Fusion Categories Research Articles

Noninvertible Symmetries, Anomalies, and Scattering Amplitudes

We show that crossing symmetry of S matrices is modified in certain theories with noninvertible symmetries or anomalies. Focusing on integrable flows to gapped phases in two dimensions, we find that S matrices derived previously from the bootstrap approach are incompatible with noninvertible symmetries along the flow. We present consistent alternatives, which, however, violate standard crossing symmetry and obey modified rules dictated by fusion categories. We extend these rules to theories with discrete anomalies. Published by the American Physical Society 2024

Unitary braided-enriched monoidal categories

Braided-enriched monoidal categories were introduced in the work of Morrison–Penneys, where they were characterized using braided central functors. The recent work of Kong–Yuan–Zhang–Zheng and Dell extended this characterization to an equivalence of 2-categories. Since their introduction, braided-enriched fusion categories have been used to describe certain phenomena in topologically ordered systems in theoretical condensed matter physics. While these systems are unitary, there was previously no general notion of unitarity for enriched categories in the literature. We supply the notion of unitarity for enriched categories and braided-enriched monoidal categories and extend the above 2-equivalence to the unitary setting.

Structure constants, Isaacs property and extended Haagerup fusion categories

This paper extends the Isaacs property, originally introduced by the second author, to noncommutative fusion rings through the use of a Fourier transform. It aligns this property with a subsequent categorical equivalent for spherical fusion categories described by Etingof et al., particularly in the pseudounitary case. Within the commutative framework, the Isaacs property is situated strictly between the integrality of structure constants and the 1- Frobenius properties. Furthermore, the paper demonstrates that the Extended Haagerup fusion categories (denoted as E H i ) do not possess the Isaacs property, thus providing a negative answer to questions posed by the aforementioned researchers. It also reaffirms that E H 1 lacks a braiding structure.

Fusion categories over non-algebraically closed fields

Several complications arise when attempting to work with fusion categories over arbitrary fields. Here we describe some of the new phenomena that occur when the field is not algebraically closed, and we adapt tools such as the Frobenius-Perron dimension in order to accommodate these new effects.

CFT Correlators and Mapping Class Group Averages

Mapping class group averages appear in the study of 3D gravity partition functions. In this paper, we work with 3D topological field theories to establish a bulk-boundary correspondence between such averages and correlators of 2D rational CFTs whose chiral mapping class group representations are irreducible and satisfy a finiteness property. We show that Ising-type modular fusion categories satisfy these properties on surfaces with or without field insertions, extending results in [1], and we comment on the absence of invertible global symmetries in the examples we consider.

Braided Zestings of Verlinde Modular Categories and Their Modular Data

Zesting of braided fusion categories is a procedure that can be used to obtain new modular categories from a modular category with non-trivial invertible objects. In this paper, we classify and construct all possible braided zesting data for modular categories associated with quantum groups at roots of unity. We produce closed formulas, based on the root system of the associated Lie algebra, for the modular data of these new modular categories.

Reshetikhin–Turaev TQFTs Close Under Generalised Orbifolds

We specialise the construction of orbifold graph TQFTs introduced in Carqueville et al. (Orbifold graph TQFTs) to Reshetikhin–Turaev defect TQFTs. We explain that the modular fusion category CA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}_\\mathcal {A}$$\\end{document} constructed in Mulevičius and Runkel (Quant Topol 13(3):459–523, 2023. https://doi.org/10.4171/QT/170) from an orbifold datum A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document} in a given modular fusion category C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}$$\\end{document} is a special case of the Wilson line ribbon categories introduced as part of the general theory of orbifold graph TQFTs. Using this, we prove that the Reshetikhin–Turaev TQFT obtained from CA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}_\\mathcal {A}$$\\end{document} is equivalent to the orbifold of the TQFT for C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}$$\\end{document} with respect to the orbifold datum A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}.

Cluster State as a Noninvertible Symmetry-Protected Topological Phase.

