Group Of Gauge Transformations Research Articles

Extended phase space in general gauge theories

In a recent paper, it was shown that in diffeomorphism-invariant theories, the symplectic vector fields induced by spacetime diffeomorphisms are integrable if one introduces an extended phase space. In this paper we extend the notion of extended phase space to all gauge theories with arbitrary combinations of internal and spacetime local symmetries. We formulate this in terms of a corresponding Atiyah Lie algebroid, a geometric object derived from a principal bundle which features internal symmetries and diffeomorphisms on an equal footing. In this language, gauge transformations are understood as morphisms between Atiyah Lie algebroids that preserve the geometric structures encoded therein. The extended configuration space of a gauge theory can subsequently be understood as the space of pairs (φ,Φ), where φ is a Lie algebroid morphism and Φ is a field configuration in the non-extended sense. Starting from this data, we outline a very powerful, manifestly geometric approach to the extended phase space. Using this approach, we find that the action of the group of gauge transformations and diffeomorphisms on the symplectic geometry of any covariant theory is integrable. We motivate our construction by carefully examining the need for extended phase space in Chern-Simons gauge theories and display its usefulness by re-computing the charge algebra. We also describe the implementation of the configuration algebroid in Einstein-Yang-Mills theories.

The Group Law for the New Internal-Spacetime Mapping between the Group of Internal Electromagnetic Gauge Transformations and the Groups LB1 and LB2 of Spacetime Tetrad Transformations

The Group Law for the New Internal-Spacetime Mapping between the Group of Internal Electromagnetic Gauge Transformations and the Groups LB1 and LB2 of Spacetime Tetrad Transformations

Langevin dynamic for the 2D Yang–Mills measure

We define a natural state space and Markov process associated to the stochastic Yang–Mills heat flow in two dimensions.To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both well-defined and satisfy good continuity properties. The desired state space is obtained as the corresponding space of orbits under this group action and is shown to be a Polish space when equipped with a natural Hausdorff metric.To construct the Markov process we show that the stochastic Yang–Mills heat flow takes values in our space of connections and use the “DeTurck trick” of introducing a time dependent gauge transformation to show invariance, in law, of the solution under gauge transformations.Our main tool for solving for the Yang–Mills heat flow is the theory of regularity structures and along the way we also develop a “basis-free” framework for applying the theory of regularity structures in the context of vector-valued noise – this provides a conceptual framework for interpreting several previous constructions and we expect this framework to be of independent interest.

An analytic application of Geometric Invariant Theory II: Coarse moduli spaces

In [5] the authors constructed a classifying space for polystable holomorphic vector bundles on a compact Kähler manifold using analytic GIT theory. The aim of this article is to show that this classifying space taken in the weakly normal category is a coarse moduli space in the sense of complex geometry when the topology is fixed as induced by the space of Hermite-Einstein connections modulo the group of unitary gauge transformations.

Dirac structures on the space of connections

We shall give a twisted Dirac structure on the space of irreducible connections on a SU(n)-bundle over a three-manifold, and give a family of twisted Dirac structures on the space of irreducible connections on the trivial SU(n)-bundle over a four-manifold. The twist is described by the Cartan 3-form on the space of connections. It vanishes over the subspace of flat connections. So the spaces of flat connections are endowed with ( non-twisted ) Dirac structures. The Dirac structure on the space of flat connections over the three-manifold is obtained as the boundary restriction of a corresponding Dirac structure over the four-manifold. We discuss also the action of the group of gauge transformations over these Dirac structures.

Singular gauge transformations in geometrodynamics

The new tetrads introduced previously for non-null electromagnetic fields in Einstein–Maxwell spacetimes enable a direct link to the local electromagnetic gauge group of transformations. Due to the peculiar elements in the construction of these new tetrads, a direct connection can be established between the local group of electromagnetic gauge transformations and local groups of tetrad transformations on two different local and orthogonal planes of eigenvectors of the Einstein–Maxwell stress–energy tensor. These tetrad vectors are gauge dependent. It is an interesting and relevant problem to study if there are local gauge transformations that can map on the timelike-spacelike plane, the timelike and the spacelike vectors into the intersection of the local light cone and the plane itself. How many of these local gauge transformations exist and how the mathematics and the geometry of these particular transformations play out. These local gauge transformations would be singular and it is important to identify them.

