Showing papers on "Frame bundle published in 2000"
TL;DR: The notion of stable isomorphism of bundle gerbes is considered in this article, which has the consequence that the stable isomorphic classes of the bundle gerbe over a manifold M are in bijective correspondence with H3(M, ℤ).
Abstract: The notion of stable isomorphism of bundle gerbes is considered. It has the consequence that the stable isomorphism classes of bundle gerbes over a manifold M are in bijective correspondence with H3(M, ℤ). Stable isomorphism sheds light on the local theory of bundle gerbes and enables a classifying theory for bundle gerbes to be developed using results of Gajer on B[Copf ]× bundles.
164 citations
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TL;DR: In this article, the notion of a bundle 2-gerbe connection and 2-curving are introduced and it is shown that there is a class in $H^{4}(M;\Z)$ associated to any bundle 2 -gerbe.
Abstract: This thesis reviews the theory of bundle gerbes and then examines the higher dimensional notion of a bundle 2-gerbe. The notion of a bundle 2-gerbe connection and 2-curving are introduced and it is shown that there is a class in $H^{4}(M;\Z)$ associated to any bundle 2-gerbe.
101 citations
TL;DR: In this paper, the Atiyah-Patodi-Singer index theory construction of the bundle of fermionic Fock spaces parameterized by vector potentials in odd space dimensions is presented.
Abstract: This paper reviews recent work on a new geometric object called a bundle gerbe and discusses some new examples arising in quantum field theory. One application is to an Atiyah–Patodi–Singer index theory construction of the bundle of fermionic Fock spaces parameterized by vector potentials in odd space dimensions and a proof that this leads in a simple manner to the known Schwinger terms (Mickelsson–Faddeev cocycle) for the gauge group action. This gives an explicit computation of the Dixmier–Douady class of the associated bundle gerbe. The method also works in other cases of fermions in external fields (external gravitational field, for example) provided that the APS theorem can be applied; however, we have worked out the details only in the case of vector potentials. Another example, in which the bundle gerbe curvature plays a role, arises from the WZW model on Riemann surfaces. A further example is the "existence of string structures" question. We conclude by showing how global Hamiltonian anomalies fit within this framework.
78 citations
TL;DR: In this article, a quantization of the supercommutative algebra of all smooth sections of the dual Grassmann algebra bundle of an arbitrarily given vector bundle E (equipped with a fiber metric) over a symplectic manifold M will be deformed by a series of bidifferential operators having first order super-commutator proportional to the Rothstein superbracket.
Abstract: On every split supermanifold equipped with the Rothstein super-Poisson bracket we construct a deformation quantization by means of a Fedosov-type procedure. In other words, the supercommutative algebra of all smooth sections of the dual Grassmann algebra bundle of an arbitrarily given vector bundle E (equipped with a fiber metric) over a symplectic manifold M will be deformed by a series of bidifferential operators having first order supercommutator proportional to the Rothstein superbracket. Moreover, we discuss two constructions related to the above result, namely the quantized BRST-cohomology for a locally free Hamiltonian Lie group action and the classical BRST cohomology in the general coisotropic (or reducible) case without using a ‘ghosts of ghosts’ scheme.
62 citations
TL;DR: In this article, it was shown that covariant field theory for sections of π: E→M lifts in a natural way to the bundle of vertically adapted linear frames LπE.
Abstract: We show that covariant field theory for sections of π : E→M lifts in a natural way to the bundle of vertically adapted linear frames LπE. Our analysis is based on the fact that LπE is a principal fiber bundle over the bundle of 1-jets J1π. On LπE the canonical soldering 1-forms play the role of the contact structure of J1π. A lifted Lagrangian L: LπE→R is used to construct modified soldering 1-forms, which we refer to as the Cartan–Hamilton–Poincare 1-forms. These 1-forms on LπE pass to the quotient to define the standard Cartan–Hamilton–Poincare m-form on J1π. We derive generalized Hamilton–Jacobi and Hamilton equations on LπE, and show that the Hamilton–Jacobi and canonical equations of Caratheodory–Rund and de Donder–Weyl are obtained as special cases.
45 citations
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TL;DR: In this article, a holomorphic G-bundle over a del Pezzo surface of degree d between 1 and 6 with rational double points was constructed, where G is a reductive group which is an appropriate conformal form of the simply connected complex linear group whose coroot lattice is isomorphic to the primitive cohomology of the minimal resolution of X.
