Invariant Connections in a Non-Abelian Principal Bundle

Invariance conditions for gauge fields under smooth group actions are interpreted in terms of invariant connections on principal bundles. A classification of group actions on bundles as automorphisms projecting to an action on a base manifold with a sufficiently regular orbit structure is given in terms of group homorphisms and a generalization of Wang’s theorem classifying invariant connections is derived. Illustrative examples on compactified Minkowski space are given.

Using the invariant geometrical interpretation of gauge and Higgs fields, a simple derivation is given of the dimensional reduction procedure. The underlying assumption with regard to the Riemannian structure, group orbits and invariant connection are clarified and the critical points of the Higgs potential are shown to have a natural geometrical interpretation.

We present a number of findings concerning groupoid dynamical systems and groupoid crossed products. The primary result is an identification of the spectrum of the groupoid crossed product when the groupoid has continuously varying abelian stabilizers and a well behaved orbit space. In this case, the spectrum of the crossed product is homeomorphic, via an induction map, to a quotient of the spectrum of the crossed product by the stabilizer group bundle. The main theorem is also generalized in the groupoid algebra case to an identification of the primitive ideal space. This generalization replaces the assumption that the orbit space is well behaved with an amenability hypothesis. We then use induction to show that the primitive ideal space of the groupoid algebra is homeomorphic to a quotient of the dual of the stabilizer group bundle. In both cases the identification is topological. We then apply these theorems in a number of examples, and examine when a groupoid algebra has Hausdorff spectrum. As a separate result, we also develop a theory of principal groupoid group bundles and locally unitary groupoid actions. We prove that such actions are characterized, up to exterior equivalence, by a cohomology class which arises from a principal bundle. Furthermore, we also demonstrate how to construct a locally unitary action from a given principal bundle. This last result uses a duality theorem for abelian group bundles which is also included as part of this thesis.

We prove local existence and uniqueness of static spherically symmetric solutions of the Einstein–Yang–Mills (EYM) equations for an arbitrary compact semisimple gauge group in the so-called regular case. By this we mean the equations obtained when the rotation group acts on the principal bundle on which the Yang–Mills connection takes its values in a particularly simple way (the only one ever considered in the literature). The boundary value problem that results for possible asymptotically flat soliton or black hole solutions is very singular and just establishing that local power series solutions exist at the center and asymptotic solutions at infinity amounts to a nontrivial algebraic problem. We discuss the possible field equations obtained for different group actions and solve the algebraic problem on how the local solutions depend on initial data at the center and at infinity.

Vacuum solutions for multidimensional gravity on the principal bundle with\nthe SU(2) structural group as the extra dimensions are found and discussed.\nThis generalizes the results of Ref. \\cite{vds3} from U(1) to the SU(2) gauge\ngroup. The spherically symmetric solution with the off-diagonal components of\nthe multidimensional metric is obtained. It is shown that two types of\nsolutions exist: the first has a wormhole-like 4D base, the second is a\ngravitational flux tube with two color and electric charges. The solution\ndepends only on two parameters: the values of the electric and magnetic fields\nat the origin. In the plane of these parameters there exists a curve separating\nthe regions with different types of solutions. An analogy with the 5D solutions\nis discussed.\n

Our purpose in this chapter is to expose, in the abstract language that we employ throughout this treatise, the fundamentals of the classical theory indicated by the subject in the title. Roughly speaking, we want to put into perspective the classical and physically, yet mathematically, important (!) theme of the so-called geometry of Yang–Mills equations. This was first advocated by I.M. Singer [1] (see also, for instance, M.F. Atiyah [1: p. 2]). Equivalently, one considers the corresponding space of solutions of the said equations, thus, by definition (see Chapt. I, Definitions 4.1 and 4.2), the space of the Yang–Mills A-connections. However, in view of the physical significance of the “gauge invariant (A-)connections” (see Atiyah’s phrasing in the epigraph above), the same space is finally divided out by the corresponding “gauge group,” so that it is, in effect, the resulting quotient space (“moduli space,” or even “orbit space”) that is under consideration.KeywordsModulus SpaceGauge GroupGauge TransformationGauge InvarianceOrbit SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Given a group $G$ and an abelian $C^*$-algebra $\mathfrak{A}$, the antihomomorphisms $\Theta\colon G\rightarrow \mathrm{Aut}(\mathfrak{A})$ are in one-to-one with those left actions $\Phi\colon G\times \mathrm{Spec}(\mathfrak{A})\rightarrow \mathrm{Spec}(\mathfrak{A})$ whose translation maps $\Phi_g$ are continuous; whereby continuities of $\Theta$ and $\Phi$ turn out to be equivalent if $\mathfrak{A}$ is unital. In particular, a left action $\phi\colon G \times X\rightarrow X$ can be uniquely extended to the spectrum of a $C^*$-subalgebra $\mathfrak{A}$ of the bounded functions on $X$ if $\phi_g^*(\mathfrak{A})\subseteq \mathfrak{A}$ holds for each $g\in G$. In the present paper, we apply this to the framework of loop quantum gravity. We show that, on the level of the configuration spaces, quantization and reduction in general do not commute, i.e., that the symmetry-reduced quantum configuration space is (strictly) larger than the quantized configuration space of the reduced classical theory. Here, the quantum-reduced space has the advantage to be completely characterized by a simple algebraic relation, whereby the quantized reduced classical space is usually hard to compute.

