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Symmetries in modern physics are a fundamental subject of high relevance that allows appreciating more deeply the physical structure of a theory. In this paper, we analyze a gauge symmetry that appears in complex holomorphic systems. We show that a complex system can be reduced to different real systems, using different gauge conditions, and the gauge transformations connect several real systems in the complex space. We prove that the space of solutions of one system is related using a gauge transformation to another one. Gauge transformations are, in some cases, canonical transformations. However, in other cases, these are more general transformations that change the symplectic structure, but there is still a map between systems. We establish a construction to extend the analysis to the quantum case using path integrals through the Batalin–Fradkin–Vilkovisky theorem and within the canonical formalism, where we show explicitly that solutions of the Schrödinger equation are gauge-related. [ABSTRACT FROM AUTHOR]

These notes provide a detailed catalog of surface charge formulas for different classes of gravity theories. The present catalog reviews and extends the existing literature on the topic. Part of the focus is on reviewing the method to compute quasi-local surface charges for gauge theories in order to clarify conceptual issues and their range of applicability. Many surface charge formulas for gravity theories are expressed in metric, tetrads-connection, Chern–Simons connection, and even BF variables. For most of them, the language of differential forms is exploited and contrasted with the more popular metric components language. The gravity theory is coupled with matter fields as scalar, Maxwell, Skyrme, Yang–Mills, and spinors. Furthermore, three examples with ready-to-download notebook codes, show the method in full action. Several new results are highlighted through the notes. [ABSTRACT FROM AUTHOR]

We argue that the origin of Poincaré gauge gravity (PGG) may be related to spontaneous violation of underlying spacetime symmetries involved and appearance of gauge fields as vector Goldstone bosons. In essence, we start with an arbitrary theory of some vector and fermion fields which possesses only global spacetime symmetries, such as Lorentz and translational invariance, in flat Minkowski space. The two vector field multiplets involved are assumed to belong, respectively, to the adjoint (Aμij) and vector (eμi) representations of the starting global Lorentz symmetry. We propose that these prototype vector fields are covariantly constrained, AμijAijμ=±MA2 and eμieiμ=±Me2, that causes a spontaneous violation of the accompanying global symmetries (MA,e are their presumed violation scales). It then follows that the only possible theory compatible with these length-preserving constraints is turned out to be the gauge invariant PGG, while the corresponding massless (pseudo)Goldstone modes are naturally collected in the emergent gauge fields of tetrads and spin-connections. In a minimal theory case being linear in a curvature we unavoidably come to the Einstein–Cartan theory. The extended theories with propagating spin-connection and tetrad modes are also considered and their possible unification with the Standard Model is briefly discussed.

The construction of gauge theories beyond the realm of Lie groups and algebras leads one to consider Lie groupoids and algebroids equipped with additional geometrical structures which, for gauge invariance of the construction, need to satisfy particular compatibility conditions. This paper is supposed to analyze these compatibilities from a mathematical perspective. In particular, we show that the compatibility of a connection with a Lie algebroid that one finds is the Cartan condition, introduced previously by A. Blaom. For the metric on the base M of a Lie algebroid equipped with any connection, we show that the compatibility suggested from gauge theories implies that the foliation induced by the Lie algebroid becomes a Riemannian foliation. Building upon a result of del Hoyo and Fernandes, we prove, furthermore, that every Lie algebroid integrating to a proper Lie groupoid admits a compatible Riemannian base. We also consider the case where the base is equipped with a compatible symplectic or generalized Riemannian structure. [ABSTRACT FROM AUTHOR]

