Showing papers on "Frame bundle published in 1996"

Strong connections on quantum principal bundles

Abstract: A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and non-strong connections are provided. In particular, such connections are constructed on a quantum deformation of the two-sphere fibrationS 2→RP 2. A certain class of strongU q (2)-connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with theq-dependent hermitian metric. A particular form of the Yang-Mills action on a trivialU q (2)-bundle is investigated. It is proved to coincide with the Yang-Mills action constructed by A. Connes and M. Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent ofq.

Conformal blocks on elliptic curves and the Knizhnik-Zamolodchikov-Bernard equations

Abstract: We give an explicit description of the vector bundle of WZW conformal blocks on elliptic curves with marked points as a subbundle of a vector bundle of Weyl group invariant vector valued theta functions on a Cartan subalgebra. We give a partly conjectural characterization of this subbundle in terms of certain vanishing conditions on affine hyperplanes. In some cases, explicit calculations are possible and confirm the conjecture. The Friedan-Shenker flat connection is calculated, and it is shown that horizontal sections are solutions of Bernard's generalization of the Knizhnik-Zamolodchikov equation.

Four-dimensional BF theory as a topological quantum field theory

Abstract: Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that four-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle. The case G=GL(4,ℝ) is especially interesting because every 4-manifold is then naturally equipped with a principal G-bundle, namely its frame bundle. In this case, the partition function of a compact oriented 4-manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of four-manifolds.

Zero-pole interpolation for meromophic matrix functions on an algebraic curve and transfer functions of 2D systems

Abstract: We formulate and solve the problem of constructing a meromorphic bundle map over a compact Riemann surface X having a prescribed zero-pole structure (including directional information). The output bundle together with the zero-pole data is prespecified while the input bundle and the bundle map are to be determined. The Riemann surface X is assumed to be (birationally) embedded as an irreducible algebraic curve in ℙ2 and both input and output bundles are assumed to be equal to the kernels of determinantal representations for X. In this setting the solution can be found as the joint transfer function of a Livsic-Kravitsky two-operator commutative vessel (2D input-output dynamical system). Also developed is the basic theory of two-operator commutative vessels and the correct analogue of the transfer function for such a system (a meromorphic bundle map between input and output bundles defined over an algebraic curve associated with the vessel) together with a state space realization, a Mittag-Leffler type interpolation theorem and the state space similarity theorem for such bundle mappings. A more abstract version of the zero-pole interpolation problem is also presented.

Generalized Fourier-Mukai transforms

Abstract: The paper sets out a generalized framework for Fourier-Mukai transforms and illustrates their use via vector bundle transforms. A Fourier-Mukai transform is, roughly, an isomorphism of derived categories of (sheaves) on smooth varieties X and Y. We show that these can only exist if the first Chern class of the varieties vanishes and, in the case of vector bundle transforms, will exist if and only if there is a bi-universal bundle on XxY which is "strongly simple" in a suitable sense. Some applications are given to abelian varieties extending the work of Mukai.

Relations between linear connections on the tangent bundle and connections on the jet bundle of a fibred manifold

Abstract: All natural operations transforming linear connections on the tangent bundle of a fibred manifold to connections on the 1-jet bundle are classified. It is proved that such operators form a 2-parameter family (with real coefficients).

On the Deformation Quantization of super-Poisson Brackets

Abstract: We show that for every vector bundle E over any given symplectic manifold M there exists an explicitly given super-Poisson bracket on the space of sections of the dual Grassmann bundle associated to E built out of symplectic structure of M, a fibre metric on E, and a connection in E compatible with the given fibre metric. Moreover, we construct a deformation quantization for this space of sections by means of a Fedosov type procedure.

Product preserving bundle functors on fibered manifolds

Abstract: To Ivan Koll a r, on the occasion of his 60th birthday. Abstract. The complete description of all product preserving bundle functors on bered manifolds in terms of natural transformations between product preserving bundle functors on manifolds is given. complete description of all product preserving bundle functors on manifolds in terms of the Weil bundles 6] by Morimoto 5]. In this note we present the complete description of all product preserving bundle functors on bered manifolds in terms of natural transformations between product preserving bundle functors on manifolds. The category of smooth manifolds and their smooth maps will be denoted by M. The category of smooth bered manifolds and their smooth bered maps will be denoted by FM. The deenition of bundle functors on a category over mani-folds can be found in the fundamental monograph by Koll a r, Michor and Slovv ak 3]. All manifolds are assumed to be nite dimensional, without boundaries and smooth, i.e. of class C 1. Maps between manifolds are assumed to be smooth, i.e. of class C 1. 1. This item consists of some examples concerning (not necessarily product preserving) bundle functors. In these examples we build a "machinary" which we use (in Item 2) to give the above mentioned full description. All constructions presented in the examples are canonical. In the rst example we show how a natural transformation : G ! H between bundle functors on manifolds induces canonically a bundle functor G H on bered manifolds.

