Showing papers on "Frame bundle published in 2002"

On the volume of a line bundle

Abstract: Using the Calabi–Yau technique to solve Monge-Ampere equations, we translate a result of T. Fujita on approximate Zariski decompositions into an analytic setting and combine this to the holomorphic Morse inequalities in order to express the volume of a line bundle as the maximum of the mean curvatures of all the singular Hermitian metrics on it, with a way to pick an element at which the maximum is reached and satisfying a singular Monge–Ampere equation. This enables us to introduce the volume of any (1,1)-class on a compact Kahler manifold, and Fujita's theorem is then extended to this context.

A reduction map for nef line bundles

Abstract: In [Ts00], H. Tsuji stated several very interesting assertions on the structure of pseudo-effective line bundles L on a projective manifold X. In particular he postulated the existence of a meromorphic “reduction map”, which essentially says that through the general point of X there is a maximal irreducible L-flat subvariety. Moreover the reduction map should be almost holomorphic, i.e. has compact fibers which do not meet the indeterminacy locus of the reduction map. The proofs of [Ts00], however, are extremely difficult to follow.

Vector Bundle Moduli Superpotentials in Heterotic Superstrings and M-Theory

Abstract: The non-perturbative superpotential generated by a heterotic superstring wrapped once around a genus-zero holomorphic curve is proportional to the Pfaffian involving the determinant of a Dirac operator on this curve. We show that the space of zero modes of this Dirac operator is the kernel of a linear mapping that is dependent on the associated vector bundle moduli. By explicitly computing the determinant of this map, one can deduce whether or not the dimension of the space of zero modes vanishes. It is shown that this information is sufficient to completely determine the Pfaffian and, hence, the non-perturbative superpotential as explicit holomorphic functions of the vector bundle moduli. This method is illustrated by a number of non-trivial examples.

On the Frobenius Integrability of Certain Holomorphic p-Forms

Abstract: The goal of this note is to exhibit the integrability properties (in the sense of the Frobenius theorem) of holomorphic p-forms with values in certain line bundles with semi-negative curvature on a compact Kahler manifold There are in fact very strong restrictions, both on the holomorphic form and on the curvature of the semi-negative line bundle In particular, these observations provide interesting information on the structure of projective manifolds which admit a contact structure: either they are Fano manifolds or, thanks to results of Kebekus-Peternell-Sommese-Wisniewski, they are biholomorphic to the projectivization of the cotangent bundle of another suitable projective manifold

Asymptotic behaviour of tame nilpotent harmonic bundles with trivial parabolic structure

Abstract: Let E be a holomorphic vector bundle. Let θ be a Higgs field, that is a holomorphic section of End (E) ⊗ Ω1,0X satisfying θ2 = 0. Let h be a pluriharmonic metric of the Higgs bundle (E, θ). The tuple (E, θ, h) is called a harmonic bundle. Let X be a complex manifold, and D be a normal crossing divisor of X. In this paper, we study the harmonic bundle (E, θ, h) over X − D. We regard D as the singularity of (E, θ, h), and we are particularly interested in the asymptotic behaviour of the harmonic bundle around D. We will see that it is similar to the asymptotic behaviour of complex variation of polarized Hodge structures, when the harmonic bundle is tame and nilpotent with the trivial parabolic structure. For example, we prove constantness of general monodromy weight filtrations, compatibility of the filtrations, norm estimates, and the purity theorem. For that purpose, we will obtain a limiting mixed twistor structure from a tame nilpotent harmonic bundle with trivial parabolic structure, on a punctured disc. It is a solution of a conjecture of Simpson.

Chern character in twisted K-theory: equivariant and holomorphic cases

Abstract: It has been argued by Witten and others that in the presence of a nontrivial B-field, D-brane charges in type IIB string theories are measured by twisted K-theory. In joint work with Bouwknegt, Carey and Murray it was proved that twisted K-theory is canonically isomorphic to bundle gerbe K-theory, whose elements are ordinary vector bundles on a principal projective unitary bundle, with an action of the bundle gerbe determined by the principal projective unitary bundle. The principal projective unitary bundle is in turn determined by the twist. In this paper, we study in more detail the Chern-Weil representative of the Chern character of bundle gerbe K-theory that was introduced previously, and we also extend it to the equivariant and holomorphic cases. Included is a discussion of interesting examples.

Cohomology and Obstructions II: Curves on K-trivial threefolds

Abstract: On a threefold with trivial canonical bundle, Kuranishi theory gives an algebro-geometry construction of the (local analytic) Hilbert scheme of curves at a smooth holomorphic curve as a gradient scheme, that is, the zero-scheme of the exterior derivative of a holomorphic function on a (finite-dimensional) polydisk. (The corresponding fact in an infinite dimensional setting was long ago discovered by physicists.) An analogous algebro-geometric construction for the holomorphic Chern-Simons functional is presented giving the local analytic moduli scheme of a vector bundle. An analogous gradient scheme construction for Brill-Noether loci on ample divisors is also given. Finally, using a structure theorem of Donagi-Markman, we present a new formulation of the Abel-Jacobi mapping into the intermediate Jacobian of a threefold with trivial canonical bundle.