Abstract Brout–Englert–Higgs physics is one of the most central and successful parts of the standard model. It is also part of a multitude of beyond-the-standard-model scenarios. The aim of this review is to describe the field-theoretical foundations of Brout–Englert–Higgs physics, and to show how the usual phenomenology arises from it. This requires to give a precise and gauge-invariant meaning to the underlying physics. This is complicated by the fact that concepts like the Higgs vacuum expectation value or the separation between confinement and the Brout–Englert–Higgs effect lose their meaning beyond perturbation theory. This is addressed by carefully constructing the corresponding theory space and the quantum phase diagram of theories with elementary Higgs fields and gauge interactions. The physical spectrum needs then to be also given in terms of gauge-invariant, i.e. composite, states. Using gauge-invariant perturbation theory, as developed by Fröhlich, Morchio, and Strocchi, it is possible to rederive conventional perturbation theory in this framework. This derivation explicitly shows why the description of the standard model in terms of the unphysical, gauge-dependent, elementary states of the Higgs and W -bosons and Z -boson, but also of the elementary fermions, is adequate and successful. These are unavoidable consequences of the field theory underlying the standard model, from which the usual picture emerges. The validity of this emergence can only be tested non-perturbatively. Such tests, in particular using lattice gauge theory, will be reviewed as well. They fully confirm the underlying mechanisms. In this course it will be seen that the structure of the standard model is very special, and qualitative changes occur beyond it. The extension beyond the standard model will therefore also be reviewed. Particular attention will be given to structural differences arising for phenomenology. Again, non-perturbative tests of these results will be reviewed. Finally, to make this review self-contained a brief discussion of issues like the triviality and hierarchy problem, and how they fit into a fundamental field-theoretical formulation, is included. [ABSTRACT FROM AUTHOR]

Abstract The physical, observable spectrum in gauge theories is made up from gauge-invariant states. The Fröhlich–Morchio–Strocchi mechanism allows in the standard model to map these states to the gauge-dependent elementary W , Z and Higgs states. This is no longer necessarily the case in theories with a more general gauge group and Higgs sector. We classify and predict the physical spectrum for a wide range of such theories, with special emphasis on GUT-like cases, and show that discrepancies between the spectrum of elementary fields and physical particles frequently arise. [ABSTRACT FROM AUTHOR]

Asymptotic safety generalizes asymptotic freedom and could contribute to understanding physics beyond the Standard Model. It is a candidate scenario to provide an ultraviolet extension for the effective quantum field theory of gravity through an interacting fixed point of the Renormalization Group. Recently, asymptotic safety has been established in specific gauge-Yukawa models in four dimensions in perturbation theory, providing a starting point for asymptotically safe model building. Moreover, an asymptotically safe fixed point might even be induced in the Standard Model under the impact of quantum fluctuations of gravity in the vicinity of the Planck scale. This review contains an overview of the key concepts of asymptotic safety, its application to matter and gravity models, exploring potential phenomenological implications and highlighting open questions.

We review the construction of the gauge theory for semi-simple Lie groups by Utiyama in “Invariant Theoretical Interpretation of Interaction”[1]. It is shown an auxiliary field A μ a x must be introduced in order to keep the system of fields ϕ A (x) invariant under a transformation group depending on n parameters ϵ a (x). This auxiliary field interacts with ϕ through the covariant derivative ∇ μ ϕ A. We determine the transformation law for A μ a under the x μ-dependent Lie group and calculate the field strength F μν a (x). Moreover, we specify the conserved current J a μ related to the invariance of the complete system. The paper ends with the application of the general theory to the cases of the charged particle in an electromagnetic field and of the Yang-Mills potential under isotopic spin space transformations; we briefly address the matter of the gravitational field as a gauge theory; finally, we comment on the extension of Utiyama's theory for LA = LA [Aa μ ; ∂ ν A a μ ; ∂ ρ ∂ ν A a μ] (x). [ABSTRACT FROM AUTHOR]

Abstract In gauge theories, the physical, experimentally observable spectrum consists only of gauge-invariant states. This spectrum can be different from the elementary spectrum even at weak coupling and in the presence of the Brout–Englert–Higgs effect. We demonstrate this for an SU (3) gauge theory with a single fundamental Higgs, a toy theory for grand-unified theories. The manifestly gauge-invariant approach of lattice gauge theory is used to determine the spectrum in four different channels. It is found to be qualitatively different from the elementary one, and especially from the one predicted by standard perturbation theory. The result can be understood in terms of the Fröhlich–Morchio–Strocchi mechanism. In fact, we find that analytic methods based on this mechanism, a gauge-invariant extension of perturbation theory, correctly determine the spectrum, and give already at leading order a reasonably good quantitative description. Together with previous results this supports that this approach is the analytic method of choice for theories with a Brout–Englert–Higgs effect. [ABSTRACT FROM AUTHOR]