Dynamical symmetry breaking in Abelian geometrodynamics

A new tetrad is introduced within the framework of geometrodynamics for non-null electromagnetic fields. This tetrad diagonalizes the Einstein-Maxwell stress-energy tensor, any stress-energy tensor in a local and covariant way, and allows for maximum simplification of the expression of the electromagnetic field, in any curved four-dimensional Lorentzian spacetime, allowing for the identification of its degrees of freedom in two local scalars. New isomorphisms are proved. A new internal-spacetime mapping is established using these new tetrads. It is possible to map the local group of electromagnetic gauge transformations into the transformation groups of tetrad vectors on two local orthogonal planes. The planes that diagonalize the stress-energy tensor. We will discuss through a first order perturbative formulation the local loss of symmetry when a source of electromagnetic and gravitational field interacts with an agent that perturbs the original geometry associated to the source. The loss of symmetry will be manifested by the tilting of these planes under the influence of an external agent. In this strict sense the original local symmetry will be lost, however a new symmetry will arise. The purpose of this report is to show that the geometrical manifestation of local gauge symmetries is dynamic.

Signature-causality reflection generated by Abelian gauge transformations

In this paper, we want to better understand the causality reflection that arises under a subset of Abelian local gauge transformations in geometrodynamics. We proved in previous papers that in Einstein–Maxwell spacetimes, there exist two local orthogonal planes of gauge symmetry at every spacetime point for non-null electromagnetic fields. Every vector in these planes is an eigenvector of the Einstein–Maxwell stress–energy tensor. The vectors that span these local orthogonal planes are dependent on electromagnetic gauge. The local group of Abelian electromagnetic gauge transformations has been proved isomorphic to the local groups of tetrad transformations in these planes. We called LB1 the local group of tetrad transformations made up of SO(1, 1) plus two different kinds of discrete transformations. One of the discrete transformations is the full inversion two by two which is a Lorentz transformation. The other discrete transformation is given by a matrix with zeroes on the diagonal and ones off-diagonal two by two, a reflection. The group LB1 is realized on this plane, we call this plane one, and is spanned by the time-like and one space-like vectors. The other local orthogonal plane is plane two and the local group of tetrad transformations, we call this LB2, which is just SO(2). The local group of Abelian electromagnetic gauge transformations is isomorphic to both LB1 and LB2, independently. It has already been proved that a subset of local electromagnetic gauge transformations that leave the electromagnetic tensor invariant induces a change in sign in the norm of the tetrad vectors that span the local plane one. The reason is that one of the discrete transformations on the local plane one that belongs to the group LB1 is not a Lorentz transformation, it is a flip or reflection. It is precisely on this kind of discrete transformation that we have an interest since it has the effect of changing the signature and the causality. This effect has never been noticed before.

Pre-symplectic structures on the space of connections

Let X be a four-manifold with boundary three-manifold M. We shall describe (i) a pre-symplectic structure on the space A(X) of connections on the bundle X×SU(n) that comes from the canonical symplectic structure on the cotangent space T⁎A(X), and (ii) a pre-symplectic structure on the space A0♭(M) of flat connections on M×SU(n) that have null charge. These two structures are related by the boundary restriction map. We discuss also the Hamiltonian features of the space of connections A(X) together with the action of the group of gauge transformations.

Conditions of smoothness of moduli spaces of flat connections and of character varieties

We use gauge theoretic and algebraic methods to examine sufficient conditions for smooth points on the moduli space of flat connections on a compact manifold and on the character variety of a finitely generated and presented group. We give a complete proof of the slice theorem for the action of the group of gauge transformations on the space of flat connections. Consequently, the slice is smooth if the second cohomology of the manifold with coefficients in the semisimple part of the adjoint bundle vanishes. On the other hand, we find that the smoothness of the slice for the character variety of a finitely generated and presented group depends not only on the second group cohomology but also on the relation module of the presentation. However, when there is a single relator or if there is no relation among the relators in the presentation, our condition reduces to the minimality of the second group cohomology. This is also verified using Fox calculus. Finally, we compare the conditions of smoothness in the two approaches.

Picard groups of Poisson manifolds

For a Poisson manifold $M$ we develop systematic methods to compute its Picard group $\mathrm{Pic}(M)$, i.e., its group of self Morita equivalences. We establish a precise relationship between $\mathrm{Pic}(M)$, and the group of gauge transformations up to Poisson diffeomorphisms showing, in particular, that their connected components of the identity coincide; this allows us to introduce the Picard Lie algebra of $M$ and to study its basic properties. Our methods lead to the proof of a conjecture from “Poisson sigma models and symplectic groupoids, in Quantization of Singular Symplectic Quotients” [A.S. Cattaneo, G. Felder, Progress in Mathematics 198 (2001), 41] stating that $\mathrm{Pic}(\mathfrak{g}^*)$, for any compact simple Lie algebra agrees with the group of outer automorphisms of $\mathfrak{g}$.