Abstract: Given a del Pezzo surface of degree d between 1 and 6, possibly with rational double points, we construct a "tautological" holomorphic G-bundle over X, where G is a reductive group which is an appropriate conformal form of the simply connected complex linear group whose coroot lattice is isomorphic to the primitive cohomology of the minimal resolution of X For example, in case d=3 and X is a smooth cubic surface, the rank 27 vector bundle over X associated to the G-bundle constructed above and the standard 27-dimensional representation of E_6 is a direct sum of the line bundles associated to the 27 lines on X We also discuss the restriction of the G-bundle to smooth hyperplane sections
36 citations
TL;DR: In this paper, a general method for the reduction of quantum systems with symmetry was proposed, where reduced quantum systems are set up as quantum systems on the associated vector bundles over a Riemannian manifold M admitting a compact Lie group G as an isometry group.
Abstract: This paper deals with a general method for the reduction of quantum systems with symmetry. For a Riemannian manifold M admitting a compact Lie group G as an isometry group, the quotient space Q=M/G is not a smooth manifold in general but stratified into a collection of smooth manifolds of various dimensions. If the action of the compact group G is free, M is made into a principal fiber bundle with structure group G. In this case, reduced quantum systems are set up as quantum systems on the associated vector bundles over Q=M/G. This idea of reduction fails, if the action of G on M is not free. However, the Peter–Weyl theorem works well for reducing quantum systems on M. When applied to the space of wave functions on M, the Peter–Weyl theorem provides the decomposition of the space of wave functions into spaces of equivariant functions on M, which are interpreted as Hilbert spaces for reduced quantum systems on Q. The concept of connection on a principal fiber bundle is generalized to be defined well on the...
31 citations
31 citations
TL;DR: In this paper, the authors consider the problem of determining a connection on a vector bundle over a compact Riemannian manifold with boundary from the known parallel transport between boundary points along geodesics.
Abstract: We consider the problem of determining a connection on a vector bundle over a compact Riemannian manifold with boundary from the known parallel transport between boundary points along geodesics. The main result is the local uniqueness theorem: if two connections r 0 and r 00 are C-close to a given connection r whose curvature tensor is su‐ciently small, then coincidence of parallel transports with respect to r 0 and r 00 implies existence of an automorphism of the bundle which is identical on the boundary and transforms r 0 to r 00 . A linearized version of the problem is also considered.
27 citations
13 Dec 2000
TL;DR: In this article, the authors characterized the triples (X, L, H) consisting of line bundles L and H on a complex projective manifold X, such that for some positive integer k, the k-th holomorphic jet bundle of L, J(k) (x, L), is isomorphic to a direct sum H +... + H.
Abstract: We characterize the triples (X, L, H), consisting of line bundles L and H on a complex projective manifold X, such that for some positive integer k, the k-th holomorphic jet bundle of L, J(k) (X, L), is isomorphic to a direct sum H + . . . + H.
27 citations
TL;DR: In this paper, the concept of affine bundle of contact elements of type A on M was introduced and some affine properties of this bundle were described. But these properties were restricted to the case of Weil functors.
Abstract: . For every r-th order Weil functor T(A), we introduce
the underliyng k-th order Weil functors T(Ak), k=1,...,r-1. We
deduce that T(A)M -> T(Ar-1)M is an affine bundle for every
manifold M. Generalizing the classical concept of contakt
element by C. Ehresmann, we define the bundle of contact
elements of type A on M and we describe some affine properties
of this bundle.
TL;DR: In this article, the fibre derivative of a bundle map is studied in detail, and a geometric construction useful to solve the Euler-Lagrange equation for a singular lagrangian is presented.
TL;DR: In this article, the generalized Hitchin systems studied by Bottacin and Markman are defined on spaces of stable pairs consisting of a vector bundle and a form-valued meromorphic endomorphism of the bundle.
Abstract: We exhibit natural Darboux coordinates for the generalized Hitchin systems studied by Bottacin and Markman. These systems are defined on spaces of stable pairs consisting of a vector bundle and a form-valued meromorphic endomorphism of the bundle. In special cases (genus zero, genus one), the bundles are rigid and one has the rational, trigonometric and elliptic Gaudin systems. Explicit formulae are given in these cases.
TL;DR: In this paper, a self consistent gauge theory of classical Lagrangian mechanics based on the bundle of affine scalars over the configuration manifold is proposed, and the set-up of the manifold is analyzed.