In the two papers of this series, we initiate the development of a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate mathematical tools to describe local symmetries when gauge transformations are combined with space–time transformations. Here, we outline the basis of the program and, as a first step, show how to (re)formulate Noether’s theorem about the connection between symmetries and conservation laws in this approach. [ABSTRACT FROM AUTHOR]

Three of the four fundamental forces in nature are described by so-called gauge theories, which include the effects of both relativity and quantum mechanics. Gravity, on the other hand, is described by General Relativity, and the lack of a well-behaved quantum theory - believed to be relevant at the centre of black holes, and at the Big Bang itself - remains a notorious unsolved problem. Recently a new correspondence, the double copy, has been discovered between scattering amplitudes (quantities related to the probability for particles to interact) in gravity, and their gauge theory counterparts. This has subsequently been extended to other quantities, providing gauge theory analogues of e.g. black holes. We here review current research on the double copy, and describe some possible applications. [ABSTRACT FROM AUTHOR]

Motivated by a demand for explicit genus 1 Belyi maps from theoretical physics, we give an efficient method of explicitly computing genus one Belyi maps by (1) composing covering maps from elliptic curves to the Riemann sphere with simpler (univariate) genus zero Belyi maps, as well as by (2) composing further with isogenies between elliptic curves. The computed examples of genus 1 Belyi maps has doubly-periodic dessins d’enfant that are listed in the physics literature as so-called brane-tilings in the context of quiver gauge theories. [ABSTRACT FROM AUTHOR]

We compute the Entanglement Entropy (EE) of a bipartition in 2D pure non-abelian U(N) gauge theory. We obtain a general expression for EE on an arbitrary Riemann surface. We find that due to area-preserving diffeomorphism symmetry EE does not depend on the size of the subsystem, but only on the number of disjoint intervals defining the bipartition. In the strong coupling limit on a torus we find that the scaling of the EE at small temperature is given by S(T)−S(0)=O(mgapTe−mgapT), which is similar to the scaling for the matter fields recently derived in literature. In the large N limit we compute all of the Renyi entropies and identify the Douglas–Kazakov phase transition.

In this paper, we study spontaneous symmetry breaking of gauge theories in a Lorentz violating background. Here, Lorentz symmetry will be broken down to its subgroup using the formalism of very special relativity. The breaking of the Lorentz symmetry will modify the gauge theories, and this will in turn modify the Higgs mechanisms for such theories. We will explicitly demonstrate that the different Hilbert spaces in various gauges of this theory can be related to each other through the gaugeon formalism. We will also discuss the FFBRST transformation for this theory, and observe that gaugeon formalism can be obtained from same. Thus, by making the BRST parameter finite and field dependent, we can relate different Hilbert spaces in different gauges for a gauge theory with spontaneous symmetry breaking in VSR. [ABSTRACT FROM AUTHOR]

We show how the BRST identities can be used to control the renormalization of the background gauge in QCD, in spite of the fact that certain one-particle reducible graphs have to be omitted in general. We obtain the all order renormalized effective action for the background field theory. [ABSTRACT FROM AUTHOR]

The chiral balance is broken as a result of an anomaly relation for the axial current in a strongly magnetized electron-positron gas in QED. The pair creation process, due to the decay of longitudinal axial photons (out of light cone), induces an imbalanced chirality and contributes to a chiral current along the external magnetic field B. In our case, a pseudo-vector longitudinal mode generates the chiral charge density. A chiral chemical potential is not necessary. Only an electromagnetic chemical potential is introduced, but the chiral effect is valid even when it vanishes. A nonzero fermion mass is assumed, which is usually considered vanishing. The introduction of a chiral chemical potential in the frame of electroweak theory at finite temperature and density is discussed. It is not compatible with the Standard Model, since the spontaneous symmetry breaking prevents the conservation of the weak neutral charge. [ABSTRACT FROM AUTHOR]

