Rigid Symmetries Research Articles

Homotopy Double Copy of Noncommutative Gauge Theories

We discuss the double-copy formulation of Moyal–Weyl-type noncommutative gauge theories from the homotopy algebraic perspective of factorisations of L∞-algebras. We define new noncommutative scalar field theories with rigid colour symmetries taking the role of the zeroth copy, where the deformed colour algebra plays the role of a kinematic algebra; some of these theories have a trivial classical limit but exhibit colour–kinematics duality, from which we construct the double copy theory explicitly. We show that noncommutative gauge theories exhibit a twisted form of colour–kinematics duality, which we use to show that their double copies match with the commutative case. We illustrate this explicitly for Chern–Simons theory, and for Yang–Mills theory where we obtain a modified Kawai–Lewellen–Tye relationship whose momentum kernel is linked to a binoncommutative biadjoint scalar theory. We reinterpret rank-one noncommutative gauge theories as double copy theories and discuss how our findings tie in with recent discussions of Moyal–Weyl deformations of self–dual Yang–Mills theory and gravity.

Theorem on extensive spectral degeneracy for systems with rigid higher symmetries in general dimensions

We establish, in the spirit of the Lieb-Schultz-Mattis theorem, lower bounds on the spectral degeneracy of quantum systems with higher (Gauge Like) symmetries with rather generic physical boundary conditions in an arbitrary number of spatial dimensions. Contrary to applying twists or equivalent adiabatic operations, we exploit the effects of modified boundary conditions. When a general choice of boundary geometry is immaterial in approaching the thermodynamic limit, systems that exhibit non-commuting Gauge Like symmetries, such as the orbital compass model, must have an exponential (in the size of the boundary) degeneracy of each of their spectral levels. We briefly discuss why, in spite of the proven large degeneracy associated with infrared-ultraviolet mixing, some systems may still exhibit conventional physical behaviors, i.e., of those of systems with non-extensive degeneracies, due to entropic "order by disorder" type effects.

Carrollian and Galilean conformal higher-spin algebras in any dimensions

We present higher-spin algebras containing a Poincaré subalgebra and with the same set of generators as the Lie algebras that are relevant to Vasiliev’s equations in any space-time dimension D ≥ 3. Given these properties, they can be considered either as candidate rigid symmetries for higher-spin gauge theories in Minkowski space or as Carrollian conformal higher-spin symmetries in one less dimension. We build these Lie algebras as quotients of the universal enveloping algebra of mathfrak{iso}left(1,D-1right) and we show how to recover them as Inönü-Wigner contractions of the rigid symmetries of higher-spin gauge theories in Anti de Sitter space or, equivalently, of relativistic conformal higher-spin symmetries. We use the same techniques to also define higher-spin algebras with the same set of generators and containing a Galilean conformal subalgebra, to be interpreted as non-relativistic limits of the conformal symmetries of a free scalar field. We begin by showing that the known flat-space higher-spin algebras in three dimensions can be obtained as quotients of the universal enveloping algebra of mathfrak{iso}left(1,2right) and then we extend the analysis along the same lines to a generic number of space-time dimensions. We also discuss the peculiarities that emerge for D = 5.

Emergent gauge symmetries: Yang-Mills theory

Gauge symmetries remove unphysical states and guarantee that field theories are free from the pathologies associated with these states. In this work we find a set of general conditions that guarantee the removal of unphysical states in field theories describing interacting vector fields. These conditions are obtained through the extension of a mechanism for the emergence of gauge symmetries proposed in a previous article [C. Barcel\'o J. High Energy Phys. 10 (2016) 084] in order to account for non-Abelian gauge symmetries, and are the following: low-energy Lorentz invariance, emergence of massless vector fields describable by an action quadratic in those fields and their derivatives, and self-coupling to a conserved current associated with specific rigid symmetries. Using a bootstrapping procedure, we prove that these conditions are equivalent to the emergence of gauge symmetries and, therefore, guarantee that any theory satisfying them must be equivalent to a Yang-Mills theory at low energies.

