Showing papers on "Frame bundle published in 2005"

T-Duality for Torus Bundles with H-Fluxes via Noncommutative Topology

Abstract: It is known that the T-dual of a circle bundle with H-flux (given by a Neveu-Schwarz 3-form) is the T-dual circle bundle with dual H-flux. However, it is also known that torus bundles with H-flux do not necessarily have a T-dual which is a torus bundle. A big puzzle has been to explain these mysterious “missing T-duals.” Here we show that this problem is resolved using noncommutative topology. It turns out that every principal T2-bundle with H-flux does indeed have a T-dual, but in the missing cases (which we characterize), the T-dual is non-classical and is a bundle of noncommutative tori. The duality comes with an isomorphism of twisted K-theories, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced by an isomorphism of twisted cyclic homology.

Nonabelian Bundle Gerbes, Their Differential Geometry and Gauge Theory

Abstract: 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.

Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories

Abstract: We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in H4(BG, ℤ) for a compact semi-simple Lie group G. The Chern-Simons bundle 2-gerbe realises differential geometrically the Cheeger-Simons invariant. We apply these notions to refine the Dijkgraaf-Witten correspondence between three dimensional Chern-Simons functionals and Wess-Zumino-Witten models associated to the group G. We do this by introducing a lifting to the level of bundle gerbes of the natural map from H4(BG, ℤ) to H3(G, ℤ). The notion of a multiplicative bundle gerbe accounts geometrically for the subtleties in this correspondence for non-simply connected Lie groups. The implications for Wess-Zumino-Witten models are also discussed.

Noncommutative Riemannian and Spin Geometry of the Standard q-Sphere

Abstract: We study the quantum sphere Open image in new window as a quantum Riemannian manifold in the quantum frame bundle approach We exhibit its 2-dimensional cotangent bundle as a direct sum Ω0,1⊕Ω1,0 in a double complex We find the natural metric, volume form, Hodge * operator, Laplace and Maxwell operators and projective module structure We show that the q-monopole as spin connection induces a natural Levi-Civita type connection and find its Ricci curvature and q-Dirac operator Open image in new window We find the possibility of an antisymmetric volume form quantum correction to the Ricci curvature and Lichnerowicz-type formulae for Open image in new window We also remark on the geometric q-Borel-Weil-Bott construction

On null Lagrangians

Abstract: We consider multiple-integral variational problems where the Lagrangian function, defined on a frame bundle, is homogeneous. We construct, on the corresponding sphere bundle, a canonical Lagrangian form with the property that it is closed exactly when the Lagrangian is null. We also provide a straightforward characterization of null Lagrangians as sums of determinants of total derivatives. We describe the correspondence between Lagrangians on frame bundles and those on jet bundles: under this correspondence, the canonical Lagrangian form becomes the fundamental Lepage equivalent. We also use this correspondence to show that, for a single-determinant null Lagrangian, the fundamental Lepage equivalent and the Caratheodory form are identical.

Moduli for decorated tuples of sheaves and representation spaces for quivers

Abstract: We extend the scope of a former paper to vector bundle problems involving more than one vector bundle. As the main application, we obtain the solution of the well-known moduli problems of vector bundles associated with general quivers.

A note on Einstein–Sasaki metrics in D ⩾ 7

Abstract: In this paper, we obtain new non-singular Einstein–Sasaki spaces in dimensions D ≥ 7. The local construction involves taking a circle bundle over a (D − 1)-dimensional Einstein–Kahler metric that is itself constructed as a complex line bundle over a product of Einstein–Kahler spaces. In general, the resulting Einstein–Sasaki spaces are singular, but if parameters in the local solutions satisfy appropriate rationality conditions, the metrics extend smoothly onto complete and non-singular compact manifolds. The seven-dimensional space, whose base is a complex line bundle over S2 × S2, is discussed in detail since it has relevance in terms of the AdS/CFT correspondence.

Duality and triple structures

Abstract: We recall the basic theory of double vector bundles and the canonical pairing of their duals, introduced by the author and by Konieczna and Urbanski. We then show that the relationship between a double vector bundle and its two duals can be understood simply in terms of an associated cotangent triple vector bundle structure. In particular, we show that the dihedral group of the triangle acts on this triple via forms of the isomorphisms R, introduced by the author and Ping Xu. We then consider the three duals of a general triple vector bundle and show that the corresponding group is neither the dihedral group of the square nor the symmetry group on four symbols.

Loop groups, Kaluza-Klein reduction and M-theory

Abstract: We show that the data of a principal G-bundle over a principal circle bundle is equivalent to that of a U(1) LG-bundle over the base of the circle bundle. We apply this to the Kaluza-Klein reduction of M-theory to IIA and show that certain generalized characteristic classes of the loop group bundle encode the Bianchi identities of the antisymmetric tensor fields of IIA supergravity. We further show that the low dimensional characteristic classes of the central extension of the loop group encode the Bianchi identities of massive IIA, thereby adding support to the conjectures of [1].