We examine the renormalization of the first order formulation of the Yang–Mills theory, by using the BRST identities. These preserve the gauge invariance of the theory and enable a recursive proof of renormalizability to higher orders in perturbation theory. The renormalization involves non-linear mixings as well as re-scalings of the fields and sources, which lead to a renormalized action at all orders. [ABSTRACT FROM AUTHOR]

From pure Yang-Mills action for the group in four Euclidean dimensions we obtain a gravity theory in the first order formalism. Besides the Einstein-Hilbert term, the effective gravity has a cosmological constant term, a curvature squared term, a torsion squared term and a matter sector. To obtain such geometrodynamical theory, asymptotic freedom and the Gribov parameter (soft BRST symmetry breaking) are crucial. Particularly, Newton and cosmological constant are related to these parameters and they also run as functions of the energy scale. One-loop computations are performed and the results are interpreted. [ABSTRACT FROM AUTHOR]

Yang-Mills theories supplemented by an additional coordinate constraint, which is solved and substituted in the original Lagrangian, provide examples of the so-called Nambu models, in the case where such constraints arise from spontaneous Lorentz symmetry breaking. Some explicit calculations have shown that, after additional conditions are imposed, Nambu models are capable of reproducing the original gauge theories, thus making Lorentz violation unobservable and allowing the interpretation of the corresponding massless gauge bosons as the Goldstone bosons arising from the spontaneous symmetry breaking. A natural question posed by this approach in the realm of gauge theories is to determine under which conditions the recovery of an arbitrary gauge theory from the corresponding Nambu model, defined by a general constraint over the coordinates, becomes possible. We refer to these theories as extended Nambu models (ENM) and emphasize the fact that the defining coordinate constraint is not treated as a standard gauge fixing term. At this level, the mechanism for generating the constraint is irrelevant and the case of spontaneous Lorentz symmetry breaking is taken only as a motivation, which naturally bring this problem under consideration. Using a nonperturbative Hamiltonian analysis we prove that the ENM yields the original gauge theory after we demand current conservation for all time, together with the imposition of the Gauss laws constraints as initial conditions upon the dynamics of the ENM. The Nambu models yielding electrodynamics, Yang-Mills theories and linearized gravity are particular examples of our general approach. [ABSTRACT FROM AUTHOR]

We investigate linearized gauge theories on globally hyperbolic spacetimes in the BRST formalism. A consistent definition of the classical phase space and of its Cauchy surface analogue is proposed. We prove that it is isomorphic to the phase space in the 'subsidiary condition' approach of Hack and Schenkel in the case of Maxwell, Yang-Mills, and Rarita-Schwinger fields. Defining Hadamard states in the BRST formalism in a standard way, their existence in the Maxwell and Yang-Mills case is concluded from known results in the subsidiary condition (or Gupta-Bleuler) formalism. Within our framework, we also formulate criteria for non-degeneracy of the phase space in terms of BRST cohomology and discuss special cases. These include an example in the Yang-Mills case, where degeneracy is not related to a non-trivial topology of the Cauchy surface. [ABSTRACT FROM AUTHOR]

We investigate lattice and continuous quantum gauge theories on the Euclidean plane with a structure group that is replaced by a H -algebra. H -algebras are non-commutative analogues of groups and contain the class of Voiculescu's dual groups. We are interested in non-commutative analogues of random gauge fields, which we describe through the random Holonomy that they induce. We propose a general definition of a Holonomy Field with symmetries displaying the structure of a H -algebra and construct such a field starting from a quantum Lévy process on a H -algebra in the category of probability spaces. We call them Quantum Holonomy Fields. We also consider the more abstract case of an H -algebra in a given algebraic category. This yields the notion of a Categorical Holonomy Field. As an application, we define higher dimensional generalizations of the so-called Master Field on the plane. [ABSTRACT FROM AUTHOR]

We follow up an earlier attempt to compute the Yang-Mills Lagrangian density from first principles. In that work, the Lagrangian density emerged replete with a Feynman-'t Hooft gauge fixing term. In this note we find that similar methods may be applied to produce the concomitant ghost term. Our methods are elementary and entirely and straightforwardly algebraic. Insofar as one of our first principles in the earlier computation was the Schwinger Action Principle, which is a differential version of the Feynman path integral, our computation here may be viewed as a differential version of the Faddeev-Popov functional integral approach to generating the ghost Lagrangian. As such, it avoids all measure theoretic difficulties and ambiguities, though at the price of generality. [ABSTRACT FROM AUTHOR]