General Relativity, Mental Causation, and Energy Conservation

The conservation of energy and momentum have been viewed as undermining Cartesian mental causation since the 1690s. Modern discussions of the topic tend to use mid-nineteenth century physics, neglecting both locality and Noether’s theorem and its converse. The relevance of General Relativity (GR) has rarely been considered. But a few authors have proposed that the non-localizability of gravitational energy and consequent lack of physically meaningful local conservation laws answers the conservation objection to mental causation: conservation already fails in GR, so there is nothing for minds to violate. This paper is motivated by two ideas. First, one might take seriously the fact that GR formally has an infinity of rigid symmetries of the action and hence, by Noether’s first theorem, an infinity of conserved energies-momenta (thus answering Schrödinger’s 1918 false-negative objection). Second, Sean Carroll has asked (rhetorically) how one should modify the Dirac–Maxwell–Einstein equations to describe mental causation. This paper uses the generalized Bianchi identities to show that General Relativity tends to exclude, not facilitate, such Cartesian mental causation. In the simplest case, Cartesian mental influence must be spatio-temporally constant, and hence 0. The difficulty may diminish for more complicated models. Its persuasiveness is also affected by larger world-view considerations. The new general relativistic objection provides some support for realism about gravitational energy-momentum in GR (taking pseudotensor laws seriously). Such realism also might help to answer an objection to theories of causation involving conserved quantities, because energies-momenta would be conserved even in GR.

Perspective and Geometry in the Roman Painted Gardens

Garden painting is a very precise genre that is distinct from landscape painting. Present in Roman villas but also in tombs of the same period, it responds to some specific compositional rules. The structure of the representation, realized for the most part with the intent to break through the wall to obtain an illusory effect of expansion of space, responds to precise geometric characteristics. The article relates the structure, also geometric, of the Roman gardens with their pictorial representation, which derives from it its justification. In both cases, in fact, the composition is guided by the point of view under which the garden (real or imaginary) must be observed, and in both cases, there is a will to bend nature within rigidly geometric and symmetrical schemas. The sense of perspective recreated in the painted gardens offers at first glance a feeling of naturalness into the garden, which on closer observation reveals the artificiality of a unified and controlled point of view, of rigid symmetries and geometric relationships between the parts.

Perspective and Geometry in the Roman Painted Gardens

Garden painting is a very precise genre that is distinct from landscape painting. Present in Roman villas but also in tombs of the same period, it responds to some specific compositional rules. The structure of the representation, realized for the most part with the intent to break through the wall to obtain an illusory effect of expansion of space, responds to precise geometric characteristics. The article relates the structure, also geometric, of the Roman gardens with their pictorial representation, which derives from it its justification. In both cases, in fact, the composition is guided by the point of view under which the garden (real or imaginary) must be observed, and in both cases, there is a will to bend nature within rigidly geometric and symmetrical schemas. The sense of perspective recreated in the painted gardens offers at first glance a feeling of naturalness into the garden, which on closer observation reveals the artificiality of a unified and controlled point of view, of rigid symmetries and geometric relationships between the parts.

Three-charge superstrata with internal excitations

We construct a new family of three-charge -BPS smooth solutions that have the same charges as the supersymmetric D1D5P black hole and are non-invariant under rotations of the compact manifold. We work in type IIB string theory on T4 and we show how the supergravity and BPS equations reduce to a linear system, arranged in two ‘layers’ of partial differential equations. We then build two solutions of our system of equations: the first is a superdescendant three-charge solution, obtained by acting with rigid symmetries on a seed two-charge solution, on which we can perform a non-trivial check for the system of equations; the second is a new superstratum solution that has both internal and external excitations. We then describe the CFT heavy states dual to these new geometries.

Universal quantum computation using fractal symmetry-protected cluster phases

We show that 2D fractal subsystem symmetry-protected topological phases may serve as resources for universal measurement-based quantum computation. This is demonstrated explicitly for two cluster models known to lie within fractal symmetry-protected topological phases, and computational universality is shown to persist throughout those phases. One of the models considered is simply the cluster model on the honeycomb lattice in one limit. We discuss the importance of rigid subsystem symmetries, as opposed to global or $(\text{D}-1)$-form symmetries, in this context.

Noether symmetries for fields and branes in backgrounds with Killing vectors

Within the formulation of classical fields and branes in curved spacetime, we first discuss the Belinfante construction of an improved energy-momentum tensor density in curved backgrounds, in analogy to the case of flat spacetime. A novelty of our approach is the use of Noether identities for generally covariant theories as a guiding instrument. In this part of the paper, our basic results, which hold irrespective of the background being dynamical or fixed, agree with those previously obtained in Gamboa Saravi (2004 J. Phys. A: Math. Gen. 37 9573). We illustrate with the example of Maxwell theory the role played by the Belinfante tensor in the construction of the Noether conserved currents and symmetry generators associated with the Killing symmetries of the background, which become Noether symmetres for the matter Lagrangian. We also give some details on the existing obstructions to the realization of part of these symmetries in phase space. Next we study extended objects (p-branes) either in a dynamic or in a fixed background, and obtain the Noether identities associated both with target spacetime and world volume diffeomorphisms. We show that in field theory as well as with extended objects, the Killing symmetries of the background become ordinary rigid Noether symmetries of the theory in this fixed background.

