Higher-form Symmetries Research Articles

Generalized Symmetries in F-theory and the Topology of Elliptic Fibrations

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.

Dynamical consequences of 1-form symmetries and the exceptional Argyres-Douglas theories

Higher-form symmetries have proved useful in constraining the dynamics of a number of quantum field theories. In the context of the Argyres-Douglas (AD) theories of the (G, G′) type, we find that the 1-form symmetries are invariant under the Higgs branch flow, and that they are captured by the non-Higgsable sector at a generic point on the Higgs branch of the AD theory in question. As a consequence, dimensional reduction of an AD theory with a non-trivial 1-form symmetry to 3d leads to a free sector. We utilize these observations, along with other results, to propose systematically the mirror theories for the AD theories of the (An, Em) type. As a by-product of these findings, we discover many important results: the Flip-Flip duality for all T[G] theories with simply-laced group G, including the exceptional ones; the class mathcal{S} descriptions of exceptional affine Dynkin diagram such that all gauge groups are special unitary; the universality of the mirror theories for {D}_{h_G^{vee }} (G) with {h}_G^{vee } the dual Coxeter number of G; and the triviality of the 2-group structure in the (An, Em) theories.

Comments on classical AdS flux vacua with scale separation

AdS flux vacua with a parametric separation between the AdS and KK scales have been conjectured to be in the Swampland. We study flux compactifications of massive IIA supergravity with O6 planes which are claimed to allow moduli-stabilised and scale separated AdS3 and AdS4 vacua at arbitrary weak coupling and large volume. A recent refinement of the AdS Distance Conjecture is shown to be inconsistent with this class of AdS3 vacua because the requisite discrete higher form symmetries are absent. We further perform a tree-level study of non-perturbative decays for the nonsupersymmetric versions of the AdS3 solutions, and find that the vacua are stable within this approximation. Finally, we provide an initial investigation of the would-be dual CFT2s and CFT3s. We study roughly a dozen different models and find for all AdS4 DGKT-type vacua that the dual operators to the lightest scalars have integer dimensions. For the putative CFT2 dual theories of the AdS3 vacua we find no integer dimensions for the operators.

Piezosuperconductivity: Novel effects in noncentrosymmetric superconductors

We study novel effects in non-centrosymmetric superconductors arising from their unique coupling of Cooper-pair condensate and elasticity. We show that although the much discussed Lifshitz coupling is not observable in a uniform bulk state, it strikingly endows dislocations with a fractional magnetic flux. We also predict a generation of voltage-free strain by a DC current in a P and T-breaking Josephson junction. Viewing superconductors through the lens of higher-form symmetries we identify the Lifshitz coupling as a chemical potential for the approximately conserved winding number, drawing an analogy with pyroelectric insulators.

Coulomb and Higgs branches from canonical singularities. Part I. Hypersurfaces with smooth Calabi-Yau resolutions

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 [1] 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.

Effective description of non-equilibrium currents in cold magnetized plasma

The dynamics of cold strongly magnetized plasma -- traditionally the domain of force-free electrodynamics -- has recently been reformulated in terms of symmetries and effective field theory, where the degrees of freedom are the momentum and magnetic flux carried by a fluid of cold strings. In physical applications where the electron mass can be neglected one might expect the presence of extra light charged modes -- electrons in the lowest Landau level -- propagating parallel to the magnetic field lines. We construct an effective description of such electric charges, describing their interaction with plasma degrees of freedom in terms of a new collective mode that can be thought of as a bosonization of the electric charge density along each field line. In this framework QED phenomena such as charged pair production and the axial anomaly are described at the classical level. Formally, our construction corresponds to gauging a particular part of the higher form symmetry associated with magnetic flux conservation. We study some simple applications of our effective theory, showing that the scattering of magnetosonic modes generically creates particles and that the rotating Michel monopole is now surrounded by a cloud of electric charge.

Exotic invertible phases with higher-group symmetries

We investigate a family of invertible phases of matter with higher-dimensional exotic excitations in even spacetime dimensions, which includes and generalizes the Kitaev’s chain in 1+1d. The excitation has \mathbb{Z}_2ℤ2 higher-form symmetry that mixes with the spacetime Lorentz symmetry to form a higher group spacetime symmetry. We focus on the invertible exotic loop topological phase in 3+1d. This invertible phase is protected by the \mathbb{Z}_2ℤ2 one-form symmetry and the time-reversal symmetry, and has surface thermal Hall conductance not realized in conventional time-reversal symmetric ordinary bosonic systems without local fermion particles and the exotic loops. We describe a UV realization of the invertible exotic loop topological order using the SO(3)_-SO(3)− gauge theory with unit discrete theta parameter, which enjoys the same spacetime two-group symmetry. We discuss several applications including the analogue of ``fermionization’’ for ordinary bosonic theories with \mathbb{Z}_2ℤ2 non-anomalous internal higher-form symmetry and time-reversal symmetry.

