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Frame bundle

About: Frame bundle is a research topic. Over the lifetime, 1600 publications have been published within this topic receiving 23049 citations.


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TL;DR: In this article, a spin structure on the loop space of a manifold is defined, which is a lift of the structure group of the looped frame bundle from the loop group to its universal central extension.
Abstract: Classically, a spin structure on the loop space of a manifold is a lift of the structure group of the looped frame bundle from the loop group to its universal central extension. Heuristically, the loop space of a manifold is spin if and only if the manifold itself is a string manifold, against which it is well-known that only the if-part is true in general. In this article we develop a new version of spin structures on loop spaces that exists if and only if the manifold is string, as desired. This new version consists of a classical spin structure plus a certain fusion product related to loops of frames in the manifold. We use the lifting gerbe theory of Carey-Murray, recent results of Stolz-Teichner on loop spaces, and some own results about string geometry and Brylinski-McLaughlin transgression.

14 citations

Journal ArticleDOI
01 Aug 2002
TL;DR: In this article, a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (E1, E2,ϕ) was constructed, where E1 and E2 are holomorphic vector bundles over a fixed compact Riemann surface.
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.

14 citations

Journal ArticleDOI
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.

14 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the projective Poincare bundle is stable with respect to any polarization and its restriction to the moduli space of stable vector bundles on an irreducible smooth projective curve of genus g ≥ 3 defined over the complex numbers.
Abstract: Let X be an irreducible smooth projective curve of genus g ≥ 3 defined over the complex numbers, and let ℳ ξ denote the moduli space of stable vector bundles on X of rank n and determinant ξ, where ξ is a fixed line bundle of degree d. If n and d have a common divisor, then there is no universal vector bundle on X × ℳ ξ . We prove that there is a projective bundle on X × ℳ ξ with the property that its restriction to X × {E} is isomorphic to P(E) for all E ∈ ℳ ξ and that this bundle (called the projective Poincare bundle) is stable with respect to any polarization; moreover its restriction to {x} × ℳ ξ is also stable for any x ∈ X. We also prove stability results for bundles induced from the projective Poincare bundle by homomorphisms PGL(n) → H for any reductive H. We further show that there is a projective Picard bundle on a certain open subset ℳ′ of ℳ ξ for any d > n(g−1) and that this bundle is also stable. Also, we obtain new results on the stability of the Picard bundle even when n and d are coprime.

14 citations

Journal ArticleDOI
01 Mar 1988
TL;DR: Toru et al. as discussed by the authors proved that the space r(X) of all cross sections of p: X -B is an 12-manifold, provided ps = id.
Abstract: Let B be a nondiscrete compactum, Y a separable complete metrizable ANR with no isolated point and p: X -B a locally trivial bundle with fiber Y admitting a section. It is proved that the space r(X) of all cross sections of p: X -B is an 12-manifold. 0. Introduction. Through the paper, spaces are separable metrizable and maps are continuous. Let p: X -* B be a locally trivial bundle with fiber Y, that is, each point b E B has a neighborhood U and a homeomorphism p: U x Y -, p-1 (U) such that p'p = ru, the projection to U. A map s: B -* X is called a cross section of p: X -* B provided ps = id. The space of all cross sections of p: X -* B with compact-open topology is denoted by 17(X). Then 17(X) is a closed subspace of the space C(B, X) of all maps from B into X. If B is compact and d is a compatible metric for X, the topology of 17(X) (and C(B, X)) is induced by the sup-metric d(f, g) = sup{d(f (b), g(b)) I b E B}. A manifold modeled on Hilbert space 12 is called an 12-manifold. In this note, we prove the following MAIN THEOREM. Let B be a nondiscrete compactum, Y a complete metrizable ANR with no isolated point and p: X B a locally trivial bundle with fiber Y admitting a section. Then 17(X) is an 12-manifold. For the trivial bundle 7rB: B x Y -* B, the space F(B x Y) can be regarded as the space C(B, Y). Thus the space C(B, Y) is an 12-manifold if B is a nondiscrete compactum and Y is a complete-metrizable ANR with no isolated point. This is a generalization of Eells-Geoghegan-Torunczyk's result [E, Ge, To1]. The author would like to thank Doug Curtis for helpful comments. 1. Preliminaries. Our proof is based on the following: TORU?NCZYK'S CHARACTERIZATION THEOREM FOR 12-MANIFOLDS [TO2] (CF. [To3]). A complete-metrizable ANR X is an 12-manifold if and only if X has the discrete approximation property, that is, for each map f: eDnEN I' X of the free union of n-cells (n E N) into X and each map E: X -+ (0, 1) there is a Received by the editors February 27, 1987 and, in revised form, April 22, 1987. Presented to the Mathematical Society of Japan, April 2, 1988. 1980 Mathematics Subject Classification. Primary 58D15, 57N20, 55F10.

14 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20236
202214
20214
202012
201911
201811