Can Chern-Simons or Rarita-Schwinger be a Volkov-Akulov Goldstone?

We study three-dimensional non-linear models of vector and vector-spinor Goldstone fields associated with the spontaneous breaking of certain higher-spin counterparts of supersymmetry whose Lagrangians are of a Volkov-Akulov type. Goldstone fields in these models transform non-linearly under the spontaneously broken rigid symmetries. We find that the leading term in the action of the vector Goldstone model is the Abelian Chern-Simons action whose gauge symmetry is broken by a quartic term. As a result, the model has a propagating degree of freedom which, in a decoupling limit, is a quartic Galileon scalar field. The vector-spinor goldstino model turns out to be a non-linear generalization of the three-dimensional Rarita-Schwinger action. In contrast to the vector Goldstone case, this non-linear model retains the gauge symmetry of the Rarita-Schwinger action and eventually reduces to the latter by a non-linear field redefinition. We thus find that the free Rarita-Schwinger action is invariant under a hidden rigid super-symmetry generated by fermionic vector-spinor operators and acting non-linearly on the Rarita-Schwinger goldstino.

On a functional equation related to a pair of hedgehogs with congruent projections

Hedgehogs are geometrical objects that describe the Minkowski differences of arbitrary convex bodies in the Euclidean space En. We prove that two hedgehogs in En, n≥3, coincide up to a translation and a reflection in the origin, provided that their projections onto any two-dimensional plane are directly congruent and have no direct rigid motion symmetries. Our result is a consequence of a more general analytic statement about the solutions of a functional equation in which the support functions of hedgehogs are replaced with two arbitrary twice continuously differentiable functions on the unit sphere.

Gauging without initial symmetry

The gauge principle is at the heart of a good part of fundamental physics: Starting with a group G of so-called rigid symmetries of a functional defined over space-time Sigma, the original functional is extended appropriately by additional Lie(G)-valued 1-form gauge fields so as to lift the symmetry to Maps(Sigma,G). Physically relevant quantities are then to be obtained as the quotient of the solutions to the Euler-Lagrange equations by these gauge symmetries. In this article we show that one can construct a gauge theory for a standard sigma model in arbitrary space-time dimensions where the target metric is not invariant with respect to any rigid symmetry group, but satisfies a much weaker condition: It is sufficient to find a collection of vector fields v_a on the target M satisfying the extended Killing equation v_{a(i;j)}=0 for some connection acting on the index a. For regular foliations this is equivalent to requiring the conormal bundle to the leaves with its induced metric to be invariant under leaf-preserving diffeomorphisms of M, which in turn generalizes Riemannian submersions to which the notion reduces for smooth leaf spaces M/~. The resulting gauge theory has the usual quotient effect with respect to the original ungauged theory: in this way, much more general orbits can be factored out than usually considered. In some cases these are orbits that do not correspond to an initial symmetry, but still can be generated by a finite dimensional Lie group G. Then the presented gauging procedure leads to an ordinary gauge theory with Lie algebra valued 1-form gauge fields, but showing an unconventional transformation law. In general, however, one finds that the notion of an ordinary structural Lie group is too restrictive and should be replaced by the much more general notion of a structural Lie groupoid.

Spontaneous breaking of nilpotent symmetry in boundary BLG theory

We exploit boundary term to preserve the supersymmetric gauge invariance of Bagger–Lambert–Gustavsson (BLG) theory. The fermionic rigid BRST and anti-BRST symmetries are studied in linear and nonlinear gauges. Remarkably, for Delbourgo–Jarvis–Baulieu–Thierry–Mieg (DJBTM) type gauge the spontaneous breaking of BRST symmetry occurs in the BLG theory. The responsible guy for such spontaneous breaking is ghost–antighost condensation. Further, we discuss the ghost–antighost condensates in the modified maximally Abelian (MMA) gauge in the BLG theory.