Given a principal bundle over a closed manifold, \(G \rightarrow P \rightarrow M\), let \(P^{Ad} \rightarrow M\) be the associated adjoint bundle. Gruher and Salvatore (Proc Lond Math Soc 96(3), 78106 2008) showed that the Thom spectrum \((P^\mathrm{Ad})^{-TM}\) is a ring spectrum whose corresponding product in homology is a Chas-Sullivan type string topology product. We refer to this spectrum as the “string topology spectrum of P”, \( \mathcal {S}(P)\). In the universal case when P is contractible, \(\mathcal {S}(P) \simeq LM^{-TM}\) where LM is the free loop space of the manifold. This ring spectrum was introduced by the authors in Cohen et al. (Math Annalen 324, 773–798 2002) as a homotopy theoretic realization of the Chas-Sullivan string topology of M. The main purpose of this paper is to introduce an action of the gauge group of the principal bundle, \(\mathcal {G}(P)\) on the string topology spectrum \(\mathcal {S}(P)\), and to study this action in detail. Indeed we study the entire group of units and the induced representation \(\mathcal {G}(P) \rightarrow GL_1(\mathcal {S}(P))\). We show that this group of units is the group of homotopy automorphisms of the fiberwise suspension spectrum of P. More generally we describe the homotopy type of the group of homotopy automorphisms of any E-line bundle for any ring spectrum E. We import some of the basic ideas of gauge theory, such as the action of the gauge group on the space of connections to the setting of E-line bundles over a manifold and do explicit calculations. We end by discussing a functorial perspective, which describes a sense in which the string topology spectrum \(\mathcal {S}(P)\) of a principal bundle is the “linearization” of the gauge group \(\mathcal {G}(P)\).

The gravitational interaction, as described by the Einstein–Cartan theory, is shown to emerge as the by-product of the spontaneous symmetry breaking of a gauge symmetry in a pre-geometric four-dimensional spacetime. Starting from a formulation a ` la Yang–Mills on an SO ( 1 , 4 ) or SO ( 3 , 2 ) principal bundle and not accounting for a spacetime metric, the Einstein–Hilbert action is recovered after the identification of the effective spacetime metric and spin connection for the residual SO ( 1 , 3 ) gauge symmetry of the spontaneously broken phase—i.e. the stabiliser of the SO ( 1 , 4 ) or SO ( 3 , 2 ) gauge group. Thus, the two fundamental tenets of general relativity, i.e. diffeomorphism invariance and the equivalence principle, can arise from a more fundamental gauge principle. The two mass parameters that characterise Einstein gravity, namely the Planck mass and the cosmological constant, are likewise shown to be emergent. The phase transition from the unbroken to the spontaneously broken phase is expected to happen close to the Planck temperature. This is conjectured to be dynamically driven by a scalar field that implements a Higgs mechanism, hence providing mass to new particles, with consequences for cosmology and high-energy physics. The couplings of gravity to matter are discussed after drawing up a dictionary that interconnects pre-geometric and effective geometric quantities. In the unbroken phase where the fundamental gauge symmetry is restored, the theory is potentially power-counting renormalisable without matter, offering a novel path towards a UV completion of Einstein gravity.

We show how characteristic classes determine equivariant prequantization bundles over the space of connections on a principal bundle. These bundles are shown to generalize the Chern-Simons line bundles to arbitrary dimensions. Our result applies to arbitrary bundles, and it is studied the action of both the gauge group and the automorphisms group. The action of the elements in the connected component of the identity of the group generalizes known results in the literature. The action of the elements not connected with the identity is shown to be determined by a characteristic class by using differential characters and equivariant cohomology. We extend our results to the space of Riemannian metrics and the actions of diffeomorphisms. In dimension 2, a ${\Gamma}_{M}$-equivariant prequantization bundle of the Weil-Petersson symplectic form on the Teichm\"uller space is obtained, where ${\Gamma}_{M}$ is the mapping class group of the surface M.

