Geometrical approach to the reduction of gauge theories with spontaneously broken symmetry

The reduction of a theory with gauge group G to a theory which is gauge invariant with respect to a subgroup H of G is formulated in a geometrical language. It is assumed that among the physical fields considered as cross-sections of fibre bundles with structure group G there exists a section of the fibre bundle with fibre isomorphic to G/H — a Higgs field. The investigation of the broken gauge symmetry is based on the reduction theorem for structure groups of principal fibre bundles. The reduction of fields and their covariant derivatives is studied.     

Quantized Abelian Principal Connections on Lorentzian Manifolds

We construct a covariant functor from a category of Abelian principal bundles over globally hyperbolic spacetimes to a category of *-algebras that describes quantized principal connections. We work within an appropriate differential geometric setting by using the bundle of connections and we study the full gauge group, namely the group of vertical principal bundle automorphisms. Properties of our functor are investigated in detail and, similar to earlier works, it is found that due to topological obstructions the locality property of locally covariant quantum field theory is violated. Furthermore, we prove that, for Abelian structure groups containing a nontrivial compact factor, the gauge invariant Borchers-Uhlmann algebra of the vector dual of the bundle of connections is not separating on gauge equivalence classes of principal connections. We introduce a topological generalization of the concept of locally covariant quantum fields. As examples, we construct for the category of principal U(1)-bundles two natural transformations from singular homology functors to the quantum field theory functor that can be interpreted as the Chern class and the electric charge. In this case we also prove that the electric charges can be consistently set to zero, which yields another quantum field theory functor that satisfies all axioms of locally covariant quantum field theory.     

Projective and volume-preserving bundle structures involved in the formulation ofA(4) gauge theories

The bundle structures required by volume-preserving and related projective properties are developed and discussed in the context ofA(4) gauge theories which may be taken as the proper framework for Poincaré gauge theories. The results of this paper include methods for extending both tensors and connections to a principal fiber bundle havingG1(4,R)xG1(4,R) as its structure group. This bundle structure is shown to be a natural arena for the generalized (±) covariant differentiation utilized by Einstein for his extended gravitational theories involving nonsymmetric connections. In particular, it is shown that this generalized (±) covariant differentiation is actually a special case of ordinary covariant differentiation with respect to a connection on theG1(4,R) xG1(4,R) bundle. These results are discussed in relation to certain properties of generalized gravitational theories based on a nonsymmetric connection which include the metric affine theories of Hehl et al. and the general requirement that it should be possible to formulate well-defined local conservation laws. In terms of the extended bundle structure considered in this paper, it is found that physically distinct particle number type conservation expressions could exist for certain given types of matter currents.     

The field copy problem: to what extent do curvature (gauge field) and its covariant derivatives determine connection (gauge potential)?

We show that a connection of a principal bundle is determined up to (global) gauge equivalence by the curvature and its covariant derivatives provided that the infinitesimal holonomy group is of constant dimension and the base space is simply connected. If the dimension of the infinitesimal holonomy group varies, there may be obstructions of a topological nature to the existence of a global or even local gauge equivalence between two connections whose curvatures and covariant derivatives of curvature agree everywhere. These obstructions are analyzed and illustrated by examples.     

A grand superspace for unified field theories

Agrand superspace is proposed as the phase space for gauge field theories with a fixed structure groupG over a fixed space-time manifoldM. This superspace incorporatesall principal fiber bundles with these data. This phase space is the space of isomorphism classes ofall connections onall G-principal fiber bundles overM (fixedG andM). The justification for choosing this grand superspace for the phase space is that the space-time and the structure group are determinants of the physical theory, but the principal fiber bundle with the givenG andM is not. Grand superspace is studied in terms of a natural universal principal fiber bundle overM, canonically associated withM alone, and with a natural universal connection on this bundle. This bundle and its connection are universal in the sense that all connections on allG-principal fiber bundles (anyG) overM can be recovered from this universal bundle and its universal connection by a canonical construction. WhenG is Abelian, grand superspace is shown to be an Abelian group. Various subspaces of grand superspace consisting of the isomorphism classes of flat connections and of Yang-Mills connections are also discussed.     