Quantum Gauge Transformations and Braided Structure on Quantum Principal Bundles

Abstract: It is shown that every quantum principal bundle is braided, in the sense that there exists an intrinsic braid operator twisting the functions on the bundle A detailed algebraic analysis of this operator is performed In particular, it turns out that the braiding admits a natural extension to the level of arbitrary differential forms on the bundle Applications of the formalism to the study of quantum gauge transformations are presented

The Differential Geometry of Cosserat Media

Abstract: A geometric description of generalized Cosserat continua is developed in terms of non-holonomic frame bundles of second order. A non-holonomic G-structure is constructed by using the smooth uniformity of the material. The theory of linear connections in frame bundles permits to express the inhomogeneity by means of some tensor fields.

On the CR structure of the tangent sphere bundle

Abstract: We adopt the methods of pseudohermitian geometry (cf. [16]) to study the tangent sphere bundle U(M) over a Riemannian manifold M . If M is an elliptic space form of sectional curvature 1 then U(M) is shown to be globally pseudo-Einstein (in the sense of J. M. Lee, [12]).

Skew Symmetric Bundle Maps on Space-time

Abstract: We study the \Lie Algebra" of the group of Gauge Transforma- tions of Space-time. We obtain topological invariants arising from this Lie Algebra. Our methods give us fresh mathematical points of view on Lorentz Transformations, orientation conventions, the Doppler shift, Pauli matrices, Electro-Magnetic Duality Rotation, Poynting vectors, and the Energy Mo- mentum Tensor T. LetM be a space-time andT (M) its tangent bundle. ThusM is a 4-dimensional manifold with a nondegenerate inner producth ;i on T (M) of index + ++. We study the space of bundle maps F : T (M)! T (M) which are skew symmetric with respect to the metric, i.e. hFv;vi =0f or allv2 Tx(M )a nd allx2 M. A skew symmetric F has invariant planes and eigenvector lines in each Tx(M). We give necessary and sucient conditions as to when these plane systems and line systems form subbundles in Theorem 7.3. Also we determine the space of those F which give the same underlying structure. This is done by introducing the bundle map TF = FF 1 (tr F 2 )I : T (M)! T (M). Then the space of skew symmetric F which give rise to the same T is homeomorphic to Map(M;S 1 ), the space of maps of M into the circle S 1 . (See Theorem 7.11.) We also show that the space of skew symmetric F has a natural complexica- tion. (see Propositions 2.2 and 2.3) This leads to an equivalence between the F and vector elds on the complexied tangent bundle T (M)C. The complexied study leads to several beautiful relations which link our subject matter to Cliord Algebras and Quaternions. (See Corollaries 4.6 and 4.7 and Theorem 4.8.) We naturally nd many points of contact with Physics, especially classical electromag- netism. These considerations frequently govern our choice of notation. The physical motivations and remarks will be explored in the Scholia; and the mathematical mo- tivations and links will be found in the Remarks.

Local formulae for Stiefel-Whitney classes

Abstract: We construct an explicit Cech cocycle representing the k-th Stiefel-Whitney class of a vector bundle. This construction involves only the transition functions of the bundle. We also give local formulae for the secondary Stiefel-Whitney classes. These may be useful in determining whether the Stiefel-Whitney numbers of a flat bundle are zero.

Torsion-free connections on higher order frame bundles

Abstract: We deduce that torsion-free connections on the r-th order frame bundle P r M of a manifold M can be identified with certain reductions of P r+1 M. They are also interpreted as splittings of T*M into the bundle of all (1,r+l)-covelocities on M. Finally we determine all natural operators transforming torsion-free connections on P 1 M into torsion-free connections on P 2 M.

The Equations of Structure of an N-Linear Connection in the Bundle of Accelerations

Abstract: The study of higher order Lagrange spaces founded on the notion of bundle of velocities of order k has been recently given by Radu Miron and author in [2]-[5]. The bundle of acceleration correspond in this study to k = 2. In this paper we shall give the equations of structure of an N-linear connection in the bundle of accelerations.

On Parallel Transport in Quantum Bundles over Robertson-Walker Spacetimes

Abstract: A recently-developed theory of quantum general relativity provides a propagator for free-falling particles in curved spacetimes. These propagators are constructed by parallel-transporting quantum states within a quantum bundle associated to the Poincare frame bundle. We consider such parallel transport in the case that the spacetime is a classical Robertson-Walker universe. An explicit integral formula is developed which expresses the propagators for parallel transport between any two points of such a spacetime. The integrals in this formula are evaluated in closed form for a particular spatially-flat model.

On the spannedness of adjunction of vector bundles

Abstract: Let X be a smooth projective variety of dimension n defined over the field of complex numbers and E be an ample vector bundle on X. The linear system Kx + c](E) is called the adjoint bundle for vector bundle E. The nefness and the ampleness of the adjoint bundle Kx + cl(E) have been investigated by several authors ([ABW], [F], [P1], [YZ]). In this paper, we shall investigate the base-point freeness (spannedness) of the adjoint bundle Kx + c](E), i.e., we ask the following: