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We construct non-invertible symmetries in 6d $\mathcal{N}=(2,0)$ superconformal field theories that arise from Green-Schwarz (GS) automorphisms, which form abelian or non-abelian groups. Applied to $\mathbb{Z}_2$, $\mathbb{Z}_3$ and $S_3$ GS automorphisms, gives rise to non-invertible duality, triality and $S_3$-ality defects, respectively, once combined with stacking symmetry protected topological phases (SPTs) and gauging 2-form symmetries. We derive the defects and their fusion rules from two distinct perspectives: from half-space gauging as well as from the Symmetry Topological Field Theory (SymTFT). This is the first concrete construction of symmetry defects in 6d forming a fusion 5-category whose fusions are intrinsically non-invertible and non-abelian., Comment: 40 pages + appendices

We construct a (1+1)d gapless theory which has Haagerup $\mathcal{H}_3$ symmetry. The construction relies on the recent exploration of the categorical Landau paradigm applied to fusion category symmetries. First, using the Symmetry Topological Field Theory, we construct all gapped phases with Haagerup symmetry. Extending this construction to gapless phases, we study the second order phase transition between gapped phases, and determine analytically an Haagerup-symmetric conformal field theory. This is given in terms of two copies of the three-state Potts model, on which we realise the full $\mathcal{H}_3$ symmetry action and determine the relevant deformations to the $\mathcal{H}_3$-symmetric gapped phases. This continuum analysis is corroborated by a lattice model construction of the gapped and gapless phases, using the anyon chain., Comment: 11 pages + appendices, 2 figures; v2: references added

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 chracterization, 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 $d=2,3$. 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., Comment: 71 pages, v2: substantially expanded

We study the fusion 3-categorical symmetries for quantum theories in (3+1)d with self-duality defects. Such defects have been realized physically by half-space gauging in theories with 1-form symmetries $A[1]$ for an abelian group $A$, and have found applications in the continuum and the lattice. These fusion 3-categories will be called (generalized) Tambara-Yamagami fusion 3-categories $(\mathbf{3TY})$. We consider the Brauer-Picard and Picard 4-groupoids to construct these categories using a 3-categorical version of the extension theory introduced by Etingof, Nikshych and Ostrik. These two 4-groupoids correspond to the construction of duality defects either directly in 4d, or from the 5d Symmetry Topological Field Theory (SymTFT). The Witt group of non-degenerate braided fusion 1-categories naturally appears in the aforementioned 4-groupoids and represents enrichments of standard duality defects by (2+1)d TFTs. Our main objective is to study graded extensions of the fusion 3-category $\mathbf{3Vect}(A[1])$. Firstly, we use invertible bimodule 3-categories and the Brauer-Picard 4-groupoid. Secondly, we use that the Brauer-Picard 4-groupoid of $\mathbf{3Vect}(A[1])$ can be identified with the Picard 4-groupoid of its Drinfeld center. Moreover, the Drinfeld center of $\mathbf{3Vect}(A[1])$, which represents topological defects of the SymTFT, is completely described by a sylleptic strongly fusion 2-category formed by topological surface defects of the SymTFT. These are classified by a finite abelian group equipped with an alternating 2-form. We relate the Picard 4-groupoid of the corresponding braided fusion 3-categories with a generalized Witt group constructed from certain graded braided fusion 1-categories using a twisted Deligne tensor product. We perform explicit computations for $\mathbb{Z}/2$ and $\mathbb{Z}/4$ graded $\mathbf{3TY}$ categories., Comment: 48 pages, comments welcome

We study gapless phases in (3+1)d in the presence of 1-form and non-invertible duality symmetries. Using the Symmetry Topological Field Theory (SymTFT) approach, we classify the gapless symmetry-protected (gSPT) phases in these setups, with particular focus on intrinsically gSPTs (igSPTs). These are symmetry protected critical points which cannot be deformed to a trivially gapped phase without spontaneously breaking the symmetry. Although these are by now well-known in (1+1)d, we demonstrate their existence in (3+1)d gauge theories. Here, they have a clear physical interpretation in terms of an obstruction to confinement, even though the full 1-form symmetry does not suffer from 't Hooft anomalies. These igSPT phases provide a new way to realize 1-form symmetries in CFTs, that has no analog for gapped phases.The SymTFT approach allows for a direct generalization from invertible symmetries to non-invertible duality symmetries, for which we study gSPT and igSPT phases as well. We accompany these theoretical results with concrete physical examples realizing such phases and explain how obstruction to confinement is detected at the level of symmetric deformations., Comment: 39 pages, 2 appendices

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. 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 0-form symmetry, and gauging the diagonal 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 = $\mathbb{Z_2}$ and more generally any finite abelian group, for which the resulting non-invertible symmetries and their gapped phases already reveal an immensely rich structure., Comment: 85 pages

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 $\mathsf{Rep}(S_3)$ formed by representations of the permutation group $S_3$, but our procedure can be applied to any fusion category symmetry., Comment: 76 pages + appendices; v2: references added, v3: minor changes

Recent years have seen the concept of global symmetry extended to non-invertible (or categorical) symmetries, for which composition of symmetry generators is not necessarily invertible. Such non-invertible symmetries lead to a generalization of the standard Landau paradigm. In this work we substantiate this framework by providing a (1+1)d lattice model, whose gapped phases and phase transitions can only be explained by symmetry breaking of non-invertible symmetries., Comment: 4.5 pages + appendices, v2: references added