Tetrads in SU(3) × SU(2) × U(1) Yang–Mills geometrodynamics

The relationship between gauge and gravity amounts to understanding the underlying new geometrical local structures. These structures are new tetrads specially devised for Yang–Mills theories, Abelian and non-Abelian in four-dimensional Lorentzian curved spacetimes. In the present paper, a new tetrad is introduced for the Yang–Mills [Formula: see text] formulation. These new tetrads establish a link between local groups of gauge transformations and local groups of spacetime transformations that we previously called LB1 and LB2. New theorems are proved regarding isomorphisms between local internal [Formula: see text] groups and local tensor products of spacetime LB1 and LB2 groups of transformations. These new tetrads define at every point in spacetime two orthogonal planes that we called blades or planes one and two. These are the local planes of covariant diagonalization of the stress–energy tensor. These tetrads are gauge dependent. Tetrad local gauge transformations leave the tetrads inside the local original planes without leaving them. These local tetrad gauge transformations enable the possibility to connect local gauge groups Abelian or non-Abelian with local groups of tetrad transformations. On the local plane one, the Abelian group [Formula: see text] of gauge transformations was already proved to be isomorphic to the tetrad local group of transformations LB1, for example. LB1 is [Formula: see text] plus two different kinds of discrete transformations. On the local orthogonal plane two [Formula: see text] is isomorphic to LB2 which is just [Formula: see text]. That is, we proved that LB1 is isomorphic to [Formula: see text] which is a remarkable result since a noncompact group plus two discrete transformations is isomorphic to a compact group. These new tetrads have displayed manifestly and nontrivially the coupling between Yang–Mills fields and gravity. The new tetrads and the stress–energy tensor allow for the introduction of three new local gauge invariant objects. Using these new gauge invariant objects and in addition a new general local duality transformation, a new algorithm for the gauge invariant diagonalization of the Yang–Mills stress–energy tensor is developed as an application. This is a paper about grand Standard Model gauge theories — General Relativity gravity unification and grand group unification in four-dimensional curved Lorentzian spacetimes.

Normalization in Lie algebras via mould calculus and applications

We establish Ecalle's mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to reduce the Lie-theoretic problem to a mould equation, the solutions of which are remarkably explicit and can be fully described by means of a gauge transformation group. The dynamical applications include the construction of Poincar{\'e}-Dulac formal normal forms for a vector field around an equilibrium point, a formal infinite-order multiphase averaging procedure for vector fields with fast angular variables (Hamiltonian or not), or the construction of Birkhoff normal forms both in classical and quantum situations. As a by-product we obtain, in the case of harmonic oscillators, the convergence of the quantum Birkhoff form to the classical one, without any Diophantine hypothesis on the frequencies of the unperturbed Hamiltonians.

Admissible boundary conditions for Hamiltonian field theories

In the setting of a multisymplectic formalism for Hamiltonian theories on manifolds with boundary a class of admissible boundary conditions based on the principle of preservation of the gauge symmetries of the theory is presented. Such admissible conditions are characterized as those boundary conditions determined by Lagrangian submanifolds on the space of fields at the boundary lying in the zero level set of the moment map of the group of gauge transformations at the boundary. The examples of gauge covariant fields and pure Yang–Mills theories are analyzed.

Equations of motion as constraints: superselection rules, Ward identities

The meaning of local observables is poorly understood in gauge theories, not to speak of quantum gravity. As a step towards a better understanding we study asymptotic (infrared) transformations in local quantum physics. Our observables are smeared by test functions, at first vanishing at infinity. In this context we show that the equations of motion can be seen as constraints, which generate a group, the group of space and time dependent gauge transformations. This is one of the main points of the paper. Infrared nontrivial effects are captured allowing test functions which do not vanish at infinity. These extended operators generate a larger group. The quotient of the two groups generate superselection sectors, which differentiate different infrared sectors. The BMS group changes the super-selection sector, a result long known for its Lorentz subgroup. It is hence spontaneously broken. Ward identities implied by the gauge invariance of the S-matrix generalize the standard results and lead to charge conservation and low energy theorems. Their validity does not require Lorentz invariance.

English

We define algebraic families of (all) morphisms which are purely algebraic analogs of quantum families of (all) maps introduced by P. M. Soltan. Also, algebraic families of (all) isomorphisms are introduced. By using these notions we construct two classes of Hopf-algebras which may be interpreted as the quantum group of all maps from a finite space to a quantum group, and the quantum group of all automorphisms of a finite noncommutative (NC) space. As special cases three classes of NC objects are introduced: quantum group of gauge transformations, Pontryagin dual of a quantum group, and Galois-Hopf-algebra of an algebra extension.