A unified approach to conservation laws in general relativity,gauge theories,and elementary particle physics

A unified treatment of conservation laws in general relativity, gauge theories, and elementary particle physics is formulated in the setting of principal fiber bundles. The group AUT(P) is introduced as the general gauge transformation group that covers space-time coordinate transformations. A set of master equations is exhibited for any Lagrangian density generally covariant with respect to AUT(P). The symmetry group for elementary particle theory is shown to be the structure group of the bundle only in the special case when the gauge potential is flat and the space-time is simply connected. In the general case, the symmetry group is reduced to the symmetry group of the gauge potential. This natural mechanism for a reduction of the symmetry group is speculated on as a model for spontaneous symmetry breaking.This essay received an honorable mention from the Gravity Research Foundation for the year 1981-Ed.Partially supported by a grant from the National Science Foundation.     

Twisted-product categorical bundles

Categorical bundles provide a natural framework for gauge theories involving multiple gauge groups. Unlike the case of traditional bundles there are distinct notions of triviality, and hence also of local triviality, for categorical bundles. We study categorical principal bundles that are product bundles in the categorical sense, developing the relationship between functorial sections of such bundles and trivializations. We construct functorial cocycles with values in categorical groups using a suitable family of locally defined functors on the object space of the base category. Categorical product bundles being too rigid to give a widely applicable model for local triviality, we introduce the notion of a twisted-product categorical bundle. We relate such bundles to decorated categorical bundles that contain more information, specifically parallel transport data.     

Normal frames for general connections on differentiable fibre bundles

The theory of frames normal for general connections on differentiable bundles is developed. Links with the existing theory of frames normal for covariant derivative operators (linear connections) in vector bundles are revealed. The existence of bundle coordinates normal at a given point and/or along injective horizontal path is proved. A necessary and sufficient condition of existence of bundle coordinates normal along injective horizontal mappings is derived.     

A Review About Invariance Induced Gravity: Gravity and Spin from Local Conformal-Affine Symmetry

In this review paper, we discuss how gravity and spin can be obtained as the realization of the local Conformal-Affine group of symmetry transformations. In particular, we show how gravitation is a gauge theory which can be obtained starting from some local invariance as the Poincaré local symmetry. We review previous results where the inhomogeneous connection coefficients, transforming under the Lorentz group, give rise to gravitational gauge potentials which can be used to define covariant derivatives accommodating minimal couplings of matter, gauge fields (and then spin connections). After we show, in a self-contained approach, how the tetrads and the Lorentz group can be used to induce the spacetime metric and then the Invariance Induced Gravity can be directly obtained both in holonomic and anholonomic pictures. Besides, we show how tensor valued connection forms act as auxiliary dynamical fields associated with the dilation, special conformal and deformation (shear) degrees of freedom, inherent to the bundle manifold. As a result, this allows to determine the bundle curvature of the theory and then to construct boundary topological invariants which give rise to a prototype (source free) gravitational Lagrangian. Finally, the Bianchi identities, the covariant field equations and the gauge currents are obtained determining completely the dynamics.     

Generalized Hamiltonian formalism of (2+1)-dimensional non-linear \sigma-model in polynomial formulation

We investigate the canonical structure of the (2+1)-dimensional non-linear model in a polynomial formulation. A current density defined in the non-linear model is a vector field, which satisfies a formal flatness (or pure gauge) condition. It is the polynomial formulation in which the vector field is regarded as a dynamic variable on which the flatness condition is imposed as a constraint condition by introducing a Lagrange multiplier field. The model so formulated has gauge symmetry under a transformation of the Lagrange multiplier field. We construct the generalized Hamiltonian formalism of the model explicitly by using the Dirac method for constrained systems. We derive three types of the pre-gauge-fixing Hamiltonian systems: In the first system the current algebra is realized as the fundamental Dirac Brackets. The second one manifests the similar canonical structure as the Chern-Simons or BF theories. In the last one there appears an interesting interaction as the dynamic variables are coupled to their conjugate momenta via the covariant derivative. Received: 29 September 1998 / Published online: 14 January 1999     

Covariant extensions and the nonsymmetric unified field

The problem of generally covariant extension of Lorentz invariant field equations, by means of covariant derivatives extracted from the nonsymmetric unified field, is considered. It is shown that the contracted curvature tensor can be expressed in terms of a covariant gauge derivative which contains the gauge derivative corresponding to minimal coupling, if the universal constantp, characterizing the nonsymmetric theory, is fixed in terms of Planck's constant and the elementary quantum of charge. By this choice the spinor representation of the linear connection becomes closely related to the spinor affinity used by Infeld and Van Der Waerden in their generally covariant formulation of Dirac's equation.     

