Showing papers on "Frame bundle published in 1998"

Hopf algebras, cyclic cohomology and the transverse index theory

Abstract: We present the solution of a longstanding internal problem of noncommutative geometry, namely the computation of the index of a transversally elliptic operator on an arbitrary foliation. The new and crucial ingredient is a certain Hopf algebra associated to the transverse frame bundle. Its cyclic cohomology is defined and shown to be canonically isomorphic to the Gelfand-Fuks cohomology.

Geometric Quantization of Vector Bundles

Abstract: I repeat my definition for quantization of a vector bundle. For the case of Toeplitz and geometric quantization of a compact Kaehler Manifold, I give a construction for quantizing any smooth vector bundle which depends functorially on a choice of connection on the bundle.

On the Riemann-Hilbert Problems

Abstract: We discuss some topological aspects of the Riemann-Hilbert transmission problem and Riemann-Hilbert monodromy problem on Riemann surfaces In particular, we describe the construction of a holomorphic vector bundle starting from the given representation of the fundamental group and investigate the local behaviour of connexions on this bundle We give formulae for the partial indices of the Riemann-Hilbert transmission problem in the three-dimensional case in terms of the correspoding vector bundle on the Riemann sphere

Torsional topological invariants (and their relevance for real life)

Abstract: The existence of topological invariants analogous to Chern/Pontryagin classes for a standard SO(D) or SU(N) connection, but constructed out of the torsion tensor, is discussed. These invariants exhibit many of the features of the Chern/Pontryagin invariants: they can be expressed as integrals over the manifold of local densities and take integer values on compact spaces without boundary; their spectrum is determined by the homotopy groups πD−1(SO(D)) and πD−1(SO(D+1)). These invariants are not solely determined by the connection bundle but depend also on the bundle of local orthonormal frames on the tangent space of the manifold. It is shown that in spacetimes with nonvanishing torsion there can occur topologically stable configurations associated with the frame bundle which are independent of the curvature. Explicit examples of topologically stable configurations carrying nonvanishing instanton number in four and eight dimensions are given, and they can be conjectured to exist in dimension 4k. It is also...

Spectral Covers, Charged Matter and Bundle Cohomology

Abstract: We consider four dimensional heterotic compactifications on smooth elliptic Calabi-Yau threefolds. Using spectral cover techniques, we study bundle cohomology groups corresponding to charged matter multiplets. The analysis shows that in generic situations, the resulting charged matter spectrum is stable under deformations of the vector bundle.

Complex manifolds with split tangent bundle

Abstract: Let X be a compact Kaehler manifold We expect that any direct sum decomposition of the tangent bundle T(X) comes from a splitting of the universal covering space of X as a product of manifolds, in such a way that the given decomposition of T(X) lifts to the canonical decomposition of the tangent bundle of a product We prove this assertion when X is a Kaehler-Einstein manifold or a Kaehler surface Simple examples show that the Kaehler hypothesis is necessary

Singular hermitian metrics on vector bundles

Abstract: We introduce a notion of singular hermitian metrics (s.h.m.) for holomorphic vector bundles and define positivity in view of $L^2$-estimates. Associated with a suitably positive s.h.m. there is a (coherent) sheaf 0-th kernel of a certain $d''$-complex. We prove a vanishing theorem for the cohomology of this sheaf. All this generalizes to the case of higher rank known results of Nadel for the case of line bundles. We introduce a new semi-positivity notion, $t$-nefness, for vector bundles, establish some of its basic properties and prove that on curves it coincides with ordinary nefness. We particularize the results on s.h.m. to the case of vector bundles of the form $E=F \otimes L$, where $F$ is a $t$-nef vector bundle and $L$ is a positive (in the sense of currents) line bundle. As applications we generalize to the higher rank case 1) Kawamata-Viehweg Vanishing Theorem, 2) the effective results concerning the global generation of jets for the adjoint to powers of ample line bundles, and 3) Matsusaka Big Theorem made effective.