New Tetrads in Riemannian Geometry and New Ensuing Results in Group Theory, Gauge Theory and Fundamental Physics in Particle Physics, General Relativity and Astrophysics

A new tetrad is introduced within the framework of geometrodynamics for non-null electromagnetic fields. This tetrad diagonalizes the electromagnetic stress-energy tensor and allows for maximum simplification of the expression of the electromagnetic field. The Einstein-Maxwell equations will also be simplified. New group isomorphisms are proved. The local group of electromagnetic gauge transformations is isomorphic to the new group LB1. LB1 is the group of local tetrad transformations comprised by SO(1,1) plus two different kinds of discrete transformations. The local group of electromagnetic gauge transformations is also isomorphic to the local group of tetrad transformations LB2, which is SO(2), as well. Therefore, we proved that LB1 is isomorphic to LB2. These group results amount to proving that the no-go theorems of the sixties like the S. Coleman- J. Mandula, the S. Weinberg or L. ORaifeartagh versions are incorrect. Not because of their internal logic, but because of the assumptions made at the outset of all these versions. These new tetrads are useful in astrophysics spacetime evolution algorithms since they introduce maximum simplification in all relevant objects, specially in stress-energy tensors.

A note on asymptotic symmetries and soft-photon theorem

We use the asymptotic data at conformal null-infinity $\mathscr{I}$ to formulate Weinberg's soft-photon theorem for Abelian gauge theories with massless charged particles. We show that the angle-dependent gauge transformations at $\mathscr{I}$ are not merely a gauge redundancy, instead they are genuine symmetries of the radiative phase space. In the presence of these symmetries, Poisson bracket between the gauge potentials is not well-defined. This does not pose an obstacle for the quantization of the radiative phase space, which proceeds by treating the conjugate electric field as the fundamental variable. Denoting by $\mathcal{G}_+$ and $\mathcal{G}_-$ as the group of gauge transformations at $\mathscr{I}^+$ and $\mathscr{I}^-$ respectively, Strominger has shown that a certain diagonal subgroup $ \mathcal{G}_{diag} \subset \mathcal{G}_+ \times \mathcal{G}_-$ is the symmetry of the S-matrix and Weinberg's soft-photon theorem is the corresponding Ward identity. We give a systematic derivation of this result for Abelian gauge theories with massless charged particles. Our derivation is a slight generalization of the existing derivations since it is applicable even when the bulk spacetime is not exactly flat, but is only "almost" Minkowskian.

Norm Continuous Unitary Representations of Lie Algebras of Smooth Sections

Let K→X be a smooth Lie algebra bundle over a σ-compact manifold X whose typical fiber is the compact Lie algebra k. We give a complete description of the irreducible bounded (i.e., norm continuous) unitary representations of the Frechet–Lie algebra Γ(K) of all smooth sections of K, and of the LF-Lie algebra Γ_c(K) of compactly supported smooth sections. For Γ(K), irreducible bounded unitary representations are finite tensor products of so-called evaluation representations, hence in particular finite dimensional. For Γ_c(K), bounded unitary irreducible (factor) representations are possibly infinite tensor products of evaluation representations, which reduces the classification problem to results of Glimm and Powers on irreducible (factor) representations of UHF C^∗-algebras. The key part in our proof is the result that every irreducible bounded unitary representation of a Lie algebra of the form k⊗A, where A is a unital real complete continuous inverse algebra, is a finite product of evaluation representations. On the group level, our results cover in particular the bounded unitary representations of the identity component Gau(P)_0 of the group of smooth gauge transformations of a principal fiber bundle P→X with compact base and structure group, and the groups SU_n(A)_0 with A a complete involutive commutative continuous inverse algebra.

MAPPING CLASS GROUP AND U(1) CHERN–SIMONS THEORY ON CLOSED ORIENTABLE SURFACES

U(1) Chern–Simons theory is quantized canonically on manifolds of the form [Formula: see text], where Σ is a closed orientable surface. In particular, we investigate the role of mapping class group of Σ in the process of quantization. We show that, by requiring the quantum states to form representation of the holonomy group and the large gauge transformation group, both of which are deformed by quantum effect, the mapping class group can be consistently represented, provided the Chern–Simons parameter k satisfies an interesting quantization condition. The representations of all the discrete groups are unique, up to an arbitrary sub-representation of the mapping class group. Also, we find a k↔1/k duality of the representations.