The fate of discrete 1-form symmetries in 6d

Recently introduced generalized global symmetries have been useful in order to understand non-perturbative aspects of quantum field theories in four and lower dimensions. In this paper we focus on 1-form symmetries of weakly coupled 6d supersymmetric gauge theories coupled to dynamical tensor multiplets. We study the consistency of global 1-form symmetries corresponding to the center of the gauge groups, or subgroups thereof, by activating their background fields, which makes the instanton density fractional. In 6d, an instanton background for a given gauge theory sources BPS strings via tadpole cancelation. The non-trivial 1-form symmetry background configurations contribute to the charge of the BPS strings. However, Dirac quantization imposes restrictions on the consistent 1-form backgrounds, since they can in general lead to and induce fractional charges, thus making (part of) the putative higher-form symmetry inconsistent. This gives explicit criteria to determine whether the discrete 1-form symmetries are realized. We implement these criteria in concrete examples originating from string compactifications. We also corroborate this by finding that a non-trivial fractional contribution is related to states which explicitly break the global 1-form symmetry appearing as massive excitations of the 6d BPS strings. For 6d theories consistently coupled to gravity, this hints at a symmetry breaking tower of states. When the fractional contributions are absent, the F-theory realization of the theories points to the gauging of the 1-form symmetry via the presence of non-trivial Mordell--Weil torsion.

Notes on Confinement on R3 × S1: From Yang–Mills, Super-Yang–Mills, and QCD (adj) to QCD(F)

This is a pedagogical introduction to the physics of confinement on R3×S1, using SU(2) Yang–Mills with massive or massless adjoint fermions as the prime example; we also add fundamental flavours to conclude. The small-S1 limit is remarkable, allowing for controlled semiclassical determination of the nonperturbative physics in these, mostly non-supersymmetric, theories. We begin by reviewing the Polyakov confinement mechanism on R3. Moving on to R3×S1, we show how introducing adjoint fermions stabilizes center symmetry, leading to abelianization and semiclassical calculability. We explain how monopole–instantons and twisted monopole–instantons arise. We describe the role of various novel topological excitations in extending Polyakov’s confinement to the locally four-dimensional case, discuss the nature of the confining string, and the θ-angle dependence. We study the global symmetry realization and, when available, present evidence for the absence of phase transitions as a function of the S1 size. As our aim is not to cover all work on the subject, but to prepare the interested reader for its study, we also include brief descriptions of topics not covered in detail: the necessity for analytic continuation of path integrals, the study of more general theories, and the ’t Hooft anomalies involving higher-form symmetries.

Reflections on the matter of 3DN=1vacua and localSpin(7)compactifications

We use Higgs bundles to study the 3d $\mathcal{N} = 1$ vacua obtained from M-theory compactified on a local $Spin(7)$ space given as a four-manifold $M_4$ of ADE singularities with further generic enhancements in the singularity type along one-dimensional subspaces. There can be strong quantum corrections to the massless degrees of freedom in the low energy effective field theory, but topologically robust quantities such as "parity" anomalies are still calculable. We show how geometric reflections of the compactification space descend to 3d reflections and discrete symmetries. The "parity" anomalies of the effective field theory descend from topological data of the compactification. The geometric perspective also allows us to track various perturbative and non-perturbative corrections to the 3d effective field theory. We also provide some explicit constructions of well-known 3d theories, including those which arise as edge modes of 4d topological insulators, and 3d $\mathcal{N} = 1$ analogs of grand unified theories. An additional result of our analysis is that we are able to track the spectrum of extended objects and their transformations under higher-form symmetries.

Higher-form symmetries and their anomalies in M-/F-theory duality

We explore higher-form symmetries of M- and F-theory compactified on elliptic fibrations, determined by the topology of their asymptotic boundaries. The underlying geometric structures are shown to be equivalent to known characterizations of the gauge group topology in F-theory via Mordell--Weil torsion and string junctions. We further study dimensional reductions of the 11d Chern--Simons term in the presence of torsional boundary $G_4$-fluxes, which encode background gauge fields of center 1-form symmetries in the lower-dimensional effective gauge theory. We find contributions that can be interpreted as 't Hooft anomalies involving the 1-form symmetry which originate from a fractionalization of the instanton number of non-Abelian gauge theories in F-/M-theory compactifications to 8d/7d and 6d/5d.