Self-duality of Coble's quartic hypersurface and applications

Abstract: The moduli space M0 of semi-stable rank 2 vector bundles with fixed trivial determinant over a non-hyperelliptic curve C of genus 3 is isomorphic to a quartic hypersurface in P 7 (Coble’s quartic). We show that M0 is self-dual and that its polar map associates to a stable bundle E ∈ M0 a bundle F which is characterized by dim H 0 (C, E ⊗ F) = 4. The projective

Classical gauge theory of gravity

Abstract: The classical theory of gravity is formulated as a gauge theory on a frame bundle with spontaneous symmetry breaking caused by the existence of Dirac fermionic fields. The pseudo-Riemannian metric (tetrad field) is the corresponding Higgs field. We consider two variants of this theory. In the first variant, gravity is represented by the pseudo-Riemannian metric as in general relativity theory; in the second variant, it is represented by the effective metric as in the Logunov relativistic theory of gravity. The configuration space, Dirac operator, and Lagrangians are constructed for both variants.

A criterion for the existence of a flat connection on a parabolic vector bundle.

Abstract: We define holomorphic connection on a parabolic vector bundle over a Riemann surface and prove that a parabolic vector bundle admits a holomorphic connection if and only if each direct summand of it is of parabolic degree zero. This is a generalization to the para- bolic context of a well-known result of Weil which says that a holomorphic vector bundle on a Riemann surface admits a holomorphic connection if and only if every direct summand of it is of degree zero. 2000 Mathematics Subject Classification. 14H60, 32L05

Morita Equivalence of Fedosov Star Products and Deformed Hermitian Vector Bundles

Abstract: Based on the usual Fedosov construction of star products for a symplectic manifold M, we give a simple geometric construction of a bimodule deformation for the sections of a vector bundle over M starting with a symplectic connection on M and a connection for E. In the case of a line bundle, this gives a Morita equivalence bimodule, and the relation between the characteristic classes of the Morita equivalent star products can be found very easily within this framework. Moreover, we also discuss the case of a Hermitian vector bundle and give a Fedosov construction of the deformation of the Hermitian fiber metric.

Hypercomplex manifolds and hyperholomorphic bundles

Abstract: Using twistor techniques we shall show that there is a hypercomplex structure in the neighbourhood of the zero section of the tangent bundle TX of any complex manifold X with a real-analytic torsion-free connection compatible with the complex structure whose curvature is of type (1, 1). The zero section is totally geodesic and the Obata connection restricts to the given connection on the zero section.We also prove an analogous result for vector bundles: any vector bundle with real-analytic connection whose curvature is of type (1, 1) over X can be extended to a hyperholomorphic bundle over a neighbourhood of the zero section of TX.

Noncommutative Ricci curvature and Dirac operator on $C_q[SL_2]$ at roots of unity

Abstract: We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group $C_q[SL_2]$, using a recent frame bundle formulation. We find that its covariant Ricci curvature is essentially proportional to the metric (i.e. an Einstein space). We compute the Dirac operator and find for $q$ an odd $r$'th root of unity that its eigenvalues are given by $q$-integers $[m]_q$ for $m=0,1,...,r-1$ offset by the constant background curvature. We fully solve the Dirac equation for $r=3$.

Fourier decompositions of loop bundles

Abstract: In this paper we investigate bundles whose structure group is the loop group LU(n). These bundles are classified by maps to the loop space of the classifying space, LBU(n). Our main result is to give a necessary and sufficient criterion for there to exist a Fourier type decomposition of such a bundle �. This is essentially a decomposition ofasLC, whereis a finite dimensional subbundle ofand LC is the space of functions, C 1 (S 1 , C). The criterion is a reduction of the structure group to the finite rank unitary group U(n) viewed as the subgroup of LU(n) consisting of constant loops. Next we study the case where one starts with an n dimensional bundle � ! M classified by a map f : M ! BU(n) from which one constructs a loop bundle L� ! LM classified by Lf : LM ! LBU(n). The tangent bundle of LM is such a bundle. We then show how to twist such a bundle by elements of the automorphism group of the pull back ofover LM via the map LM ! M that evaluates a loop at a basepoint. Given a connection on �, we view the associated parallel transport operator as an element of this gauge group and show that twisting the loop bundle by such an operator satisfies the criterion and admits a Fourier decomposition.

The determinant bundle on the moduli space of stable triples over a curve

Abstract: We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (E1, E2,ϕ), where E1 and E2 are holomorphic vector bundles over a fixed compact Riemann surfaceX, andϕ: E2 → E1 is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kahler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed C∞ Hermitian vector bundle over a compact Riemann surface.