An approach to gauge theory in the context of locally conformally flat space–time is described. It is discussed how there are a number of natural principal bundles associated with any given locally conformally flat space–time X. The simplest of these principal bundles is the bundle PX(G) with structure group G = U(2, 2). An 11-dimensional bundle Q with structure group a certain seven-dimensional group K is constructed by a method involving a reduction of structure group for the bundle PX(G). It is shown how the gauge groups U(1), SU(2), and SU(3) can be derived from the geometry of locally conformally flat space–time. Fock spaces of multiparticle states for the fields of the standard model are constructed in the context of bundles with these groups as structure groups. Scattering and other particle interaction processes are defined in terms of linear maps between multiparticle state spaces. A technique for computing analytically and/or computationally the masses of the elementary particles is described. This method involves the computation of a certain quantity called the integral mass spectrum for a given family of particles and then the masses of the particles in the family are determined to be the locations of the peaks of the integral mass spectrum. The method is applied successfully in the electroweak sector to the cases of the charged leptons, μ and τ, and the Z0 particle.

In this paper we exhibit a family of flat left invariant affine structures on the double Lie group of the oscillator Lie group of dimension 4, associated to each solution of classical Yang-Baxter equation given by Boucetta and Medina. On the other hand, using Koszul's method, we prove the existence of an immersion of Lie groups between the group of affine transformations of a flat affine and simply connected manifold and the classical group of affine transformations of $\mathbb{R}^n$. In the last section, for each flat left invariant affine symplectic connection on the group of affine transformations of the real line, describe for Medina-Saldarriaga-Giraldo, we determine the affine symplectomorphisms. Finally we exhibit the Hess connection, associated to a Lagrangian bi-foliation, which is flat left invariant affine.

For pure Yang-Mills theory on Minkowski space-time, formulated in functional spaces where the covariant divergence is an elliptic operator without zero modes, and for a trivial principal bundle over the fixed time Euclidean space with a compact, semisimple, connected and simply connected structure Lie group, a Green function for the covariant divergence has been found. It allows one to solve the first class constraints associated with Gauss’ laws and to identify a connection-dependent coordinatization of the trivial principal bundle. In a neighborhood of the global identity section, by using canonical coordinates of the first kind on the fibers, one has a symplectic implementation of the Lie algebra of the small gauge transformations generated by Gauss’ laws and one can make a generalized Hodge decomposition of the gauge potential one-forms based on the BRST operator. This decomposition singles out a pure gauge background connection (the BRST ghost as Maurer-Cartan one-form on the group of gauge transformations) and a transverse gauge-covariant magnetic gauge potential. After an analogous decomposition of the electric field strength into the transverse and the longitudinal part, Dirac’s observables associated with the transverse electric and magnetic components are identified as their restriction to the global identity section of the trivial principal bundle. The longitudinal part of the electric field can be re-expressed in terms of these electric and magnetic transverse parts and of the constraints without Gribov ambiguity. The physical Lagrangian, Hamiltonian, non-Abelian and topological charges have been obtained in terms of transverse Dirac’s observables, also in the presence of fermion fields, after a symplectic decoupling of the gauge degrees of freedom; one has an explicit realization of the abstract “Riemannian metric” on the orbit space. Both the Lagrangian and the Hamiltonian are nonlocal and nonpolynomial; like in the Coulomb gauge they are not Lorentz-invariant, but the invariance can be enforced on them if one introduces Wigner covariance of the observables by analyzing the various kinds of Poincare orbits of the system and by reformulating the theory on suitable spacelike hypersurfaces, following Dirac. By extending to classical relativistic field theory the problems associated with the Lorentz noncovariance of the canonical (presymplectic) center of mass for extended relativistic systems, in the sector of the field theory with P2>0 and W2≠0 one identifies a classical invariant intrinsic unit of length, determined by the Poincare Casimirs, whose quantum counterpart is the ultraviolet cutoff looked for by Dirac and Yukawa: it is the Compton wavelength of the field configuration (in an irreducible Poincare representation) multiplied by the value of its spin.

The functional quantization of gauge field theories leads to interesting infinite dimensional geometric problems from which one expects to extract some information on the original gauge theory. The geometric framework underlying a gauge field theory is essentially that of an infinite dimensional Lie group G, the gauge group, acting on an infinite dimensional manifold, the manifold of paths p. Here we shall consider a setting in which this group action gives rise to a principal bundle π : p → p/G. KeywordsGauge GroupVector BundleFundamental FormElliptic OperatorPseudodifferential OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, as well as the “underlying” diffeological space of any stack. More precisely, we consider diffeological spaces as so-called concrete sheaves and show that the Grothendieck construction sending these sheaves to stacks has a left adjoint: the functor sending any stack to its diffeological coarse moduli space. As an application, we restrict our attention to differentiable stacks and examine the geometry behind the coarse moduli space construction in terms of Lie groupoids and their principal bundles. Within this context, we define a “gerbe”, and show when a Lie groupoid is such a gerbe (or when a stack is represented by one). Additionally, we define basic differential forms for stacks and confirm in the differentiable case that these agree (under certain conditions) with basic differential forms on a representative Lie groupoid. These basic differentiable forms in turn match the diffeological forms on the orbit space.