We propose a correspondence between topological order in 2+1d and Seifert three-manifolds together with a choice of ADE gauge group $G$. Topological order in 2+1d is known to be characterized in terms of modular tensor categories (MTCs), and we thus propose a relation between MTCs and Seifert three-manifolds. The correspondence defines for every Seifert manifold and choice of $G$ a fusion category, which we conjecture to be modular whenever the Seifert manifold has trivial first homology group with coefficients in the center of $G$. The construction determines the spins of anyons and their S-matrix, and provides a constructive way to determine the R- and F-symbols from simple building blocks. We explore the possibility that this correspondence provides an alternative classification of MTCs, which is put to the test by realizing all MTCs (unitary or non-unitary) with rank $r\leq 5$ in terms of Seifert manifolds and a choice of Lie group $G$., Comment: 9 pages + appendices

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., Comment: v2: 26 pages + appendix, added a generalized superconductivity interpretation in terms of condensed, confined and deconfined charges, typos fixed

We provide a generalization of the Symmetry Topological Field Theory (SymTFT) framework to characterize phase transitions and gapless phases with categorical symmetries. The central tool is the club sandwich, which extends the SymTFT setup to include an interface between two topological orders: there is a symmetry boundary, which is gapped, and a physical boundary that may be gapless, but in addition, there is also a gapped interface in the middle. The club sandwich generalizes so-called Kennedy-Tasaki (KT) transformations. Building on the results in [1, 2] on gapped phases with categorical symmetries, we construct gapless theories describing phase transitions with non-invertible symmetries by applying suitable KT transformations on known phase transitions provided by the critical Ising model and the 3-state Potts model. We also describe in detail the order parameters in these gapless theories characterizing the phase transitions, which are generally mixtures of conventional and string-type order parameters mixed together by the action of categorical symmetries. Additionally, removing the physical boundary from the club sandwiches results in club quiches, which characterize all possible gapped boundary phases with (possibly non-invertible) symmetries that can arise on the boundary of a bulk gapped phase. We also provide a mathematical characterization of gapped boundary phases with symmetries as pivotal tensor functors whose targets are pivotal multi-fusion categories., Comment: 58 pages, v2: typos fixed

We propose a unified framework to classify gapped infra-red (IR) phases with categorical symmetries, leading to a generalized, categorical Landau paradigm. This is applicable in any dimension and gives a succinct, comprehensive, and computationally powerful approach to classifying gapped symmetric phases. The key tool is the symmetry topological field theory (SymTFT), which is a one dimension higher TFT with two boundaries, which we choose both to be topological. We illustrate the general idea for (1+1)d gapped phases with categorical symmetries and suggest higher-dimensional extensions., Comment: 4 pages + supplementary material

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 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. We also provide a mathematical description of symmetric gapped phases as 2-functors from delooping of fusion category characterizing the symmetry to Euler completion of 2-vector spaces., Comment: 141 pages, v2: corrected an omission in the analysis of TY(Z_N) gapped phases reported by A. Antinucci, references added, v3: some charges corrected in section 7.1, v4,5: referee comments addressed and references added

Recently it has been observed that branes in geometric engineering and holography have a striking connection with generalized global symmetries. In this paper we argue that branes, in a certain topological limit, not only furnish the symmetry generators, but also encode the so-called Symmetry Topological Field Theory (or SymTFT). For a $d$-dimensional QFT, this is a $(d+1)$-dimensional topological field theory, whose topological defects encode both the symmetry generators (invertible or non-invertible) and the generalized charges. Mathematically, the topological defects form the Drinfeld center of the symmetry category of the QFT. In this paper we derive the SymTFT and the Drinfeld center topological defects directly from branes. Central to the identification of these are Hanany-Witten brane configurations, which encode both topological couplings in the SymTFT and the generalized charges under the symmetries. We exemplify the general analysis with examples of QFTs realized in geometric engineering or holography., Comment: 68 pages plus appendices

What comprises a global symmetry of a Quantum Field Theory (QFT) has been vastly expanded in the past 10 years to include not only symmetries acting on higher-dimensional defects, but also most recently symmetries which do not have an inverse. The principle that enables this generalization is the identification of symmetries with topological defects in the QFT. In these lectures, we provide an introduction to generalized symmetries, with a focus on non-invertible symmetries. We begin with a brief overview of invertible generalized symmetries, including higher-form and higher-group symmetries, and then move on to non-invertible symmetries. The main idea that underlies many constructions of non-invertible symmetries is that of stacking a QFT with topological QFTs (TQFTs) and then gauging a diagonal non-anomalous global symmetry. The TQFTs become topological defects in the gauged theory called (twisted) theta defects and comprise a large class of non-invertible symmetries including condensation defects, self-duality defects, and non-invertible symmetries of gauge theories with disconnected gauge groups. We will explain the general principle and provide numerous concrete examples. Following this extensive characterization of symmetry generators, we then discuss their action on higher-charges, i.e. extended physical operators. As we will explain, even for invertible higher-form symmetries these are not only representations of the $p$-form symmetry group, but more generally what are called higher-representations. Finally, we give an introduction to the Symmetry Topological Field Theory (SymTFT) and its utility in characterizing symmetries, their gauging and generalized charges. Lectures prepared for the ICTP Trieste Spring School, April 2023., Comment: 89 pages + bibliography, v2: typos fixed and refs added