We show that the standard 1+1D Z_{2}×Z_{2} cluster model has a noninvertible global symmetry, described by the fusion category Rep(D_{8}). Therefore, the cluster state is not only a Z_{2}×Z_{2} symmetry protected topological (SPT) phase, but also a noninvertible SPT phase. We further find two new commuting Pauli Hamiltonians for the other two Rep(D_{8}) SPT phases on a tensor product Hilbert space of qubits, matching the classification in field theory and mathematics. We identify the edge modes and the local projective algebras at the interfaces between these noninvertible SPT phases. Finally, we show that there does not exist a symmetric entangler that maps between these distinct SPT states.

Non-invertible symmetries and higher representation theory II

In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the construction of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. We propose that the symmetry categories obtained by gauging higher subgroups may be defined as higher group-theoretical fusion categories, which are built from the projective higher representations of higher groups. As concrete applications we provide a unified description of the symmetry categories of gauge theories in three and four dimensions based on the Lie algebra \mathfrak{so}(N)𝔰𝔬(N), and a fully categorical description of non-invertible symmetries obtained by gauging a 1-form symmetry with a mixed ’t Hooft anomaly. We also discuss the effect of discrete torsion on symmetry categories, based a series of obstructions determined by spectral sequence arguments.

Particle-soliton degeneracies from spontaneously broken non-invertible symmetry

We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading ℤ2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.

Fusion category symmetry. Part II. Categoriosities at c = 1 and beyond

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.

Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space

We discuss the exact non-invertible Kramers-Wannier symmetry of 1+1d lattice models on a tensor product Hilbert space of qubits. This symmetry is associated with a topological defect and a conserved operator, and the latter can be presented as a matrix product operator. Importantly, unlike its continuum counterpart, the symmetry algebra involves lattice translations. Consequently, it is not described by a fusion category. In the presence of this defect, the symmetry algebra involving parity/time-reversal is realized projectively, which is reminiscent of an anomaly. Different Hamiltonians with the same lattice non-invertible symmetry can flow in their continuum limits to infinitely many different fusion categories (with different Frobenius-Schur indicators), including, as a special case, the Ising CFT. The non-invertible symmetry leads to a constraint similar to that of Lieb-Schultz-Mattis, implying that the system cannot have a unique gapped ground state. It is either in a gapless phase or in a gapped phase with three (or a multiple of three) ground states, associated with the spontaneous breaking of the lattice non-invertible symmetry.

Electric-magnetic duality and Z2 symmetry enriched Abelian lattice gauge theory

Kitaev’s quantum double model is a lattice realization of Dijkgraaf–Witten topological quantum field theory. Its topologically protected ground state space has broad applications for topological quantum computation and topological quantum memory. We investigate the Z2 symmetry enriched generalization of the model for the Abelian group in a categorical framework and present an explicit Hamiltonian construction. This model provides a lattice realization of the Z2 symmetry of the topological phase. We discuss in detail the categorical symmetry of the phase, for which the electric-magnetic (EM) duality symmetry is a special case. The aspects of symmetry defects are investigated using the G-crossed unitary braided fusion category. By determining the corresponding anyon condensation, the gapped boundaries and boundary-bulk duality are also investigated. In the last part, an explicit lattice realization of EM duality is discussed.

Diagramatics for cyclic pointed fusion categories

We give a parametrization of cyclic pointed categories associated to the cyclic group of order n in terms of n-th roots of unity. We also provide a diagramatic description of these categories by generators and relations, and use it to characterize their 2-group of automorphisms.

From the XXZ chain to the integrable Rydberg-blockade ladder via non-invertible duality defects

Strongly interacting models often possess “dualities” subtler than a one-to-one mapping of energy levels. The maps can be non-invertible, as apparent in the canonical example of Kramers and Wannier. We analyse an algebraic structure common to the XXZ spin chain and three other models: Rydberg-blockade bosons with one particle per square of a ladder, a three-state antiferromagnet, and two Ising chains coupled in a zigzag fashion. The structure yields non-invertible maps between the four models while also guaranteeing all are integrable. We construct these maps explicitly utilising topological defects coming from fusion categories and the lattice version of the orbifold construction, and use them to give explicit conformal-field-theory partition functions describing their critical regions. The Rydberg and Ising ladders also possess interesting non-invertible symmetries, with the spontaneous breaking of one in the former resulting in an unusual ground-state degeneracy.

