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arXiv:2311.16230 · Journal of High Energy Physics

Notes on gauging noninvertible symmetries. Part I. Multiplicity-free cases

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

Published Feb 21, 2024• 39 citations
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Abstract

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 $$ \textrm{Rep}\left(\mathcal{H}\right) $$ Rep H for $$ \mathcal{H} $$ 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 $$ {\mathcal{H}}^{\ast } $$ 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 $$ \textrm{Rep}\left({\mathcal{H}}_8\right) $$ 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.

DOI
10.1007/jhep02(2024)154

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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.
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.
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
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.
A bstract We study generalized symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. This paper follows our companion paper on gapped phases and anomalies associated with these symmetries. In the present work we focus on identifying fusion category symmetries, using both specialized 1+1D methods such as the modular bootstrap and (rational) conformal field theory (CFT), as well as general methods based on gauging finite symmetries, that extend to all dimensions. We apply these methods to c = 1 CFTs and uncover a rich structure. We find that even those c = 1 CFTs with only finite group-like symmetries can have continuous fusion category symmetries, and prove a Noether theorem that relates such symmetries in general to non-local conserved currents. We also use these symmetries to derive new constraints on RG flows between 1+1D CFTs.
We provide a general prescription for gauging finite non-invertible symmetries in 1+1d lattice Hamiltonian systems. Our primary example is the Rep(D _8 8 ) fusion category generated by the Kennedy-Tasaki transformation, which is the simplest anomaly-free non-invertible symmetry on a spin chain of qubits. We explicitly compute its lattice F-symbols and illustrate our prescription for a particular (non-maximal) gauging of this symmetry. In our gauging procedure, we introduce two qubits around each link, playing the role of “gauge fields” for the non-invertible symmetry, and impose novel Gauss’s laws. Similar to the Kramers-Wannier transformation for gauging an ordinary \mathbb{Z}_2 ℤ 2 , our gauging can be summarized by a gauging map, which is part of a larger, continuous non-invertible cosine symmetry.
A bstract We discuss the fermionization of fusion category symmetries in two-dimensional topological quantum field theories (TQFTs). When the symmetry of a bosonic TQFT is described by the representation category Rep( H ) of a semisimple weak Hopf algebra H , the fermionized TQFT has a superfusion category symmetry SRep( H \mathcal{H} H H u ), which is the supercategory of super representations of a weak Hopf superalgebra H \mathcal{H} H H u . The weak Hopf superalgebra H \mathcal{H} H H u depends not only on H but also on a choice of a non-anomalous ℤ 2 subgroup of Rep( H ) that is used for the fermionization. We derive a general formula for H \mathcal{H} H H u by explicitly constructing fermionic TQFTs with SRep( H \mathcal{H} H H u ) symmetry. We also construct lattice Hamiltonians of fermionic gapped phases when the symmetry is non-anomalous. As concrete examples, we compute the fermionization of finite group symmetries, the symmetries of finite gauge theories, and duality symmetries. We find that the fermionization of duality symmetries depends crucially on F -symbols of the original fusion categories. The computation of the above concrete examples suggests that our fermionization formula of fusion category symmetries can also be applied to non-topological QFTs.