Consider a 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 $q$-charges, i.e. $q$-dimensional operators. The main result of this paper is that $q$-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 $(d+1)$-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 $q$-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)., Comment: 142 pages; v2: References added

$q$-charges describe the possible actions of a generalized symmetry on $q$-dimensional operators. In Part I of this series of papers, we describe $q$-charges for invertible symmetries; while the discussion of $q$-charges for non-invertible symmetries is the topic of Part II. We argue that $q$-charges of a standard global symmetry, also known as a 0-form symmetry, correspond to the so-called $(q+1)$-representations of the 0-form symmetry group, which are natural higher-categorical generalizations of the standard notion of representations of a group. This generalizes already our understanding of possible charges under a 0-form symmetry! Just like local operators form representations of the 0-form symmetry group, higher-dimensional extended operators form higher-representations. This statement has a straightforward generalization to other invertible symmetries: $q$-charges of higher-form and higher-group symmetries are $(q+1)$-representations of the corresponding higher-groups. There is a natural extension to higher-charges of non-genuine operators (i.e. operators that are attached to higher-dimensional operators), which will be shown to be intertwiners of higher-representations. This brings into play the higher-categorical structure of higher-representations. We also discuss higher-charges of twisted sector operators (i.e. operators that appear at the boundary of topological operators of one dimension higher), including operators that appear at the boundary of condensation defects., Comment: 77 pages, v4: minor revision

We propose new $G_2$-holonomy manifolds, which geometrize the Gaiotto-Kim 4d N=1 duality domain walls of 5d N=1 theories. These domain walls interpolate between different extended Coulomb branch phases of a given 5d superconformal field theory. Our starting point is the geometric realization of such a 5d superconformal field theory and its extended Coulomb branch in terms of M-theory on a non-compact singular Calabi-Yau three-fold and its K\"ahler cone. We construct the 7-manifold that realizes the domain wall in M-theory by fibering the Calabi-Yau three-fold over a real line, whilst varying its K\"ahler parameters as prescribed by the domain wall construction. In particular this requires the Calabi-Yau fiber to pass through a canonical singularity at the locus of the domain wall. Due to the 4d N=1 supersymmetry that is preserved on the domain wall, we expect the resulting 7-manifold to have holonomy $G_2$. Indeed, for simple domain wall theories, this construction results in 7-manifolds, which are known to admit torsion-free $G_2$-holonomy metrics. We develop several generalizations to new 7-manifolds, which realize domain walls in 5d SQCD theories and walls between 5d theories which are UV-dual., Comment: 81 pages; v2: typos fixed

5d Superconformal Field Theories (SCFTs) are intrinsically strongly-coupled UV fixed points, whose realization hinges on string theoretic methods: they can be constructed by compactifying M-theory on local Calabi-Yau threefold singularities or alternatively from the world-volume of 5-brane-webs in type IIB string theory. There is a correspondence between 5-brane-webs and toric Calabi-Yau threefolds, however this breaks down when multiple 5-branes are allowed to end on a single 7-brane. In this paper, we extend this connection and provide a geometric realization of brane configurations including 7-branes. A web with 7-branes defines a so-called generalized toric polygon (GTP), which corresponds to combinatorial data that is obtained by removing vertices along external edges of a toric polygon. We identify the geometries associated to GTPs as non-toric deformations of toric Calabi-Yau threefolds and provide a precise, algebraic description of the geometry, when 7-branes are introduced along a single edge. The key ingredients in our analysis are T-branes in a type IIA frame, which includes D6-branes. We show that performing Hanany-Witten moves for the 7-branes on the type IIB side corresponds to switching on semisimple vacuum expectation values on the worldvolume of D6-branes, which in turn uplifts to complex structure deformations of the Calabi-Yau geometries. We test the proposal by computing the crepant resolutions of the deformed geometries, thereby checking consistency with the expected properties of the SCFTs.

We study generalized global symmetries and their 't Hooft anomalies in 3d N=4 superconformal field theories (SCFTs). Following some general considerations, we focus on good quiver gauge theories, comprised of balanced unitary nodes and unbalanced unitary and special unitary nodes. While the global form of the Higgs branch symmetry group may be determined from the UV Lagrangian, the global form of Coulomb branch symmetry groups and associated mixed 't Hooft anomalies are more subtle due to potential symmetry enhancement in the IR. We describe how Coulomb branch symmetry groups and their mixed 't Hooft anomalies can be deduced from the UV Lagrangian by studying center charges of various types of monopole operators, providing a concrete and unambiguous way to implement 't Hooft anomaly matching. The final expression for the symmetry group and 't Hooft anomalies has a concise form that can be easily read off from the quiver data, specifically from the positions of the unbalanced and flavor nodes with respect to the positions of the balanced nodes. We provide consistency checks by applying our method to compute symmetry groups of 3d N=4 theories corresponding to magnetic quivers of 4d Class S theories and 5d SCFTs. We are able to match these results against the flavor symmetry groups of the 4d and 5d theories computed using independent methods. Another strong consistency check is provided by comparing symmetry groups and anomalies of two theories related by 3d mirror symmetry., Comment: 79 pages, v2: Corrected an important typo reported by M. Sperling

