Sakura Schäfer-Nameki

The study of gapless phases with categorical (or so-called non-invertible) symmetries is a formidable task, in particular in higher than two space-time dimensions. In this paper we build on previous works [L. Bhardwaj et al. SciPost Phys. 19, 056 (2025); L. Bhardwaj et al. arXiv:2502.20440] on gapped phases in (2+1)d and provide a systematic framework to study phase transitions with categorical symmetries. The Symmetry Topological Field Theory (SymTFT) is, as often in these matters, the central tool. Applied to gapless theories, we need to consider the extension of the SymTFT to interfaces between topological orders, so-called “club sandwiches”, which realize generalizations of so-called Kennedy-Tasaki (KT) transformations. This requires an input phase transition for a smaller symmetry, such as the Ising transition for \mathbb{Z}_2 ℤ 2 , and the SymTFT constructs a transformation to a gapless phase with a larger categorical symmetry. We carry this out for categorical symmetries whose SymTFT is a (3+1)d Dijkgraaf-Witten (DW) theory for a finite group G G with twist - so-called all bosonic fusion 2-categories. We classify such interfaces using a physically motivated picture of generalized gauging, as well as with a complementary analysis using (bi-)module 2-categories. This is exemplified in numerous Abelian and non-Abelian DW theories, giving rise to interesting gapless phases such as intrinsically gapless symmetry protected phases (igSPTs) and spontaneous symmetry breaking phases (igSSBs) from Abelian, S_3 S 3 , and D_8 D 8 DW theories.

Symmetries in the form of actions of groups and Lie algebras have been an important tool for the analysis of physical systems for a very long time. It has been realized only relatively recently that quantum field theories admit symmetries that are described by more general mathematical structures: fusion categories and their higher-categorical generalizations.The understanding of such higher categories and the construction of concrete examples raise many mathematical challenges which are at present addressed with a wide variety of mathematical tools, including homotopy theory, the theory of C^{*} -algebras, factorization algebras and operads.An improved mathematical understanding of higher categories has an immediate impact on the classification and characterization of quantum systems which can realize these categorical structures as symmetries. Particularly note-worthy applications of these symmetries include integrable systems with fusion category symmetries, defects in topological and conformal field theories, the characterization of three-dimensional topological order and systems in quantum information theory realizing quantum codes.The workshop Higher Structures from Symmetries in Quantum Field Theory, brought a wide range of mathematical researchers together, working on a broad range of topics, from category theory to condensed matter physics, and provided an extremely fruitful interaction across disciplines. For some leading researchers from neighbouring fields, this was their first visit to Oberwolfach.

Recently, an intriguing correspondence was conjectured in [D. Gaiotto and H. Verlinde, J. High Energy Phys. 06, 163 (2025)] between Schur half-indices of pure 4d SU(2) S U ( 2 ) \mathcal{N}=2 ???? = 2 supersymmetric Yang-Mills (SYM) theory with line operator insertions and partition functions of the double scaling limit of the Sachdev-Ye-Kitaev model (DSSYK). Motivated by this, we explore a generalization to SU(N) S U ( N ) \mathcal{N}=2 ???? = 2 SYM theories. We begin by deriving the algebra of line operators, \mathcal{A}_{\text{Schur}} ???? Schur , representing it both in terms of the \mathfrak{q} ???? -Weyl algebra and \mathfrak{q} ???? -deformed harmonic oscillators, respectively. In the latter framework, the half-index admits a natural description as an expectation value in the Fock space of the oscillators. This \mathfrak{q} ???? -oscillator perspective further suggests an interpretation in terms of generalized colored chord counting, and maps the half-index to a purely combinatorial quantity. Finally, we establish a connection with the quantum Toda chain, which is an integrable model whose commuting Hamiltonians can be identified with the Wilson lines of the SU(N) S U ( N ) SYM, and their eigenfunctions correspond to the function basis appearing in the half-index.

Non-invertible categorical symmetries have emerged as a powerful tool to uncover new beyond-Landau phases of matter, both gapped and gapless, along with second order phase transitions between them. The general theory of such phases in (1+1)d has been studied using the Symmetry Topological Field Theory (SymTFT), also known as topological holography. This has unearthed the infrared (IR) structure of these phases and transitions. In this paper, we describe how the SymTFT information can be converted into an ultraviolet (UV) anyonic chain lattice model realizing, in the IR limit, these phases and transitions. In many cases, the Hilbert space of the anyonic chain is tensor product decomposable and the model can be realized as a quantum spin-chain Hamiltonian. We also describe operators acting on the lattice models that are charged under non-invertible symmetries and act as order parameters for the phases and transitions. In order to fully describe the action of non-invertible symmetries, it is crucial to understand the symmetry twisted sectors of the lattice models, which we describe in detail. Throughout the paper, we illustrate the general concepts using the symmetry category \mathsf{Rep}(S_3) ???? ???? ???? ( S 3 ) formed by representations of the permutation group S_3 S 3 , but our procedure can be applied to any fusion category symmetry.