Bundles with spherical Euler class

Abstract: In this note, we show that the holonomy group of a Riemannian connection on a k-dimensional Euclidean vector bundle is transitive on the unit sphere bundle whenever the Euler class is spherical. We extract several consequences from this, among them that this is always the case as long as does not vanish, and the base of the bundle is simply connected and rationally (k + 1)/2-connected.

Vector Bundle Moduli

Abstract: We give the general presciption for calculating the number of moduli of irreducible, stable U(n) holomorphic vector bundles with positive spectral covers over elliptically fibered Calabi–Yau threefolds. Explicit results are presented for Hirzebruch base surfaces B = F r. Vector bundle moduli appear as gauge singlet scalar fields in the effective low-energy actions of heterotic superstrings and heterotic M-theory.

Riemann-Roch-Grothendieck and torsion for foliations

Abstract: In this article we prove a Riemann-Roch-Grothendieck theorem for the characteristic classes of a flat vector bundle over a foliation whose graph is Hausdorff. We assume that the strong foliation Novikov-Shubin invariants of the flat bundle are greater than three times the codimension of the foliation. Using transgression, we define a torsion form which in the odd acyclic case determines a Haefliger cohomology class which only depends on the foliation and the flat bundle. We construct examples where this torsion class is highly non-trivial.

Note on generalized connections and affine bundles

Abstract: We develop an alternative view on the concept of connections over a vector bundle map, which consists of a horizontal lift procedure to a prolonged bundle. We further focus on prolongations to an affine bundle and introduce the concept of affineness of a generalized connection.

Fibre bundle formulation of relativistic quantum mechanics

Abstract: We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the bundle description the wavefunctions are replaced with (state) sections (covariant approach) or liftings of paths (equivalently: sections along paths) (time-dependent approach) of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation of the state sections/liftings in the (total) bundle space. The equations of these transportations turn to be the bundle versions of the corresponding relativistic wave equations. Connections between the (retarded) Green functions of these equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.

Aharonov-Bohm Effect, Flat Connections, and Green's Theorem

Abstract: The validity of Green's theorem, and hence of Stokes' theorem, when the involved vector field is differentiable but not continuously differentiable, is crucial for a theoretical explanation of the Aharonov–Bohm (A-B) effect; we review this theorem. We describe the principal bundle in which the A-B effect occurs, and give the geometrical description of the relevant connection. We study the set of gauge equivalence classes of flat connections on a product bundle with abelian structural group, and show that this set has a canonical group structure, which is isomorphic to a quotient of cohomology groups. We apply this result to the A-B bundle and calculate the holonomy groups of all flat connections.

Principal bundles on the projective line

Abstract: We classify principalG-bundles on the projective line over an arbitrary fieldk of characteristic ≠ 2 or 3, whereG is a reductive group. If such a bundle is trivial at ak-rational point, then the structure group can be reduced to a maximal torus.

On the geometry of fiber product preserving bundle functors

Abstract: Using the the description of a fiber product preserving bundle functor F in terms of Weil algebras, we deduce several geometric properties of the F-prolongations of principal and associated bundles.

Structure of Malicious Singularities

Abstract: We investigate spacetimes with their singular boundaries as noncommutative spaces. Such a space is defined by a noncommutative algebra on a transformation groupoid $\Gamma = E \times G$, where $E$ is the total space of the frame bundle over spacetime with its singular boundary, and $G$ its structural group. There is a bijective correspondence between unitary representations of the groupoid $\Gamma $ and the systems of imprimitivity of the group $G$. This allows us to apply the Mackey theorem, and deduce from it some information concerning singular fibres of the groupoid. A subgroup $K$ of $G$, from which -- according to the Mackey theorem -- the representation is induced to the whole of $G$, can be regarded as measuring the "richness" of the singularity structure.

The spinor bundle of Riemannian products

Abstract: In this note we compare the spinor bundle of a Riemannian manifold $(M=M_1\times\times M_N,g)$ with the spinor bundles of the Riemannian factors $(M_i,g_i)$ We show, that - without any holonomy conditions - the spinor bundle of $(M,g)$ for a special class of metrics is isomorphic to a bundle obtained by tensoring the spinor bundles of $(M_i,g_i)$ in an appropriate way

The Spectral Density Function for the Laplacian on High Tensor Powers of a Line Bundle

Abstract: For a symplectic manifold with quantizing line bundle, a choice of almost complex structure determines a Laplacian acting on tensor powers of the bundle. For high tensor powers Guillemin–Uribe showed that there is a well-defined cluster of low-lying eigenvalues, whose distribution is described by a spectral density function. We give an explicit computation of the spectral density function, by constructing certain quasimodes on the associated principle bundle.

An Introduction to Cocycle Super-Rigidity

Abstract: The cocycle super-rigidity theorem is a central result in the study of dynamics of semisimple Lie groups and lattices. We give an overview of the main ideas centered on this theorem and some of its most immediate applications. The emphasis will be on the topological and differentiable (as opposed to measurable) aspects of the theory.