Non-invertible symmetries have by now seen numerous constructions in higher dimensional Quantum Field Theories (QFT). In this paper we provide an in depth study of gauging 0-form symmetries in the presence of non-invertible symmetries. The starting point of our analysis is a theory with $G$ 0-form symmetry, and we propose a description of sequential partial gaugings of sub-symmetries. The gauging implements the theta-symmetry defects of the companion paper [1]. The resulting network of symmetry structures related by this gauging will be called a non-invertible symmetry web. Our formulation makes direct contact with fusion 2-categories, and we uncover numerous interesting structures such as symmetry fractionalization in this categorical setting. The complete symmetry web is derived for several groups $G$, and we propose extensions to higher dimensions. The highlight of this analysis is the complete categorical symmetry web, including non-invertible symmetries, for 3d pure gauge theories with orthogonal gauge groups and its extension to arbitrary dimensions., Comment: 115 pages

In the past year several constructions of non-invertible symmetries in Quantum Field Theory in $d\geq 3$ have appeared. In this paper we provide a unified perspective on these constructions. Central to this framework are so-called theta defects, which generalize the notion of theta-angles, and allow the construction of universal and non-universal topological symmetry defects. We complement this physical analysis by proposing a mathematical framework (based on higher-fusion categories) that converts the physical construction of non-invertible symmetries into a concrete computational scheme., Comment: 52 pages

What follows is a broad-brush overview of the recent synergistic interactions between mathematics and theoretical physics of quantum field theory and string theory. The discussion is forward-looking, suggesting potentially useful and fruitful directions and problems, some old, some new, for further development of the subject. This paper is a much extended version of the Snowmass whitepaper on physical mathematics [1].

We derive the Symmetry Topological Field Theories (SymTFTs) for 3d supersymmetric quantum field theories (QFTs) constructed in M-theory either via geometric engineering or holography. These 4d SymTFTs encode the symmetry structures of the 3d QFTs, for instance the generalized global symmetries and their 't Hooft anomalies. Using differential cohomology, we derive the SymTFT by reducing M-theory on a 7-manifold $Y_7$, which either is the link of a conical Calabi-Yau four-fold or part of an $\text{AdS}_4\times Y_7$ holographic solution. In the holographic setting we first consider the 3d $\mathcal{N}=6$ ABJ(M) theories and derive the BF-couplings, which allow the identification of the global form of the gauge group, as well as 1-form symmetry anomalies. Secondly, we compute the SymTFT for 3d $\mathcal{N}=2$ quiver gauge theories whose holographic duals are based on Sasaki-Einstein 7-manifolds of type $Y_7 = Y^{p,k}(\mathbb{C}\mathbb{P}^2)$. The SymTFT encodes 0- and 1-form symmetries, as well as potential 't Hooft anomalies between these. Furthermore, by studying the gapped boundary conditions of the SymTFT we constrain the allowed choices for $U(1)$ Chern-Simons terms in the dual field theory., Comment: 44 pages plus appendices

We give an overview of the field of quantum gravity, string theory and black holes summarizing various white papers in this subject that were submitted as part of the Snowmass process., Comment: 21 pages; v2, minor changes

We propose a systematic approach to deriving symmetry generators of Quantum Field Theories in holography. Central to this are the Gauss law constraints in the Hamiltonian quantization of Symmetry Topological Field Theories (SymTFTs), which are obtained from supergravity. In turn we realize the symmetry generators from world-volume theories of D-branes in holography. Our main focus is on non-invertible symmetries, which have emerged in the past year as a new type of symmetry in $d\geq 4$ QFTs. We exemplify our proposal in the holographic confinement setup, dual to 4d $\mathcal{N}=1$ Super-Yang Mills. In the brane-picture, the fusion of non-invertible symmetries naturally arises from the Myers effect on D-branes. In turn, their action on line defects is modeled by the Hanany-Witten effect., Comment: 10 pages; published version

It is well-known that gauging a finite 0-form symmetry in a quantum field theory leads to a dual symmetry generated by topological Wilson line defects. These are described by the representations of the 0-form symmetry group which form a 1-category. We argue that for a d-dimensional quantum field theory the full set of dual symmetries one obtains is in fact much larger and is described by a (d-1)-category, which is formed out of lower-dimensional topological quantum field theories with the same 0-form symmetry. We study in detail a 2-categorical piece of this (d-1)-category described by 2d topological quantum field theories with 0-form symmetry. We further show that the objects of this 2-category are the recently discussed 2d condensation defects constructed from higher-gauging of Wilson lines. Similarly, dual symmetries obtained by gauging any higher-form or higher-group symmetry also form a (d-1)-category formed out of lower-dimensional topological quantum field theories with that higher-form or higher-group symmetry. A particularly interesting case is that of the 2-category of dual symmetries associated to gauging of finite 2-group symmetries, as it describes non-invertible symmetries arising from gauging 0-form symmetries that act on (d-3)-form symmetries. Such non-invertible symmetries were studied recently in the literature via other methods, and our results not only agree with previous results, but our approach also provides a much simpler way of computing various properties of these non-invertible symmetries. We describe how our results can be applied to compute non-invertible symmetries of various classes of gauge theories with continuous disconnected gauge groups in various spacetime dimensions. We also discuss the 2-category formed by 2d condensation defects in any arbitrary quantum field theory., Comment: 75 pages. v2: Minor improvements