Abstract: A self consistent gauge theory of Classical Lagrangian Mechanics, based on the introduction of the bundle of affine scalars over the configuration manifold is proposed. In the resulting set-up, the...
TL;DR: In this paper, the Laplace-Beltrami operator of the tangent bundle of spacetime is constructed on the manifold of the manifold, and possible particle spectra are represented by quantum fields that have a null eigenvalue when acted upon by the LBP operator.
TL;DR: In this article, a generalized symplectic geometry on a principal bundle over the configuration space of a classical field is presented, which is obtained by breaking the symmetry of the full linear frame bundle of the field configuration space.
TL;DR: In this paper, the authors studied polystable Higgs bundles twisted by a line bundle over a compact Kahler manifold and identified the corresponding Tannaka group in the case in which the line bundle is of finite order.
Abstract: We study polystable Higgs bundles twisted by a line bundle over a compact Kahler manifold These form a Tannakian category when the first and second Chern classes of the bundle are zero In this paper we identify the corresponding Tannaka group in the case in which the line bundle is of finite order This group is described in terms of the pro-reductive completion of the fundamental group of the manifold, and the character associated to the line bundle
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TL;DR: In this paper, an intersection theory for a singular hemitian line bundle with positive curvature current on a smooth projective variety and irreducible curves on the variety was developed.
Abstract: We develop an intersection theory for a singular hemitian line bundle with positive curvature current on a smooth projective variety and irreducible curves on the variety. And we prove the existence of a natural rational fibration structure associated with the singular hermitian line bundle. Also for any pseudoeffective line bundle on a smooth projective variety, we prove the existence of a rational natural fibration structure associated with the line bundle.
We also characterize a numerically trivial singular hermitain line bundle on a smooth projective variety.
TL;DR: In this paper, the fundamental classes of degeneracy loci associated with vector bundle maps given locally by (not necessary square) matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal are described.
Abstract: We give closed-form formulas for the fundamental classes of degeneracy loci associated with vector bundle maps given locally by (not necessary square) matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal. Our description uses essentially Schur Q-polynomials of a bundle and is based on a push-forward formula for these polynomials in a Grassmann bundle, established in [P4].
TL;DR: In this paper, a notion of geometrically atomic manifold maps is introduced, and a canonical cohomology is established for any such map, where each term in the sum in equation (*) is a d-closed current.
Abstract: Let : E ! F be a smooth bundle map between vector bundles with connection on a manifold X , and let ( ) be a Chern-Weil characteristic form of either E or F . A notion of \\geometric atomicity\" for is introduced. For any such map we establish a canonical cohomology ( ) ( ) X k 0Res ;k [ k( )] = dT where k( ) = fx 2 X : dim ker( ) = kg, Res ;k is a smooth residue form along k( ), and T is a canonical L1loc-form onX . When rank E = rank F , (*) can be written ( F ) ( E) = X k>0Res ;k [ k( )] + dT: Normal sections ofHom(E;F ) (those by de nition which are transversal to the universal singularity sets k) are always geometrically atomic, and for such maps equation (*) expresses a classical formula of R. MacPherson at the level of forms and currents. Every real analytic map is geometrically atomic, no matter how misbehaved its singularities. For those where each k( ) has the expected dimension, analogous formulas are established. In all cases, each term in the sum in equation (*) is a d-closed current. Proofs entail a direct application of the methods of singular connections and of nite volume ows developed by the authors. Geometrically atomic maps prove to be generic or \\typical\" in all structured situations such as: direct sum mappings, tensor product mappings, mappings given by Cli ord multiplication, etc. In each case the methods yield new formulas. This will be done in Part II. *Research of both authors partially supported by the NSF.
Journal Article•
TL;DR: In this article, complete lifts of derivations for semitangent bundles are investigated and relations between these and lifts already known are discussed. But the main purpose of this paper is to investigate the complete lifts.
Abstract: The main purpose of this paper is to investigate the complete lifts of derivations for semitangent bundle and to discuss relations between these and lifts already known.
TL;DR: In this paper, a proof of Petri's general conjecture on the unobstructedness of linear systems on a general curve is proposed, using only the local properties of the deformation space of the pair (curve, line bundle).
Abstract: A proof of Petri's general conjecture on the unobstructedness of linear systems on a general curve is proposed, using only the local properties of the deformation space of the pair (curve, line bundle).