2D sigma models and differential Poisson algebras

We construct a two-dimensional topological sigma model whose target space is endowed with a Poisson algebra for differential forms. The model consists of an equal number of bosonic and fermionic fields of worldsheet form degrees zero and one. The action is built using exterior products and derivatives, without any reference to any worldsheet metric, and is of the covariant Hamiltonian form. The equations of motion define a universally Cartan integrable system. In addition to gauge symmetries, the model has one rigid nilpotent supersymmetry corresponding to the target space de Rham operator. The rigid and local symmetries of the action, respectively, are equivalent to the Poisson bracket being compatible with the de Rham operator and obeying graded Jacobi identities. We propose that perturbative quantization of the model yields a covariantized differential star product algebra of Kontsevich type. We comment on the resemblance to the topological A model.

Symmetries of embedded complete bipartite graphs

We characterize which automorphisms of an arbitrary complete bipartite graph Kn;m can be induced by a homeomorphism of some embedding of the graph in S 3 . 1. Introduction. Knowing the symmetries of a molecule helps to pre- dict its chemical behavior. Chemists use the point group as a way to rep- resent the rigid symmetries of a molecule. However, molecules which are large enough to be exible (such as long polymers) or have pieces which

THE GAUGING OF TWO-DIMENSIONAL BOSONIC SIGMA MODELS ON WORLD-SHEETS WITH DEFECTS

We extend our analysis of the gauging of rigid symmetries in bosonic two-dimensional sigma models with Wess–Zumino terms in the action to the case of world-sheets with defects. A structure that permits a non-anomalous coupling of such sigma models to world-sheet gauge fields of arbitrary topology is analyzed, together with obstructions to its existence, and the classification of its inequivalent choices.

The Jackiw–Pi model: Classical theory

The massive even-parity non-Abelian gauge model in three space–time dimensions proposed by Jackiw and Pi is studied at the tree-level. The propagators are computed and the spectrum consistency is analyzed, besides, the symmetries of the model are collected and established through BRS invariance and Slavnov–Taylor identity. In the Landau gauge, thanks to the antighost equations and the Slavnov–Taylor identity, two rigid symmetries are identified by means of Ward identities. It is presented here a promising path for perturbatively quantization of the Jackiw–Pi model and a hint concerning its possible quantum scale invariance is also pointed out.

Local BRST cohomologyin(non-)Lagrangian field theory

Some general theorems are established on the local BRST cohomology for not necessarily Lagrangian gauge theories. Particular attention is given to the BRST groups with direct physical interpretation. Among other things, the groups of rigid symmetries and conservation laws are shown to be still connected, though less tightly than in the Lagrangian theory. The connection is provided by the elements of another local BRST cohomology group whose elements are identified with Lagrange structures. This extends the cohomological formulation of the Noether theorem beyond the scope of Lagrangian dynamics. We show that each integrable Lagrange structure gives rise to a Lie bracket in the space of conservation laws, which generalizes the Dickey bracket of conserved currents known in Lagrangian field theory. We study the issues of existence and uniqueness of the local BRST complex associated with a given set of field equations endowed with a compatible Lagrange structure. Contrary to the usual BV formalism, such a complex does not always exist for non-Lagrangian dynamics, and when exists it is by no means unique. The ambiguity and obstructions are controlled by certain cohomology classes, which are all explicitly identified.

Global Gauge Anomalies in Two-Dimensional Bosonic Sigma Models

We revisit the gauging of rigid symmetries in two-dimensional bosonic sigma models with a Wess-Zumino term in the action. Such a term is related to a background closed 3-form H on the target space. More exactly, the sigma-model Feynman amplitudes of classical fields are associated to a bundle gerbe with connection of curvature H over the target space. Under conditions that were unraveled more than twenty years ago, the classical amplitudes may be coupled to the topologically trivial gauge fields of the symmetry group in a way which assures infinitesimal gauge invariance. We show that the resulting gauged Wess-Zumino amplitudes may, nevertheless, exhibit global gauge anomalies that we fully classify. The general results are illustrated on the example of the WZW and the coset models of conformal field theory. The latter are shown to be inconsistent in the presence of global anomalies. We introduce a notion of equivariant gerbes that allow an anomaly-free coupling of the Wess-Zumino amplitudes to all gauge fields, including the ones in non-trivial principal bundles. Obstructions to the existence of equivariant gerbes and their classification are discussed. The choice of different equivariant structures on the same bundle gerbe gives rise to a new type of discrete-torsion ambiguities in the gauged amplitudes. An explicit construction of gerbes equivariant with respect to the adjoint symmetries over compact simply connected simple Lie groups is given.