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Physical review. B./Physical review. B

Anomalies of ( 1 + 1 ) -dimensional categorical symmetries

Carolyn Zhang, Clay Córdova

Published Jul 24, 2024• 60 citations
Publisher

Abstract

We present a general approach for detecting when a fusion category symmetry is anomalous, based on the existence of a special kind of Lagrangian algebra of the corresponding Drinfeld center. The Drinfeld center of a fusion category $\mathcal{A}$ describes a $(2+1)$-dimensional topological order whose gapped boundaries enumerate all $(1+1)$-dimensional gapped phases with the fusion category symmetry, which may be spontaneously broken. There always exists a gapped boundary, given by the electric Lagrangian algebra, that describes a phase with $\mathcal{A}$ fully spontaneously broken. The symmetry defects of this boundary can be identified with the objects in $\mathcal{A}$. We observe that if there exists a different gapped boundary, given by a magnetic Lagrangian algebra, then there exists a gapped phase where $\mathcal{A}$ is not spontaneously broken at all, which means that $\mathcal{A}$ is not anomalous. In certain cases, we show that requiring the existence of such a magnetic Lagrangian algebra leads to highly computable obstructions to $\mathcal{A}$ being anomaly free. As an application, we consider the Drinfeld centers of ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N}$ Tambara-Yamagami fusion categories and recover known results from the study of fiber functors.

DOI
10.1103/physrevb.110.035155

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We present a general approach for detecting when a fusion category symmetry is anomalous, based on the existence of a special kind of Lagrangian algebra of the corresponding Drinfeld center. The Drinfeld center of a fusion category A\mathcal{A}A describes a (2+1)(2+1)(2+1)-dimensional topological order whose gapped boundaries enumerate all (1+1)(1+1)(1+1)-dimensional gapped phases with the fusion category symmetry, which may be spontaneously broken. There always exists a gapped boundary, given by the electric Lagrangian algebra, that describes a phase with A\mathcal{A}A fully spontaneously broken. The symmetry defects of this boundary can be identified with the objects in A\mathcal{A}A. We observe that if there exists a different gapped boundary, given by a magnetic Lagrangian algebra, then there exists a gapped phase where A\mathcal{A}A is not spontaneously broken at all, which means that A\mathcal{A}A is not anomalous. In certain cases, we show that requiring the existence of such a magnetic Lagrangian algebra leads to highly computable obstructions to A\mathcal{A}A being anomaly free. As an application, we consider the Drinfeld centers of ZN\ifmmode×\else\texttimes\fiZN{\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N}ZN​\ifmmode×\else\texttimes\fiZN​ Tambara-Yamagami fusion categories and recover known results from the study of fiber functors.
Anomalies of (1+1)D(1+1)D(1+1)D categorical symmetries
We present a general approach for detecting when a fusion category symmetry is anomalous, based on the existence of a special kind of Lagrangian algebra of the corresponding Drinfeld center. The Drinfeld center of a fusion category A\mathcal{A}A describes a (2+1)D(2+1)D(2+1)D topological order whose gapped boundaries enumerate all (1+1)D(1+1)D(1+1)D gapped phases with the fusion category symmetry, which may be spontaneously broken. There always exists a gapped boundary, given by the \emph{electric} Lagrangian algebra, that describes a phase with A\mathcal{A}A fully spontaneously broken. The symmetry defects of this boundary can be identified with the objects in A\mathcal{A}A. We observe that if there exists a different gapped boundary, given by a \emph{magnetic} Lagrangian algebra, then there exists a gapped phase where A\mathcal{A}A is not spontaneously broken at all, which means that A\mathcal{A}A is not anomalous. In certain cases, we show that requiring the existence of such a magnetic Lagrangian algebra leads to highly computable obstructions to A\mathcal{A}A being anomaly-free. As an application, we consider the Drinfeld centers of ZN×ZN\mathbb{Z}_N\times\mathbb{Z}_NZN​×ZN​ Tambara-Yamagami fusion categories and recover known results from the study of fiber functors.
We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe 't Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to "ungauge" the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.
A bstract We investigate symmetry topological field theories (SymTFTs) of non-abelian and non-invertible symmetries and the different Lagrangian algebras associated with a given Drinfeld center. For several examples we analyze the condensable algebras of the Drinfeld center to identify the intrinsically gapless symmetry protected topological (igSPT) phases. In the previous work, the relation between igSPT phases and resolving anomalies by embedding an anomalous symmetry inside a larger fusion category was demonstrated. Here we present more examples of this mechanism that involve both group-like and categorical symmetries.
In a (1+1)-dimensional QFT with non-invertible symmetry, the fusion rules of topological lines are often the most accessible data, yet they do not fully determine whether the symmetry is anomaly-free, i.e. whether it admits trivially gapped phases. Such phases are characterized by fiber functors of the corresponding fusion category. Building on Tannaka duality, we extract fiber functors from the fusion ring alone by solving simultaneously for monoidal structure maps and tensor automorphisms. This allows us to find categorifications admitting a fiber functor without solving the pentagon equations directly. We test this procedure on the known Tambara--Yamagami ring of Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2Z2​×Z2​. Our main result concerns the rank-11 fusion ring shared by the representation categories of the two extraspecial groups of order 27, which has a multiplicity 3 channel and is relevant to self-triality. In both cases we find more than one fiber functor, i.e. several non-invertible SPT phases. We use the explicit data of these fiber functors to derive the FFF-symbols of the two representation categories in a compact form depending on a non-degenerate alternating bicharacter and an invertible object εεε. Using the reconstructed data, we compute the anomalous algebra on the interface between two different SPT phases.
We propose a framework for fusion category symmetry on the (1+1)D lattice in the infinite-volume limit by giving a formal interpretation of SymTFT decompositions. Our approach is based on axiomatizing physical boundary subalgebra of quasi-local observables, and applying ideas from algebraic quantum field theory to derive the expected categorical structures. We show that given a physical boundary subalgebra BBB of a quasi-local algebra AAA, there is a canonical fusion category C\mathcal{C}C that acts on AAA by bimodules and whose fusion ring acts by locality preserving quantum channels on the quasi-local algebra such that BBB is recovered as the fixed point operators. We show that a fusion category can be realized as symmetries on a tensor product quasi-local algebra if and only if all of its objects have integer dimensions, and that it admits an ``on-site" action on a tensor product spin chain if and only if it admits a fiber functor. We give a formal definition of a topological symmetric state, and prove two anomaly enforced gaplessness theorems, one for internal categorical symmetries and one for anomalous duality channels. Using the first, we show that for any fusion category C\mathcal{C}C with no fiber functor there always exist gapless pure symmetric states on an anyon chain.
We propose a general framework to characterize gapped infra-red (IR) phases of theories with non-invertible (or categorical) symmetries. In this paper we focus on (1+1)d gapped phases with fusion category symmetries. The approach that we propose uses the Symmetry Topological Field Theory (SymTFT) as a key input: associated to a field theory in d d spacetime dimensions, the SymTFT lives in one dimension higher and admits a gapped boundary, which realizes the categorical symmetries. It also admits a second, physical, boundary, which is generically not gapped. Upon interval compactification of the SymTFT by colliding the gapped and physical boundaries, we regain the original theory. In this paper, we realize gapped symmetric phases by choosing the physical boundary to be a gapped boundary condition as well. This set-up provides computational power to determine the number of vacua, the symmetry breaking pattern, and the action of the symmetry on the vacua. The SymTFT also manifestly encodes the order parameters for these gapped phases, thus providing a generalized, categorical Landau paradigm for (1+1)d gapped phases. We find that for non-invertible symmetries the order parameters involve multiplets containing both untwisted and twisted sector local operators, and hence can be interpreted as mixtures of conventional and string order parameters. We also observe that spontaneous breaking of non-invertible symmetries can lead to vacua that are physically distinguishable: unlike the standard symmetries described by groups, non-invertible symmetries can have different actions on different vacua of an irreducible gapped phase. This leads to the presence of relative Euler terms between physically distinct vacua. Along with the physical description of symmetric gapped phases, we also provide a mathematical one as pivotal 2-functors whose source 2-category is the delooping of the fusion category characterizing the symmetry and the target 2-category is the Euler completion of 2-vector spaces.