbifold theory at radius R to that at radius 2/ R . At R = 2 \sqrt{2} 2​ 2 , which corresponds to two decoupled Ising CFTs (Ising 2 in short), the theory is self-dual under gauging the Rep( H 8 ) symmetry. This implies the existence of a topological defect line in the Ising 2 CFT obtained from half-space gauging of the Rep( H 8 ) symmetry, which commutes with the c = 1 Virasoro algebra but does not preserve the fully extended chiral algebra. We bootstrap its action on the c = 1 Virasoro primary operators, and find that there are no relevant or marginal operators preserving it. Mathematically, the new topological line combines with the Rep( H 8 ) symmetry to form a bigger fusion category which is a ℤ 2 -extension of Rep( H 8 ). We solve the pentagon equations including the additional topological line and find 8 solutions, where two of them are realized in the Ising 2 CFT. Finally, we show that the torus partition functions of the Monster 2 CFT and Ising × Monster CFT are also invariant under gauging the Rep( H 8 ) symmetry.
Non-invertible dualities/symmetries have become an important tool in the study of quantum field theories and quantum lattice models in recent years. One of the most studied examples is non-invertible dualities obtained by gauging a discrete group. When the physical system has more global symmetries than the gauged symmetry, it has not been thoroughly investigated how those global symmetries transform under non-invertible duality. In this paper, we study the change of global symmetries under non-invertible duality of gauging a discrete group G G in the context of (1+1)-dimensional quantum lattice models. We obtain the global symmetries of the dual model by focusing on different Hilbert space sectors determined by the \mathrm{Rep}(G) R e p ( G ) symmetry. We provide general conjectures of global symmetries of the dual model forming an algebraic ring of the double cosets. We present concrete examples of the XXZ models and the duals, providing strong evidence for the conjectures.
A novel kind of generalized global symmetries is uncovered in a large class of familiar 3+1-dimensional gauge theories, including the free Maxwell theory and Yang-Mills gauge theories. These new symmetries, known as the non-invertible global symmetries, do not have inverses and thus go beyond the traditional paradigm of (anti-)unitary transformations. They are implemented by topological duality defects, generalizing the Kramers-Wannier duality defects in 1+1 dimensions. Remarkably, the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase and hence gives new constraints on renormalization group flows.