Fusion category symmetry. Part I. Anomaly in-flow and gapped phases

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.

$\alpha$-induction for bi-unitary connections

The tensor functor called \alpha -induction produces a new unitary fusion category from a Frobenius algebra object, or a Q -system, in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of N to M arising from a subfactor N\subset M of finite index and finite depth, which gives a braided fusion category of endomorphisms of N . It is also understood in terms of Ocneanu’s graphical calculus. We study this \alpha -induction for bi-unitary connections, which provides a characterization of finite-dimensional nondegenerate commuting squares, and present certain 4 -tensors appearing in recent studies of 2 -dimensional topological order. We show that the resulting \alpha -induced bi-unitary connections are flat if we start with a commutative Frobenius algebra, or a local Q -system. Examples related to chiral conformal field theory and the Dynkin diagrams are presented.

Enriched string-net models and their excitations

Boundaries of Walker-Wang models have been used to construct commuting projector models which realize chiral unitary modular tensor categories (UMTCs) as boundary excitations. Given a UMTC A representing the Witt class of an anomaly, the article \cite{MR4640433} gave a commuting projector model associated to an A-enriched unitary fusion category X on a 2D boundary of the 3D Walker-Wang model associated to A. That article claimed that the boundary excitations were given by the enriched center/Müger centralizer ZA(X) of A in Z(X).In this article, we give a rigorous treatment of this 2D boundary model, and we verify this assertion using topological quantum field theory (TQFT) techniques, including skein modules and a certain semisimple algebra whose representation category describes boundary excitations. We also use TQFT techniques to show the 3D bulk point excitations of the Walker-Wang bulk are given by the Müger center Z2(A), and we construct bulk-to-boundary hopping operators Z2(A)→ZA(X) reflecting how the UMTC of boundary excitations ZA(X) is symmetric-braided enriched in Z2(A).This article also includes a self-contained comprehensive review of the Levin-Wen string net model from a unitary tensor category viewpoint, as opposed to the skeletal 6j symbol viewpoint.

Detecting algebra objects from NIM-reps in pointed, near-group and quantum group-like fusion categories

In this article we study the possible Morita equivalence classes of algebras in three families of fusion categories (pointed, near-group and (A1,l)12) by studying the Non-negative Integer Matrix representations (NIM-reps) of their underlying fusion ring, and compare these results with existing classification results of algebra objects. Also, in an appendix we include a test of the exponents conjecture for modular tensor categories of rank up to 4.

Gauging non-invertible symmetries: topological interfaces and generalized orbifold groupoid in 2d QFT

Gauging is a powerful operation on symmetries in quantum field theory (QFT), as it connects distinct theories and also reveals hidden structures in a given theory. We initiate a systematic investigation of gauging discrete generalized symmetries in two-dimensional QFT. Such symmetries are described by topological defect lines (TDLs) which obey fusion rules that are non-invertible in general. Despite this seemingly exotic feature, all well-known properties in gauging invertible symmetries carry over to this general setting, which greatly enhances both the scope and the power of gauging. This is established by formulating generalized gauging in terms of topological interfaces between QFTs, which explains the physical picture for the mathematical concept of algebra objects and associated module categories over fusion categories that encapsulate the algebraic properties of generalized symmetries and their gaugings. This perspective also provides simple physical derivations of well-known mathematical theorems in category theory from basic axiomatic properties of QFT in the presence of such interfaces. We discuss a bootstrap-type analysis to classify such topological interfaces and thus the possible generalized gaugings and demonstrate the procedure in concrete examples of fusion categories. Moreover we present a number of examples to illustrate generalized gauging and its properties in concrete conformal field theories (CFTs). In particular, we identify the generalized orbifold groupoid that captures the structure of fusion between topological interfaces (equivalently sequential gaugings) as well as a plethora of new self-dualities in CFTs under generalized gaugings.