We propose the general idea that 't Hooft anomalies of generalized global symmetries can be understood in terms of the properties of solitonic defects, which generically are non-topological defects. The defining property of such defects is that they act as sources for background fields of generalized symmetries. 't Hooft anomalies arise when solitonic defects are charged under these generalized symmetries. We illustrate this idea for several kinds of anomalies in various spacetime dimensions. A systematic exploration is performed in 3d for 0-form, 1-form, and 2-group symmetries, whose 't Hooft anomalies are related to two special types of solitonic defects, namely vortex line defects and monopole operators. This analysis is supplemented with detailed computations of such anomalies in a large class of 3d gauge theories. Central to this computation is the determination of the gauge and 0-form charges of a variety of monopole operators: these involve standard gauge monopole operators, but also fractional gauge monopole operators, as well as monopole operators for 0-form symmetries. The charges of these monopole operators mainly receive contributions from Chern-Simons terms and fermions in the matter content. Along the way, we interpret the vanishing of the global gauge and ABJ anomalies, which are anomalies not captured by local anomaly polynomials, as the requirement that gauge monopole operators and mixed monopole operators for 0-form and gauge symmetries have non-fractional integer charges., Comment: 85 pages

We sketch a procedure to capture general non-invertible symmetries of a d-dimensional quantum field theory in the data of a higher-category, which captures the local properties of topological defects associated to the symmetries. We also discuss fusions of topological defects, which involve condensations/gaugings of higher-categorical symmetries localized on the worldvolumes of topological defects. Recently some fusions of topological defects were discussed in the literature where the dimension of topological defects seems to jump under fusion. This is not possible in the standard description of higher-categories. We explain that the dimension-changing fusions are understood as higher-morphisms of the higher-category describing the symmetry. We also discuss how a 0-form sub-symmetry of a higher-categorical symmetry can be gauged and describe the higher-categorical symmetry of the theory obtained after gauging. This provides a procedure for constructing non-invertible higher-categorical symmetries starting from invertible higher-form or higher-group symmetries and gauging a 0-form symmetry. We illustrate this procedure by constructing non-invertible 2-categorical symmetries in 4d gauge theories and non-invertible 3-categorical symmetries in 5d and 6d theories. We check some of the results obtained using our approach against the results obtained using a recently proposed approach based on 't Hooft anomalies., Comment: 104 pages. v2: Removed the distinction between local and global fusion to improve clarity. Added more details on the computation of fusion rules. v3: Minor improvements

Quantum Field Theories engineered in M-theory can have 2-group symmetries, mixing 0-form and 1-form symmetry backgrounds in non-trivial ways. In this paper we develop methods for determining the 2-group structure from the boundary geometry of the M-theory background. We illustrate these methods in the case of 5d theories arising from M-theory on ordinary and generalised toric Calabi-Yau cones, including cases in which the resulting theory is non-Lagrangian. Our results confirm and elucidate previous results on 2-groups from geometric engineering., Comment: 27 pages

We realize higher-form symmetries in F-theory compactifications on non-compact elliptically fibered Calabi-Yau manifolds. Central to this endeavour is the topology of the boundary of the non-compact elliptic fibration, as well as the explicit construction of relative 2-cycles in terms of Lefschetz thimbles. We apply the analysis to a variety of elliptic fibrations, including geometries where the discriminant of the elliptic fibration intersects the boundary. We provide a concrete realization of the 1-form symmetry group by constructing the associated charged line operator from the elliptic fibration. As an application we compute the symmetry topological field theories in the case of elliptic three-folds, which correspond to mixed anomalies in 5d and 6d theories., Comment: 57 pages, tikz figures, added clarifications

A relative theory is a boundary condition of a higher-dimensional topological quantum field theory (TQFT), and carries a non-trivial defect group formed by mutually non-local defects living in the relative theory. Prime examples are 6d N=(2,0) theories that are boundary conditions of 7d TQFTs, with the defect group arising from surface defects. In this paper, we study codimension-two defects in 6d N=(2,0) theories, and find that the line defects living inside these codimension-two defects are mutually non-local and hence also form a defect group. Thus, codimension-two defects in a 6d N=(2,0) theory are relative defects living inside a relative theory. These relative defects provide boundary conditions for topological defects of the 7d bulk TQFT. A codimension-two defect carrying a non-trivial defect group acts as an irregular puncture when used in the construction of 4d N=2 Class S theories. The defect group associated to such an irregular puncture provides extra "trapped" contributions to the 1-form symmetries of the resulting Class S theories. We determine the defect groups associated to large classes of both conformal and non-conformal irregular punctures. Along the way, we discover many new classes of irregular punctures. A key role in the analysis of defect groups is played by two different geometric descriptions of the punctures in Type IIB string theory: one provided by isolated hypersurface singularities in Calabi-Yau threefolds, and the other provided by ALE fibrations with monodromies., Comment: 125 pages

We determine the $d+1$ dimensional topological field theory, which encodes the higher-form symmetries and their 't Hooft anomalies for $d$-dimensional QFTs obtained by compactifying M-theory on a non-compact space $X$. The resulting theory, which we call the Symmetry TFT, or SymTFT for short, is derived by reducing the topological sector of 11d supergravity on the boundary $\partial X$ of the space $X$. Central to this endeavour is a reformulation of supergravity in terms of differential cohomology, which allows the inclusion of torsion in cohomology of the space $\partial X$, which in turn gives rise to the background fields for discrete (in particular higher-form) symmetries. We apply this framework to 7d super-Yang Mills where $X= \mathbb{C}^2/\Gamma_{ADE}$, as well as the Sasaki-Einstein links of Calabi-Yau three-fold cones that give rise to 5d superconformal field theories. This M-theory analysis is complemented with a IIB 5-brane web approach, where we derive the SymTFTs from the asymptotics of the 5-brane webs. Our methods apply to both Lagrangian and non-Lagrangian theories, and allow for many generalisations., Comment: 61 pages, 3 figures; updated references

Compactification of M-theory and of IIB string theory on threefold canonical singularities gives rise to superconformal field theories (SCFTs) in 5d and 4d, respectively. The resolutions and deformations of the singularities encode salient features of the SCFTs and of their moduli spaces. In this paper, we build on Part 0 of this series (arXiv:2007.15600) and further explore the physics of SCFTs arising from isolated hypersurface singularities. We study in detail these canonical isolated hypersurface singularities that admit a smooth Calabi-Yau (crepant) resolution. Their 5d and 4d physics is discussed and their 3d reduction and mirrors (the magnetic quivers) are determined in many cases. As an explorative tool, we provide a Mathematica code which computes key quantities for any canonical isolated hypersurface singularity, including the 5d rank, the 4d Coulomb branch spectrum and central charges, higher-form symmetries in 4d and 5d, and crepant resolutions., Comment: Ancillary mathematica code. 45 pages plus appendix; added references and clarifications

We uncover 2-group symmetries in 6d superconformal field theories. These symmetries arise when the discrete 1-form symmetry and continuous flavor symmetry group of a theory mix with each other. We classify all 6d superconformal field theories with such 2-group symmetries. The approach taken in 6d is applicable more generally, with minor modifications to include dimension specific operators (such as instantons in 5d and monopoles in 3d), and we provide a discussion of the dimension-independent aspects of the analysis. We include an ancillary mathematica code for computing 2-group symmetries, once the dimension specific input is provided. We also discuss a mixed 't Hooft anomaly between discrete 0-form and 1-form symmetries in 6d., Comment: 61 pages, 1 ancillary mathematica code, v2: references and type 3' added

We study confinement in 4d N=1 theories obtained by deforming 4d N=2 theories of Class S. We argue that confinement in a vacuum of the N=1 theory is encoded in the 1-cycles of the associated N=1 curve. This curve is the spectral cover associated to a generalized Hitchin system describing the profiles of two Higgs fields over the Riemann surface upon which the 6d (2,0) theory is compactified. Using our method, we reproduce the expected properties of confinement in various classic examples, such as 4d N=1 pure Super-Yang-Mills theory and the Cachazo-Seiberg-Witten setup. More generally, this work can be viewed as providing tools for probing confinement in non-Lagrangian N=1 theories, which we illustrate by constructing an infinite class of non-Lagrangian N=1 theories that contain confining vacua. The simplest model in this class is an N=1 deformation of the N=2 theory obtained by gauging $SU(3)^3$ flavor symmetry of the $E_6$ Minahan-Nemeschansky theory., Comment: 86 pages, many tikz figures, v2: references added

2-group symmetries arise when 1-form symmetries and 0-form symmetries of a theory mix with each other under group multiplication. We discover the existence of 2-group symmetries in 5d N=1 abelian gauge theories arising on the (non-extended) Coulomb branch of 5d superconformal field theories (SCFTs), leading us to argue that the UV 5d SCFT itself admits a 2-group symmetry. Furthermore, our analysis determines the global forms of the 0-form flavor symmetry groups of 5d SCFTs, irrespective of whether or not the 5d SCFT admits a 1-form symmetry. As a concrete application of our method, we analyze 2-group symmetries of all 5d SCFTs, which reduce in the IR, after performing mass deformations, to 5d N=1 non-abelian gauge theories with simple, simply connected gauge groups. For rank-1 Seiberg theories, we check that our predictions for the flavor symmetry groups match with the superconformal and ray indices available in the literature. We also comment on the mixed 't Hooft anomaly between 1-form and 0-form symmetries arising in 5d N=1 non-abelian gauge theories and its relation to the 2-groups., Comment: 97 pages; v2: Added an additional argument for the presence of 2-group symmetry in SU(2)_0 theory in section 3.4, and clarified that the arguments of the paper take into account the charges of non-BPS states; v3: Revised Higgs branch interpretation

We study confinement in 4d $\mathcal{N}=1$ $SU(N)$ Super-Yang Mills (SYM) from a holographic point of view, focusing on the 1-form symmetry and its relation to chiral symmetry breaking. In the 5d supergravity dual, obtained by truncation of the Klebanov-Strassler solution, we identify the topological couplings that determine the 1-form symmetry and its 't Hooft anomalies. One such coupling is a mixed 0-form/1-form symmetry anomaly closely related to chiral symmetry breaking in gapped confining vacua. From the dual gravity description we also identify the infra-red (IR) 4d topological field theory (TQFT), which realises chiral symmetry breaking and matches the mixed anomaly. Finally, complementing this, we derive the chiral and mixed anomalies from the Little String Theory realization of pure SYM., Comment: 4 pages + appendix