Global 4-group symmetry and ’t Hooft anomalies in topological axion electrodynamics

Abstract We study higher-form global symmetries and a higher-group structure of a low-energy limit of (3 + 1)-dimensional axion electrodynamics in a gapped phase described by a topological action. We argue that the higher-form symmetries should have a semi-strict 4-group (3-crossed module) structure by consistency conditions of couplings of the topological action to background gauge fields for the higher-form symmetries. We find possible ’t Hooft anomalies for the 4-group global symmetry, and discuss physical consequences.

Topological axion electrodynamics and 4-group symmetry

We study higher-form symmetries and a higher group in the low energy limit of a (3+1)-dimensional axion electrodynamics with a massive axion and a massive photon. A topological field theory describing topological excitations with the axion-photon coupling, which we call a topological axion electrodynamics, is obtained in the low energy limit. Higher-form symmetries of the topological axion electrodynamics are specified by equations of motion and Bianchi identities. We find that there are induced anyons on the intersections of symmetry generators. By a link of worldlines of the anyons, we show that the worldvolume of an axionic domain wall is topologically ordered. We further specify the underlying mathematical structure elegantly describing all salient features of the theory to be a 4-group.

Remarks on compatibility between conformal symmetry and continuous higher-form symmetries

We study the compatibility between conformal symmetry, together with unitarity, and continuous higher-form symmetries. We show that the $d$-dimensional, unitary, conformal field theories are not consistent with continuous $p$-form symmetries for certain $(d,p)$, assuming that the corresponding conserved current is a conformal primary operator. We further discuss several dynamical applications of this constraint.

Operator lifetime and the force-free electrodynamic limit of magnetised holographic plasma

Using the framework of higher-form global symmetries, we examine the regime of validity of force-free electrodynamics by evaluating the lifetime of the electric field operator, which is non-conserved due to screening effects. We focus on a holographic model which has the same global symmetry as that of low energy plasma and obtain the lifetime of (non-conserved) electric flux in a strong magnetic field regime. The lifetime is inversely correlated to the magnetic field strength and thus suppressed in the strong field regime.

Topological violation of global symmetries in quantum gravity

We discuss a topological reason why global symmetries are not conserved in quantum gravity, at least when the symmetry comes from compactification of a higher form symmetry. The mechanism is purely topological and does not require any explicit breaking term in the UV Lagrangian. Local current conservation does not imply global charge conservation in a sum over geometries in the path integral. We explicitly consider the shift symmetry of an axion-like field which originates from the compactification of a p-form gauge field. Our topological construction is motivated by the brane/black-brane correspondence, brane instantons, and an idea that virtual black branes of a simple kind may be realized by surgery on spacetime manifolds.

Universal features of higher-form symmetries at phase transitions

We investigate the behavior of higher-form symmetries at various quantum phase transitions. We consider discrete 1-form symmetries, which can be either part of the generalized concept “categorical symmetry” (labelled as \tilde{Z}_N^{(1)}Z̃N(1)) introduced recently, or an explicit Z_N^{(1)}ZN(1) 1-form symmetry. We demonstrate that for many quantum phase transitions involving a Z_N^{(1)}ZN(1) or \tilde{Z}_N^{(1)}Z̃N(1) symmetry, the following expectation value \langle \left( O_\mathcal{C}\right)^2 \rangle⟨(O𝒞)2⟩ takes the form \langle \left( \log O_\mathcal{C} \right)^2 \rangle \sim - \frac{A}{\epsilon} P + b \log P⟨(logO𝒞)2⟩∼−AϵP+blogP, where O_\mathcal{C}O𝒞 is an operator defined associated with loop \mathcal{C}𝒞 (or its interior \mathcal{A}𝒜), which reduces to the Wilson loop operator for cases with an explicit Z_N^{(1)}ZN(1) 1-form symmetry. PP is the perimeter of \mathcal{C}𝒞, and the b \log PblogP term arises from the sharp corners of the loop \mathcal{C}𝒞, which is consistent with recent numerics on a particular example. bb is a universal microscopic-independent number, which in (2+1)d(2+1)d is related to the universal conductivity at the quantum phase transition. bb can be computed exactly for certain transitions using the dualities between (2+1)d(2+1)d conformal field theories developed in recent years. We also compute the "strange correlator" of O_\mathcal{C}O𝒞: S_{\mathcal{C}} = \langle 0 | O_\mathcal{C} | 1 \rangle / \langle 0 | 1 \rangleS𝒞=⟨0|O𝒞|1⟩/⟨0|1⟩ where |0\rangle|0⟩ and |1\rangle|1⟩ are many-body states with different topological nature.