Principal Bundles and the Dixmier Douady Class

Abstract: A systematic consideration of the problem of the reduction and “lifting” of the structure group of a principal bundle is made and a variety of techniques in each case are explored and related to one another. We apply these to the study of the Dixmier-Douady class in various contexts including string structures, \(\) bundles and other examples motivated by considerations from quantum field theory.

On the Noncommutative Geometry of the Endomorphism Algebra of a Vector Bundle

Abstract: In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of its complex of noncommutative differential forms.

Semistable bundles on curves and irreducible representations of the fundamental group

Abstract: This note is an attempt to generalize Bolibruch's theorem from the projective line to curves of higher genus. We show that an irreducible representation of the fundamental group of an open in a curve of higher genus has always a representation as a regular system of differential equations on a semistable bundle of degree 0. Vice-versa, we show that given such a bundle and 3 points on the curve, one can construct an irreducible representation of the curve minus the 3 points such that an associated regular system of differential equations lives on this bundle.

Stiefel—Whitney currents

Abstract: A canonically defined mod 2 linear dependency current is associated to each collection v of sections, v1,…,vm, of a real rank n vector bundle. This current is supported on the linear dependency set of v. It is defined whenever the collection v satisfies a weak measure theoretic condition called “atomicity.” Essentially any reasonable collection of sections satisfies this condition, vastly extending the usual general position hypothesis. This current is a mod 2 d-closed locally integrally flat current of degree q = n −m + 1 and hence determines a ℤ2-cohomology class. This class is shown to be well defined independent of the collection of sections. Moreover, it is the qth Stiefel-Whitney class of the vector bundle.

Stochastic Cohomology of the Frame Bundle of the Loop Space

Abstract: We study the differential forms over the frame bundle of the based loop space. They are stochastics in the sense that we put over this frame bundle a probability measure. In order to understand the curvatures phenomena which appear when we look at the Lie bracket of two horizontal vector fields, we impose some regularity assumptions over the kernels of the differential forms. This allows us to define an exterior stochastic differential derivative over these forms.

Fibre bundle formulation of nonrelativistic quantum mechanics. III. Pictures and integrals of motion

Abstract: We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one but it is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it a pure state of some quantum system is described by a state section (along paths) of a (Hilbert) fibre bundle. It's evolution is determined through the bundle (analogue of the) Schrodinger equation. Now the dynamical variables and the density operator are described via bundle morphisms (along paths). The mentioned quantities are connected by a number of relations derived in this work. In this third part of our series we investigate the bundle analogues of the conventional pictures of motion. In particular, there are found the state sections and bundle morphisms corresponding to state vectors and observables respectively. The equations of motion for these quantities are derived too. Using the results obtained, we consider from the bundle view-point problems concerning the integrals of motion. An invariant (bundle) necessary and sufficient conditions for a dynamical variable to be an integral of motion are found.

Vector bundles with holomorphic connection over a projective manifold with tangent bundle of nonnegative degree

Abstract: For a projective manifold whose tangent bundle is of nonnegative degree, a vector bundle on it with a holomorphic connection actually admits a compatible flat holomorphic connection, if the manifold satisfies certain conditions. The conditions in question are on the Harder-Narasimhan filtration of the tangent bundle, and on the Neron-Severi group.

Moduli of vector bundles on curves in positive characteristic

Abstract: Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.

Vanishing theorems for the kernel of a Dirac operator

Abstract: We obtain a vanishing theorem for the kernel of a Dirac operator on a Clifford module twisted by a sufficiently large power of a line bundle, whose curvature is non-degenerate at any point of the base manifold. In particular, if the base manifold is almost complex, we prove a vanishing theorem for the kernel of a $\spin^c$ Dirac operator twisted by a line bundle with curvature of a mixed sign. In this case we also relax the assumption of non-degeneracy of the curvature. These results are generalization of a vanishing theorem of Borthwick and Uribe. As an application we obtain a new proof of the classical Andreotti-Grauert vanishing theorem for the cohomology of a compact complex manifold with values in the sheaf of holomorphic sections of a holomorphic vector bundle, twisted by a large power of a holomorphic line bundle with curvature of a mixed sign. As another application we calculate the sign of the index of a signature operator twisted by a large power of a line bundle.

