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arXiv:2610.07165

Non-invertible SPT phases from fusion rings via Tannaka duality

Antonio Santaniello, Roberto Valandro, Massimo Zorzenon

Published Oct 5, 2026• 0 citations
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Abstract

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 $\mathbb{Z}_2\times\mathbb{Z}_2$. 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 $F$-symbols of the two representation categories in a compact form depending on a non-degenerate alternating bicharacter and an invertible object $ε$. Using the reconstructed data, we compute the anomalous algebra on the interface between two different SPT phases.

DOI
10.48550/arxiv.2610.07165

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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 investigate (1+1)d symmetry-protected topological (SPT) phases with fusion category symmetries. We emphasize that the UV description of an anomaly-free fusion category symmetry must include the fiber functor, giving rise to a local symmetry action, a charge category and a trivial phase. We construct an ``onsite'' matrix-product-operator (MPO) version of the Hopf algebra symmetry operators in a lattice model with tensor-product Hilbert space. In particular, we propose a systematic framework for classifying and constructing SPTs with non-invertible symmetries. An SPT phase corresponds to a Q-system in the charge category, such that the Q-system becomes a matrix algebra when the symmetry is forgotten. As an example, we provide an explicit microscopic realization of all three Rep†(D8)\mathsf{Rep}^\dagger(D_8)Rep†(D8​) SPT phases, including a trivial phase, and further demonstrate the S3S_3S3​-duality among these three SPT phases.
Classification of intrinsically mixed 1+11+11+1D non-invertible Rep(G)×G(G) \times G(G)×G SPT phases
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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.
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 ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N}$ Tambara-Yamagami fusion categories and recover known results from the study of fiber functors.
We consider symmetry protected topological (SPT) phases with finite non-invertible symmetry C\mathcal{C}C in 1+1d. In particular, we investigate interfaces and parameterized families of them within the framework of matrix product states. After revealing how to extract the C\mathcal{C}C-SPT invariant, we identify the algebraic structure of symmetry operators acting on the interface of two C\mathcal{C}C-SPT phases. By studying the representation theory of this algebra, we show that there must be a degenerate interface mode between different C\mathcal{C}C-SPT phases. This result generalizes the bulk-boundary correspondence for ordinary SPT phases. We then propose the classification of one-parameter families of C\mathcal{C}C-SPT states based on the explicit construction of invariants of such families. Our invariant is identified with a non-abelian generalization of the Thouless charge pump, which is the pump of a local excitation within a C\mathcal{C}C-SPT phase. Finally, by generalizing the results for one-parameter families of SPT phases, we conjecture the classification of general parameterized families of general gapped phases with finite non-invertible symmetries in both 1+1d and higher dimensions.
We discuss (1+1)d gapless phases with non-invertible global symmetries, also referred to as categorical symmetries. This includes gapless phases showing properties analogous to gapped symmetry protected topological (SPT) phases, known as gapless SPT (or gSPT) phases; and gapless phases showing properties analogous to gapped spontaneous symmetry broken (SSB) phases, that we refer to as gapless SSB (or gSSB) phases. We fit these gapless phases, along with gapped SPT and SSB phases, into a phase diagram describing possible deformations connecting them. This phase diagram is partially ordered and defines a so-called Hasse diagram. Based on these deformations, we identify gapless phases exhibiting symmetry protected criticality, that we refer to as intrinsically gapless SPT (igSPT) and intrinsically gapless SSB (igSSB) phases. This includes the first examples of igSPT and igSSB phases with non-invertible symmetries. Central to this analysis is the Symmetry Topological Field Theory (SymTFT), where each phase corresponds to a condensable algebra in the Drinfeld center of the symmetry category. On a mathematical note, gSPT phases are classified by functors between fusion categories, generalizing the fact that gapped SPT phases are classified by fiber functors; and gSSB phases are classified by functors from fusion to multi-fusion categories. Finally, our framework can be applied to understand gauging of trivially acting non-invertible symmetries, including possible patterns of decomposition arising due to such gaugings.