Gapped phases in 2+1 dimensional quantum field theories with fusion 2-categorical symmetries were recently classified and characterized using the Symmetry Topological Field Theory (SymTFT) approach [L. Bhardwaj et al., SciPost Phys. 19, 056 (2025); L. Bhardwaj et al., arXiv: 2502.20440]. In this paper, we provide a systematic lattice model construction for all such gapped phases. Specifically, we consider “all-boson type” fusion 2-category symmetries, all of which are obtainable from 0-form symmetry groups G G (possibly with an ’t Hooft anomaly) via generalized gauging—that is, by stacking with an H H -symmetric TFT and gauging a subgroup H H . The continuum classification directly informs the lattice data, such as the generalized gauging that determines the symmetry category, and the data that specifies the gapped phase. We construct commuting projector Hamiltonians and ground states applicable to any non-chiral gapped phase with such symmetries. We also describe the ground states in terms of tensor networks. In light of the length of the paper, we include a self-contained summary section presenting the main results and examples.

Categorical symmetries have recently been shown to generalize the classification of phases of matter, significantly broadening the traditional Landau paradigm. To test these predictions, we propose a simple spin chain model that encompasses all gapped phases and second-order phase transitions governed by the categorical symmetry Rep ( D 8 ) . This model not only captures the essential features of noninvertible phases but is also straightforward enough to enable practical realization. Specifically, we outline an implementation using neutral atoms trapped in optical tweezer arrays. Employing a dual-species setup and Rydberg blockade, we propose a digital simulation approach that can efficiently implement the many-body evolution in several nontrivial quantum phases.

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.

Consider a d d -dimensional quantum field theory (QFT) \mathfrak{T} ???? , with a generalized symmetry \mathcal{S} ???? , which may or may not be invertible. We study the action of \mathcal{S} ???? on generalized or q q -charges, i.e. q q -dimensional operators. The main result of this paper is that q q -charges are characterized in terms of the topological defects of the Symmetry Topological Field Theory (SymTFT) of \mathcal{S} ???? , also known as the “Sandwich Construction”. The SymTFT is a (d+1) ( d + 1 ) -dimensional topological field theory, which encodes the symmetry \mathcal{S} ???? and the physical theory in terms of its boundary conditions. Our proposal applies quite generally to any finite symmetry \mathcal{S} ???? , including non-invertible, categorical symmetries. Mathematically, the topological defects of the SymTFT form the Drinfeld Center of the symmetry category \mathcal{S} ???? . Applied to invertible symmetries, we recover the result of Part I of this series of papers. After providing general arguments for the identification of q q -charges with the topological defects of the SymTFT, we develop this program in detail for QFTs in 2d (for general fusion category symmetries) and 3d (for fusion 2-category symmetries).

We study properties of boundary conditions (BCs) in theories with categorical (or non-invertible) symmetries. We describe how the transformation properties, or (generalized) charges, of BCs are captured by topological BCs of Symmetry Topological Field Theory (SymTFT), which is a topological field theory in one higher spacetime dimension. As an application of the SymTFT characterization, we discuss the symmetry properties of boundary conditions for (1+1)d gapped and gapless phases. We provide a number of concrete examples in spacetime dimensions d=2,3 d = 2 , 3 . We furthermore expand the lattice description for (1+1)d anyon chains with categorical symmetries to include boundary conditions carrying arbitrary 1-charges under the symmetry.

We use the Symmetry Topological Field Theory (SymTFT) to study and classify gapped phases in (2+1)d for a class of categorical symmetries, referred to as being of bosonic type. The SymTFTs for these symmetries are given by twisted and untwisted (3+1)d Dijkgraaf-Witten (DW) theories for finite groups G G . A finite set of boundary conditions (BCs) of these DW theories is well-known: these simply involve imposing Dirichlet and Neumann conditions on the (3+1)d gauge fields. We refer to these as minimal BCs. The key new observation here is that for each DW theory, there exists an infinite number of other BCs, that we call non-minimal BCs. These non-minimal BCs are all obtained by a ‘theta construction’, which involves stacking the Dirichlet BC with 3d TFTs having G G 0-form symmetry, and gauging the diagonal G G symmetry. On the one hand, using the non-minimal BCs as symmetry BCs gives rise to an infinite number of non-invertible symmetries having the same SymTFT, while on the other hand, using the non-minimal BCs as physical BCs in the sandwich construction gives rise to an infinite number of (2+1)d gapped phases for each such non-invertible symmetry. Our analysis is thoroughly exemplified for G=\mathbb{Z}_2 G = ℤ 2 and more generally any finite abelian group, for which the resulting non-invertible symmetries and their gapped phases already reveal an immensely rich structure.