TL;DR: In this article, a singularity free vector field X defined on an open set in a three-dimensional Euclidean space with curl X = 0 admits a complex line bundle Fa with fiber-wise defined symplectic structure, a principal bundle % MathType!MTEF!MteF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXan
Abstract: Any singularity free vector field X defined on an open set in a three-dimensional Euclidean space with curl X= 0 admits a complex line bundle Fa with fibre-wise defined symplectic structure, a principal bundle % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!
$ {\cal P}^{a} $
and a Heisenberg group bundle. For X = const, the geometry of % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!
$ {\cal P}^{a} $
defines the Schrodinger representation of any fibre of the Heisenberg group bundle and a quantization procedure for homogeneous quadratic polynomials on the real line visualised as a transport along field lines of internal degrees of freedom in Fa which is related to signal transmission.
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TL;DR: In this article, the notion of \Gamma-linear connection
abla on the 1-jet fiber bundle J^1(T,M), and its local components were introduced.
Abstract: The paper introduces the notion of \Gamma-linear connection
abla on the 1-jet fibre bundle J^1(T,M), and presents its local components. We also describe the local Ricci and Bianchi identities of $
abla$.
TL;DR: In this paper, the authors improved the bound on the order of a torsion element in the Picard group of the base of an Abelian line bundle, and obtained an optimal bound when the degree of the line bundle d is odd and the set of residue characteristics of base does not intersect the set p dividing d, such that p≡-1 mod(4) and p \leqslant 2g - 1\), where g is the relative dimension of the Abelian scheme.
Abstract: To a symmetric, relatively ample line bundle on an Abelian scheme one can associate a linear combination of the determinant bundle and the relative canonical bundle, which is a torsion element in the Picard group of the base. We improve the bound on the order of this element found by Faltings and Chai. In particular, we obtain an optimal bound when the degree of the line bundle d is odd and the set of residue characteristics of the base does not intersect the set of primes p dividing d, such that p≡-1 mod(4) and \(p \leqslant 2g - 1\), where g is the relative dimension of the Abelian scheme. Also, we show that in some cases these torsion elements generate the entire torsion subgroup in the Picard group of the corresponding moduli stack.
01 Jan 2000
TL;DR: In this article, the authors studied polystable Higgs bundles twisted by a line bundle over a compact Kahler manifold and identified the corre-sponding Tannaka group in the case in which the line bundle is of finite order.
Abstract: We study polystable Higgs bundles twisted by a line bundle over a compact Kahler manifold. These form a Tannakian category when the first and second Chern classes of the bundle are zero. In this paper we identify the corre- sponding Tannaka group in the case in which the line bundle is of finite order. This group is described in terms of the pro-reductive completion of the fundamental group of the manifold, and the character associated to the line bundle.
TL;DR: In this article, the determinant of the push forward of a symmetric line bundle on a complex abelian fibration is defined in terms of the pull back of the relative dualizing sheaf via the zero section.
Abstract: We give an expression of the determinant of the push forward of a symmetric line bundle on a complex abelian fibration, in terms of the pull back of the relative dualizing sheaf via the zero section.
TL;DR: In this article, a canonical family of Hermitian connections in a CR-holomorphic vector bundle (E,h) over a non-degenerate real hypersurface in a complex manifold was constructed, and an existence and uniqueness result for the solution to the inhomogeneous Yang-Mills equation was proved.
Abstract: We build a canonical family {Ds } of Hermitian connections in a Hermitian CR-holomorphic vector bundle (E,h ) over a nondegenerate CR manifold M, parametrized by S ∈ Γ ∞(End (E )), S skewsymmetric. Consequently, we prove an existence and uniqueness result for the solution to the inhomogeneous Yang-Mills equation dD*R D =f on M. As an application we solve for D ∈D (E,h ) when E is either the trivial line bundle, or a locally trivial CR-holomorphic vector bundle over a nondegenerate real hypersurface in a complex manifold, or a canonical bundle over a pseudo-Einstein CR manifold.
TL;DR: In this paper, the authors extend the Shatz stratification of sheaves to arbitrary families of projective schemes and show how the Harder-Narasimhan polygon of the restriction of the tangent bundle ΘIPn to space curves reflects the geometry of these curves and their embeddings.
Abstract: We extend the Shatz stratification of sheaves to arbitrary families of projective schemes. This allows a stratification of Hilbert schemes. We investigate how the Harder-Narasimhan polygon of the restriction of the tangent bundle ΘIPn to space curves reflects the geometry of these curves and their embeddings.