The lie derivative of spinor fields: theory and applications

Abstract: Starting from the general concept of a Lie derivative of an arbitrary differentiable map, we develop a systematic theory of Lie differentiation in the framework of reductive G-structures P on a principal bundle Q. It is shown that these structures admit a canonical decomposition of the pull-back vector bundle over P. For classical G-structures, i.e. reductive G-subbundles of the linear frame bundle, such a decomposition defines an infinitesimal canonical lift. This lift extends to a prolongation Γ-structure on P. In this general geometric framework the concept of a Lie derivative of spinor fields is reviewed. On specializing to the case of the Kosmann lift, we recover Kosmann's original definition. We also show that in the case of a reductive G-structure one can introduce a "reductive Lie derivative" with respect to a certain class of generalized infinitesimal automorphisms, and, as an interesting by-product, prove a result due to Bourguignon and Gauduchon in a more general manner. Next, we give a new characterization as well as a generalization of the Killing equation, and propose a geometric reinterpretation of Penrose's Lie derivative of "spinor fields". Finally, we present an important application of the theory of the Lie derivative of spinor fields to the calculus of variations.

Krull-Schmidt reduction for principal bundles

Abstract: We prove a principal bundle analog of a theorem on vector bundles that says that a vector bundle on a projective variety is isomorphic to a unique direct sum of indecomposable vector bundles (unique up to a permutation of the direct summands).

$L^{2}$ extension for jets of holomorphic sections of a hermitian line bundle

Abstract: Let $(X, \omega)$ be a weakly pseudoconvex Kahler manifold, $Y \subset X$ a closed submanifold defined by some holomorphic section of a vector bundle over $X$, and $L$ a Hermitian line bundle satisfying certain positivity conditions. We prove that for any integer $k \geq 0$, any section of the jet sheaf $L \otimes {\cal O}_{X}/{\cal I}_{Y}^{k+1}$, which satisfies a certain $L^{2}$ condition, can be extended into a global holomorphic section of $L$ over $X$ whose $L^{2}$ growth on an arbitrary compact subset of $X$ is under control. In particular, if $Y$ is merely a point, this gives the existence of a global holomorphic function with an $L^{2}$ norm under control and with prescribed values for all its derivatives up to order $k$ at that point. This result generalizes the $L^{2}$ extension theorems of Ohsawa-Takegoshi and of Manivel to the case of jets of sections of a line bundle. A technical difficulty is to achieve uniformity in the constant appearing in the final estimate. To this end, we make use of the exponential map and of a Rauch-type comparison theorem for complete Riemannian manifolds.

Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry

Abstract: We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module there is a simplicial group NC, the nerve of the 1-category defined by the crossed module and its geometric realization |NC|. Equivalence classes of principal bundles with structure group |NC| are shown to be one-to-one with stable equivalence classes of what we call crossed module gerbes bundle gerbes. We can also associate to a crossed module a 2-category C'. Then there are two equivalent ways how to view classifying spaces of NC-bundles and hence of |NC|-bundles and crossed module bundle gerbes. We can either apply the W-construction to NC or take the nerve of the 2-category C'. We discuss the string group and string structures from this point of view. Also a simplicial principal bundle can be equipped with a simplicial connection and a B-field. It is shown how in the case of a simplicial principal NC-bundle these simplicial objects give the bundle gerbe connection and the bundle gerbe B-field.

Complex Hyperbolic Structures on Disc Bundles over Surfaces. II. Example of a Trivial Bundle

Abstract: This article is based on the methods developed in [AGG]. We construct a complex hyperbolic structure on a trivial disc bundle over a closed orientable surface $\Sigma$ (of genus 2) thus solving a long standing problem in complex hyperbolic geometry (see [Gol1, p. 583] and [Sch, p. 14]). This example answers also [Eli, Open Question 8.1] if a trivial circle bundle over a closed surface of genus >1 admits a holomorphically fillable contact structure. The constructed example M satisfies the relation $2(\chi+e)=3\tau$ which is necessary for the existence of a holomorphic section of the bundle, where $\chi=\chi\Sigma$ stands for the Euler characteristic of $\Sigma$, e=eM, for the Euler number of the bundle, and $\tau$, for the Toledo invariant. (The relation is also valid for the series of examples constructed in [AGG].) Open question: Does there exist a holomorphic section of the bundle M?

Contact Metric Geometry of the Unit Tangent Sphere Bundle

Abstract: We review some of the most interesting results concerning the interactions between the geometry of a Riemannian manifold and the one of its unit tangent sphere bundle, equipped with its natural contact metric structure.