We determine the 1-form symmetry group for any 4d N = 2 class S theory constructed by compactifying a 6d N=(2,0) SCFT on a Riemann surface with arbitrary regular untwisted and twisted punctures. The 6d theory has a group of mutually non-local dimension-2 surface operators, modulo screening. Compactifying these surface operators leads to a group of mutually non-local line operators in 4d, modulo screening and flavor charges. Complete specification of a 4d theory arising from such a compactification requires a choice of a maximal subgroup of mutually local line operators, and the 1-form symmetry group of the chosen 4d theory is identified as the Pontryagin dual of this maximal subgroup. We also comment on how to generalize our results to compactifications involving irregular punctures. Finally, to complement the analysis from 6d, we derive the 1-form symmetry from a Type IIB realization of class S theories., Comment: 62 pages, 43 TikZ figures; v2: Important typos corrected

Canonical threefold singularities in M-theory and Type IIB string theory give rise to superconformal field theories (SCFTs) in 5d and 4d, respectively. In this paper, we study canonical hypersurface singularities whose resolutions contain residual terminal singularities and/or 3-cycles. We focus on a certain class of `trinion' singularities which exhibit these properties. In Type IIB, they give rise to 4d $\mathcal{N}=2$ SCFTs that we call $D_p^b(G)$-trinions, which are marginal gaugings of three SCFTs with $G$ flavor symmetry. In order to understand the 5d physics of these trinion singularities in M-theory, we reduce these 4d and 5d SCFTs to 3d $\mathcal{N}=4$ theories, thus determining the electric and magnetic quivers (or, more generally, quiverines). In M-theory, residual terminal singularities give rise to free sectors of massless hypermultiplets, which often are discretely gauged. These free sectors appear as `ugly' components of the magnetic quiver of the 5d SCFT. The 3-cycles in the crepant resolution also give rise to free hypermultiplets, but their physics is more subtle, and their presence renders the magnetic quiver `bad'. We propose a way to redeem the badness of these quivers using a class $\mathcal{S}$ realization. We also discover new S-dualities between different $D_p^b(G)$-trinions. For instance, a certain $E_8$ gauging of the $E_8$ Minahan-Nemeschansky theory is S-dual to an $E_8$-shaped Lagrangian quiver SCFT., Comment: 95 pages; v2: added references and corrected typos; v3&v4: added references and footnotes, JHEP version

5d superconformal field theories (SCFTs) can be obtained from 6d SCFTs by circle compactification and mass deformation. Successive decoupling of hypermultiplet matter and RG-flow generates a decoupling tree of descendant 5d SCFTs. In this paper we determine the magnetic quivers and Hasse diagrams, that encode the Higgs branches of 5d SCFTs, for entire decoupling trees. Central to this undertaking is the approach in arXiv:2008.05577, which, starting from the generalized toric polygons (GTPs) dual to 5-brane webs/tropical curves, provides a systematic and succinct derivation of magnetic quivers and their Hasse diagrams. The decoupling in the GTP description is straightforward, and generalizes the standard flop transitions of curves in toric polygons. We apply this approach to a large class of 5d KK-theories, and compute the Higgs branches for their descendants. In particular we determine the decoupling tree for all rank 2 5d SCFTs. For each tree, we also identify the flavor symmetry algebras from the magnetic quivers, including non-simply-laced flavor symmetries., Comment: 82 pages, many figures, 23 tables

We describe general methods for determining higher-form symmetry groups of known 5d and 6d superconformal field theories (SCFTs), and 6d little string theories (LSTs). The 6d theories can be described as supersymmetric gauge theories in 6d which include both ordinary non-abelian 1-form gauge fields and also abelian 2-form gauge fields. Similarly, the 5d theories can also be often described as supersymmetric non-abelian gauge theories in 5d. Naively, the 1-form symmetry of these 6d and 5d theories is captured by those elements of the center of ordinary gauge group which leave the matter content of the gauge theory invariant. However, an interesting subtlety is presented by the fact that some massive BPS excitations, which includes the BPS instantons, are charged under the center of the gauge group, thus resulting in a further reduction of the 1-form symmetry. We use the geometric construction of these theories in M/F-theory to determine the charges of these BPS excitations under the center. We also provide an independent algorithm for the determination of 1-form symmetry for 5d theories that admit a generalized toric construction. The 2-form symmetry group of 6d theories, on the other hand, is captured by those elements of the center of the abelian 2-form gauge group that leave all the massive BPS string excitations invariant, which is much more straightforward to compute as it is encoded in the Green-Schwarz coupling associated to the 6d theory., Comment: 53 pages, 1 figure, v2: Examples discussing 1-form symmetry of twisted 5d KK theories added, comments about spontaneous symmetry breaking added, references added

We derive the structure of the Higgs branch of 5d superconformal field theories or gauge theories from their realization as a generalized toric polygon (or dot diagram). This approach is motivated by a dual, tropical curve decomposition of the $(p,q)$ 5-brane-web system. We define an edge coloring, which provides a decomposition of the generalized toric polygon into a refined Minkowski sum of sub-polygons, from which we compute the magnetic quiver. The Coulomb branch of the magnetic quiver is then conjecturally identified with the 5d Higgs branch. Furthermore, from partial resolutions, we identify the symplectic leaves of the Higgs branch and thereby the entire foliation structure. In the case of strictly toric polygons, this approach reduces to the description of deformations of the Calabi-Yau singularities in terms of Minkowski sums., Comment: 68 pages, 1 figure, many Tikz pictures, and 1 ancillary Mathematica notebook

Five- and four-dimensional superconformal field theories with eight supercharges arise from canonical threefold singularities in M-theory and Type IIB string theory, respectively. We study their Coulomb and Higgs branches using crepant resolutions and deformations of the singularities. We propose a relation between the resulting moduli spaces, by compactifying the theories to 3d, followed by 3d $\mathcal{N}=4$ mirror symmetry and an $S$-type gauging of an abelian flavor symmetry. In particular, we use this correspondence to determine the Higgs branch of some 5d SCFTs and their magnetic quivers from the geometry. As an application of the general framework, we observe that singularities that engineer Argyres-Douglas theories in Type IIB also give rise to rank-0 5d SCFTs in M-theory. We also compute the higher-form symmetries of the 4d and 5d SCFTs, including the one-form symmetries of generalized Argyres-Douglas theories of type $(G, G')$., Comment: 64 pages; v2: fixed typos, added references and clarifications. v3: more references added; v4: fixed refs and typos, JHEP version; v5: small corrections and updated references

We study higher-form symmetries in 5d quantum field theories, whose charged operators include extended operators such as Wilson line and 't Hooft operators. We outline criteria for the existence of higher-form symmetries both from a field theory point of view as well as from the geometric realization in M-theory on non-compact Calabi-Yau threefolds. A geometric criterion for determining the higher-form symmetry from the intersection data of the Calabi-Yau is provided, and we test it in a multitude of examples, including toric geometries. We further check that the higher-form symmetry is consistent with dualities and is invariant under flop transitions, which relate theories with the same UV-fixed point. We explore extensions to higher-form symmetries in other compactifications of M-theory, such as $G_2$-holonomy manifolds, which give rise to 4d $\mathcal{N}=1$ theories., Comment: 67 pages

The trinions $T_N$ are a class of 5d $\mathcal{N}=1$ superconformal field theories (SCFTs) realized as M-theory on $\mathbb{C}^3/\mathbb{Z}_N \times \mathbb{Z}_N$. We apply to $T_N$, as well as closely-related SCFTs that are obtained by mass deformations, a multitude of recently developed approaches to studying 5d SCFTs and their IR gauge theory descriptions. Thereby we provide a complete picture of the theories both on the Coulomb branch and Higgs branch, from various geometric points of view - toric and gluing of compact surfaces as well as combined fiber diagrams - to magnetic quivers and Hasse diagrams., Comment: 60 pages, v2: JHEP version

We study the holographic dual to $c$-extremization for 2d $(0,2)$ superconformal field theories (SCFTs) that have an AdS$_3$ dual realized in Type IIB with varying axio-dilaton, i.e. F-theory. M/F-duality implies that such AdS$_3$ solutions can be mapped to AdS$_2$ solutions in M-theory, which are holographically dual to superconformal quantum mechanics (SCQM), obtained by dimensional reduction of the 2d SCFTs. We analyze the corresponding map between holographic $c$-extremization in F-theory and $\mathcal{I}$-extremization in M-theory, where in general the latter receives corrections relative to the F-theory result., Comment: 45 pages, v2: references added

We propose a generalization of S-folds to 4d $\mathcal{N}=2$ theories. This construction is motivated by the classification of rank one 4d $\mathcal{N}=2$ super-conformal field theories (SCFTs), which we reproduce from D3-branes probing a configuration of $\mathcal{N}=2$ S-folds combined with 7-branes. The main advantage of this point of view is that realizes both Coulomb and Higgs branch flows and allows for a straight forward generalization to higher rank theories., Comment: 18 pages, v2: references added, v3: published version

We systematically analyse 5d superconformal field theories (SCFTs) obtained by dimensional reduction from 6d $\mathcal{N}=(1,0)$ SCFTs. Such theories have a realization as M-theory on a singular Calabi-Yau threefold, from which we determine the so-called combined fiber diagrams (CFD) introduced in arXiv:1906.11820, arXiv:1907.05404, arXiv:1909.09128. The CFDs are graphs that encode the superconformal flavor symmetry, BPS states, low energy descriptions, as well as descendants upon flavor matter decoupling. To obtain a 5d SCFT from 6d, there are two approaches: the first is to consider a circle-reduction combined with mass deformations. The second is to circle-reduce and decouple an entire gauge sector from the theory. The former is applicable e.g. for very Higgsable theories, whereas the latter is required to obtain a 5d SCFT from a non-very Higgsable 6d theory. In the M-theory realization the latter case corresponds to decompactification of a set of compact surfaces in the Calabi-Yau threefold. To exemplify this we consider the 5d SCFTs that descend from non-Higgsable clusters and non-minimal conformal matter theories. Finally, inspired by the quiver structure of 6d theories, we propose a gluing construction for 5d SCFTs from building blocks and their CFDs., Comment: 102 pages, 127 figures, v4: JHEP version

By incorporating higher-form symmetries, we propose a refined definition of the theories obtained by compactification of the 6d $(2,0)$ theory on a three-manifold $M_3$. This generalization is applicable to both the 3d $\mathcal{N}=2$ and $\mathcal{N}=1$ supersymmetric reductions. An observable that is sensitive to the higher-form symmetries is the Witten index, which can be computed by counting solutions to a set of Bethe equations that are determined by $M_3$. This is carried out in detail for $M_3$ a Seifert manifold, where we compute a refined version of the Witten index. In the context of the 3d-3d correspondence, we complement this analysis in the dual topological theory, and determine the refined counting of flat connections on $M_3$, which matches the Witten index computation that takes the higher-form symmetries into account., Comment: 101 pages, 11 figures