Higher-form symmetry breaking at Ising transitions

In recent years, new phases of matter that are beyond the Landau paradigm of symmetry breaking have been accumulating, and to catch up with this fast development, new notions of global symmetry are introduced. Among them, the higher-form symmetry, whose symmetry charges are spatially extended, can be used to describe topologically ordered phases as the spontaneous breaking of the symmetry, and consequently unify the unconventional and conventional phases under the same conceptual framework. However, such conceptual tools have not been put into quantitative tests except for certain solvable models, therefore limiting their usage in the more generic quantum many-body systems. In this work, we study ${\mathbb{Z}}_{2}$ higher-form symmetry in a quantum Ising model, which is dual to the global (zero-form) Ising symmetry. We compute the expectation value of the Ising disorder operator, which is a nonlocal order parameter for the higher-form symmetry, analytically in free scalar theories and through unbiased quantum Monte Carlo simulations for the interacting fixed point in $(2+1)d$. From the scaling form of this extended object, we confirm that the higher-form symmetry is indeed spontaneously broken inside the paramagnetic, or quantum disordered phase (in the Landau sense), but remains symmetric in the ferromagnetic or ordered phase. At the Ising critical point, we find that the disorder operator also obeys a ``perimeter'' law scaling with possibly multiplicative power-law corrections. We discuss examples where both the global zero-form symmetry and the dual higher-form symmetry are preserved, in systems with a codimension-1 manifold of gapless points in momentum space. These results provide nontrivial working examples of higher-form symmetry operators, including the direct computation of one-form order parameter in an interacting conformal field theory, and open the avenue for their generic implementation in quantum many-body systems.

Symmetries in Quantum Field Theory and Quantum Gravity

In this paper we use the AdS/CFT correspondence to refine and then establish a set of old conjectures about symmetries in quantum gravity. We first show that any global symmetry, discrete or continuous, in a bulk quantum gravity theory with a CFT dual would lead to an inconsistency in that CFT, and thus that there are no bulk global symmetries in AdS/CFT. We then argue that any “long-range” bulk gauge symmetry leads to a global symmetry in the boundary CFT, whose consistency requires the existence of bulk dynamical objects which transform in all finite-dimensional irreducible representations of the bulk gauge group. We mostly assume that all internal symmetry groups are compact, but we also give a general condition on CFTs, which we expect to be true quite broadly, which implies this. We extend all of these results to the case of higher-form symmetries. Finally we extend a recently proposed new motivation for the weak gravity conjecture to more general gauge groups, reproducing the “convex hull condition” of Cheung and Remmen. An essential point, which we dwell on at length, is precisely defining what we mean by gauge and global symmetries in the bulk and boundary. Quantum field theory results we meet while assembling the necessary tools include continuous global symmetries without Noether currents, new perspectives on spontaneous symmetry-breaking and ’t Hooft anomalies, a new order parameter for confinement which works in the presence of fundamental quarks, a Hamiltonian lattice formulation of gauge theories with arbitrary discrete gauge groups, an extension of the Coleman–Mandula theorem to discrete symmetries, and an improved explanation of the decay $$\pi ^0\rightarrow \gamma \gamma $$ in the standard model of particle physics. We also describe new black hole solutions of the Einstein equation in $$d+1$$ dimensions with horizon topology $${\mathbb {T}}^p\times {\mathbb {S}}^{d-p-1}$$ .

Field theories with higher-group symmetry from composite currents

Higher-form symmetries are associated with transformations that only act on extended objects, not on point particles. Typically, higher-form symmetries live alongside ordinary, point-particle (0-form), symmetries and they can be jointly described in terms of a direct product symmetry group. However, when the actions of 0-form and higher-form symmetries become entangled, a more general mathematical structure is required, related to higher categorical groups. Systems with continuous higher-group symmetry were previously constructed in a top-down manner, descending from quantum field theories with a specific mixed ’t Hooft anomaly. I show that higher-group symmetry also naturally emerges from a bottom-up, low-energy perspective, when the physical system at hand contains at least two different given, spontaneously broken symmetries. This leads generically to a hierarchy of emergent higher-form symmetries, corresponding to the Grassmann algebra of topological currents of the theory, with an underlying higher-group structure. Examples of physical systems featuring such higher-group symmetry include superfluid mixtures and variants of axion electrodynamics.