Tau functions and the twistor theory of integrable systems

We first define τ-functions as generalized cross-ratios of four points on a finite- or infinite-dimensional Grassmannian. We show how this definition can be used to construct a natural flat connection on a determinant line bundle associated with two equivariant holomorphic vector bundles over a twistor space, provided that the action of the symmetries on the bundles has the same normal form at the fixed points for the two bundles. The determinant line bundle has a natural meromorphic section of which the logarithmic covariant derivative is the logarithmic derivative of the τ-function. We establish a natural product formula for this τ-function; we show that it vanishes at the jumping lines of one bundle and has poles at the jumping lines of the other. We also show that this definition leads to standard expressions for the τ-functions of the KdV equation, the Ernst equation, and the isomonodromic deformation equations. We describe a new twistor treatment of the isomonodromic deformation equations.     

Nonabelian Bundle Gerbes,Their Differential Geometry and Gauge Theory

Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge fields needed for the construction of Yang-Mills theories with 2-form gauge potential.Acknowledgement We have benefited from discussions with L. Breen, D. Husemoller, A. Alekseev, L. Castellani, J. Kalkkinen, J. Mickelsson, R. Minasian, D. Stevenson and R. Stora.     

Elliptic operators in the functional quantisation for gauge field theories

Given a gauge theory with gauge groupG acting on a path spaceX,G andX being both infinite dimensional manifolds modelled on spaces of sections of vector bundles on a compact riemannian manifold without boundary, it is shown that when the action ofG onX is smooth, free and proper, the same ellipticity condition on an operator naturally given by the geometry of the problem yields both the existence of a principal fibre bundle structure induced by the canonical projection :XX/G and the existence of the Faddeev-Popov determinant arising in the functional quantisation of the gauge theory. This holds for certain gauge theories with anomalies like bosonic closed string theory in non-critical dimension and also holds for a class of gauge theories which includes Yang-Mills theory.     

Geometry of quantum principal bundles I

A theory of principal bundles possessing quantum structure groups and classical base manifolds is presented. Structural analysis of such quantum principal bundles is performed. A differential calculus is constructed, combining differential forms on the base manifold with an appropriate differential calculus on the structure quantum group. Relations between the calculus on the group and the calculus on the bundle are investigated. A concept of (pseudo)tensoriality is formulated. The formalism of connections is developed. In particular, operators of horizontal projection, covariant derivative and curvature are constructed and analyzed. Generalizations of the first Structure Equation and of the Bianchi identity are found. Illustrative examples are presented.     

Nonlinear realizations of symmetry groups in terms of differential geometry

Nonlinear realizations of a symmetry group G, which become linear when restricted to a subgroup H are described in terms of fibre bundles. It is shown that so-called “covariant derivatives” occuring in nonlinear Lagrangians are equivalent to the covariant derivatives of the canonical connection in the principal bundle (G, G/H, H, δ). After the specification of a cross-section of the bundle, our formulae for the covariant derivatives coincide with those obtained by other authors in a group-theoretical way. In a special case where G is a chiral group and H is its diagonal subgroup, the canonical connection induces the Riemannian connection in the tangent bundle over G/H. For G = SU(2) × SU(2) and H = SU(2) this connection coincides with the Riemannian connection on the three-dimensional sphere introduced by K. Meetz.     

The Second Noether Theorem in the formalism of jet-bundles: Symmetries and degeneration

By means of the jet-bundle formalism, the Second Noether Theorem is formulated for a general first-order Lagrangian field theory with infinitesimal local symmetries. These symmetries are implemented by a linear differential operator acting between the sections of a vector bundle and vector fields on the configuration bundle. The problem of the degeneration of the Lagrangian system is examined from a covariant and an instantaneous (i.e. space+time split) viewpoint. It is shown that in the instantaneous approach the presence of infinitesimal local symmetries leads to degeneration of the theory. Vertical local symmetries are shown to imply degeneration also in the covariant formalism. These results can be extended to higher-order Lagrangians as well.     

Gauge theory of the star product

The choice of a star product realization for non-commutative field theory can be regarded as a gauge choice in the space of all equivalent star products. With the goal of having a gauge invariant treatment, we develop tools, such as integration measures and covariant derivatives on this space. The covariant derivative can be expressed in terms of connections in the usual way giving rise to new degrees of freedom for non-commutative theories.