Einstein-Podolski-Rosen Experiment from Noncommutative Quantum Gravity

Abstract: It is shown that experiments of the Einstein-Podolski-Rosen type are the natural consequence of the groupoid approach to noncommutative unification of general relativity and quantum mechanics The geometry of this model is determined by the noncommutative algebra of complex valued, compactly supported functions (with convolution as multiplication) on the groupoid G = E x D In the model considered in the present paper E is the total space of the frame bundle over space-time, and D is the Lorentz group Correlations of the EPR type should be regarded as remnants of the totally non-local physics below the Planck threshold which is modelled by a noncommutative geometry

Euclidean supergravity in terms of Dirac eigenvalues

Abstract: It has been recently shown that the eigenvalues of the Dirac operator can be considered as dynamical variables of Euclidean gravity. The purpose of this paper is to explore the possiblity that the eigenvalues of the Dirac operator might play the same role in the case of supergravity. It is shown that for this purpose some primary constraints on covariant phase space as well as secondary constraints on the eigenspinors must be imposed. The validity of primary constraints under covariant transport is further analyzed. It is show that in the this case restrictions on the tanget bundle and on the spinor bundle of spacetime arise. The form of these restrictions is determined under some simplifying assumptions. It is also shown that manifolds with flat curvature of tangent bundle and spinor bundle and spinor bundle satisfy these restrictons and thus they support the Dirac eigenvalues as global observables.

Tensor Fields of Type (0, 2) on the Tangent Bundle of a Riemannian Manifold

Abstract: To any (0, 2)-tensor field on the tangent bundle of a Riemannian manifold, we associate a global matrix function. Based on this fact, natural tensor fields are defined and characterized, essentially by means of well-known algebraic results. In the symmetric case, this classification coincides with the one given by Kowalski–Sekizawa; in the skew-symmetric one, it does with that obtained by Janyska.

Another Kaehler structure on the tangent bundle of a space form

Abstract: Introduction It is known (see [2], [7], [17]) that the tangent bundle TM of an ndimensional Riemannian manifold (M,g) can be organized as an almost Kaehlerian manifold by using the Sasaki metric and an almost complex structure defined by the splitting of the tangent bundle to TM into the vertical and horizontal distributions VTM, HTM (the last one being determined by the Levi Civita connection on M) (see also [15], [16]). However, this structure is Kaehler only in the case where the base manifold is locally Euclidean. In [14] V. Oproiu and the present author, inspired by an idea of Calabi (see [1]) to define a hyper-Kaehler structure on the cotangent bundle of a Kaehler manifold of positive constant holomorphic curvature, have considered a Lagrangian on a Riemannian manifold (M, g), defined by a real valued smooth function depending on the energy density only. They have shown that the nonlinear connection defined by the Euler-Lagrange equations associated with the considered Lagrangian does coincide with the nonlinear connection defined by the Levi Civita connection of g and have obtained a Riemannian metric G and an almost complex structure J, defined on TM, such that (TM, G, J) is an almost Kaehler manifold (like as in the case of the Sasaki metric). Further, if (M,g) has positive constant curvature, then 1991 Mathematics Subject Classification: 53C15, 53C07, 53C55.

On complex structures in 8-dimensional vector bundles

Abstract: Necessary and sufficient conditions for the existence of a complex structure in an 8-dimensional spin vector bundle over a closed connected spin manifold of dimension 8 are given in terms of characteristic classes. The result completes the papers by Heaps [H] and Thomas [T] on the same topic.

Diffusion on the Total Space of a Vector Bundle

Abstract: In recent years, the study of the differential geometry of the total space E, of a vector bundle π : E → M, initiated by R. Miron [11], [12] has been developed by many people (see [13] and the references therein). If we take a horizontal complement of the vertical subbundle V E, we can express the geometrical objects defined on E in a more simplified form and new geometric objects can be obtained.

Elliptic surfaces with a nef line bundle of genus two

Abstract: Complex projective elliptic surfaces endowed with a numerically effective line bundle of arithmetic genus two are studied and partially classified. A key role is played by elliptic quasi-bundles, where some ideas developed by Serrano in order to study ample line bundles apply to this more general situation.