We shall review conformal gravity as a gauge natural theory and discuss the consequences of Weyl covariance on the definition of physical states. [ABSTRACT FROM AUTHOR]

We consider generally relativistic gauge transformations for the spinorial fields finding two mutually exclusive but together exhaustive classes in which fermions are placed adding supplementary information to the results obtained by Lounesto, and identifying quantities analogous to the momentum vector and the Pauli-Lubanski axial vector. We discuss how our results are similar to those obtained by Wigner by taking into account the system of Dirac field equations. We will investigate the consequences for the dynamics and in particular we shall address the problem of getting the nonrelativistic approximation in a consistent way. We are going to comment on extensions. [ABSTRACT FROM AUTHOR]

The electroweak unification group is proposed for the charged lepton flavor violating decays of the muon and tau leptons. The group is in the lepton space. The left-handed leptons and antileptons are assigned to the fundamental representation of the semi-simple group. The gauge group is spontaneously broken to , where is the hypercharge, by introducing a scalar multiplet which belongs to the triplet representation 3 of the and is singlet under . At this stage charged vector bosons of which mediate the lepton flavor violating decays acquire masses and are decoupled with one Higgs scalar . The residual group has all the features of the left-right electroweak unification group extensively studied in the literature. The probability for lepton flavor violating decays is . [ABSTRACT FROM AUTHOR]

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 Σ , the original functional is extended appropriately by additional Lie ( G ) -valued 1-form gauge fields so as to lift the symmetry to Maps ( Σ , 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. [ABSTRACT FROM AUTHOR]

A new classical 2-spinor approach to U (1) gauge theory is presented in which the usual four-potential vector field is replaced by a symmetric second rank spinor. Following a lagrangian formulation, it is shown that the four-rank spinor representing the Maxwell field tensor has a U (1) local gauge invariance in terms of the electric and magnetic field strengths. When applied to the magnetic field of a monopole, this formulation, via the irreducible representation condition for the gauge group, leads to a quantization condition differing by a factor 2 of the one predicted by Dirac without relying on any kind of singular vector potentials. Finally, the U (1) invariant spinor equations, are applied to electron magnetic resonance which has many applications in the study of materials. [ABSTRACT FROM AUTHOR]

We consider a generalization of non-relativistic Schrödinger-Higgs Lagrangian by introducing a nonstandard kinetic term. We show that this model is Galilean invariant, we construct the conserved charges associated to the symmetries and realize the algebra of the Galilean group. In addition, we study the model in the presence of a gauge field. We also show that the gauged model is Galilean invariant. Finally, we explore relations between the twin models and their solutions. [ABSTRACT FROM AUTHOR]

We propose to analyze a modified electromagnetism inspired from the gauge sector of the Standard Model extension (SME). From the point of view of a canonical formulation, we carried out the usual analysis on the constraints structure of the odd sector (Carroll-Field-Jackiw term) and a Maxwell term with an effective metric. This effective metric is obtained by a vectorial decomposition of the CPT-even term, that is absorbed in the ordinary Maxwell term. Using symplectic projector method (SPM), we obtain the dispersions relations and we have verified conditions of stability to determine the valid spectrum. [ABSTRACT FROM AUTHOR]

It is proven that the differential equation for the null-curves of the pseudo-Euclidean space ℝ defines a flat dynamical system in the sense of optimal control theory. The connection with general gauge theories is briefly discussed. [ABSTRACT FROM AUTHOR]

We study Ihara's zeta function for graphs in the context of quivers arising from gauge theories, especially under Seiberg duality transformations. The distribution of poles is studied as we proceed along the duality tree, in light of the weak and strong graph versions of the Riemann Hypothesis. As a by-product, we find a refined version of Ihara's zeta function to be the generating function for the generic superpotential of the gauge theory. [ABSTRACT FROM AUTHOR]