On Poincaré bundles of vector bundles on curves

Abstract: Let M denote the moduli space of stable vector bundles of rank n and fixed determinant of degree coprime to n on a non-singular projective curve X of genus g≥2. Denote by a universal bundle on X×M. We show that, for x,y ∈ X, x≠y, the restrictions |{x}×M and |{y}×M are stable and non-isomorphic when considered as bundles on M. Vector bundle, Poincare bundle, moduli space

Transverse totally geodesic submanifolds of the tangent bundle

Abstract: It is well-known that ifis a smooth vector field on a given Rie- mannian manifold M n thennaturally defines a submanifold �(M n ) transverse to the fibers of the tangent bundle TM n with Sasaki metric. In this paper, we are interested in transverse totally geodesic subman- ifolds of the tangent bundle. We show that a transverse submanifold N l of TM n (1 ≤ l ≤ n) can be realized locally as the image of a sub- manifold F l of M n under some vector fielddefined along F l . For such images �(F l ), the conditions to be totally geodesic are presented. We show that these conditions are not so rigid as in the case of l = n, and we treat several special cases (� of constant length, � normal to F l , M n of constant curvature, M n a Lie group anda left invariant vector field).

On the torsion-free connections on higher order frame bundles

Abstract: Using the r-jets of flows of vector fields, we show that every torsion-free linear r-th order connection on the tangent bundle of a manifold M determines a reduction of the (r+1)-st order frame bundle of M to the general linear group We deduce that this reduction coincides with another reduction constructed earlier

Manifolds with an $SU(2)$-action on the tangent bundle

Abstract: We study manifolds arising as spaces of sections of complex manifolds fibering over CP 1 with the normal bundle of each section isomorphic to O(k)⊗C n

Bundles of C*-categories and duality

Abstract: We introduce the notions of multiplier C*-category and continuous bundle of C*-categories, as the categorical analogues of the corresponding C*-algebraic notions. Every symmetric tensor C*-category with conjugates is a continuous bundle of C*-categories, with base space the spectrum of the C*-algebra associated with the identity object. We classify tensor C*-categories with fibre the dual of a compact Lie group in terms of suitable principal bundles. This also provides a classification for certain C*-algebra bundles, with fibres fixed-point algebras of O_d.

The Brill-Noether Curve of a Stable Vector Bundle on a Genus Two Curve

Abstract: Let U(r) be the moduli space of rank r vector bundles with trivial determinant on a smooth curve of genus 2 The map theta_r: U(r) -> |r Theta|, which associates to a general bundle its theta divisor, is generically finite In this paper we give a geometric interpretation of the generic fibre of theta_r

The single-leaf Frobenius Theorem with Applications

Abstract: Using the notion of Levi form of a smooth distribution, we discuss the local and the global problem of existence of one horizontal section of a smooth vector bundle endowed with a horizontal distribution.

Bundles of acceleration on Banach manifolds

Abstract: We consider an infinite-dimensional manifold M modelled on a Banach space E and we construct smooth fiber bundle structures on the tangent bundle of order two T 2 M , which consists of all smooth curves of M that agree up to their acceleration, as well as on the corresponding second-order frame bundle L 2 M . These bundles prove to be associated with respect to the identity representation of the general linear group GL ( E ) that serves as the structure group of both of them. Moreover, a bijective correspondence between linear connections on T 2 M and connection forms of L 2 M is revealed.

On the stability of the tangent bundle of a hypersurface in a Fano variety

Abstract: Let M be a complex projective Fano manifold whose Picard group is isomorphic to Z and the tangent bundle TM is semistable. Let Z ⊂ M be a smooth hypersurface of degree strictly greater than degree(TM)(dimC Z−1)/(2 dimC Z−1) and satisfying the condition that the inclusion of Z in M gives an isomorphism of Picard groups. We prove that the tangent bundle of Z is stable. A similar result is proved also for smooth complete intersections in M . The main ingredient in the proof is a vanishing result for the top cohomology of the twisted holomorphic differential forms on Z.

Numerically flat principal bundles

Abstract: Generalizing the notion of a numerically flat vector bundle over a Kahler manifold $M$, we define a numerically flat principal $G$-bundle over $M$, where $G$ is a semisimple complex algebraic group. It is proved that a principal $G$-bundle $E_G$ is numerically flat if and only if $\text{ad}(E_G)$ is numerically flat. Numerically flat bundles are also characterized using the notion of semistability.

On representations attached to semistable vector bundles on Mumford curves

Abstract: We compare two constructions that associate to a semistable vector bundle on a Mumford curve a representation of the Schottky group and the algebraic fundamental group respectively.