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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 paper, the authors describe the standard Liouville form θ and the symplectic form dθ in terms of the right Maurer Cartan form and the left moment mapping (of the right action of G on itself).
Abstract: On a cotangent bundle T*G of a Lie group G one can describe the standard Liouville form θ and the symplectic form dθ in terms of the right Maurer Cartan form and the left moment mapping (of the right action of G on itself), and also in terms of the left Maurer–Cartan form and the right moment mapping, and also the Poisson structure can be written in related quantities. This leads to a wide class of exact symplectic structures on T*G and to Poisson structures by replacing the canonical momenta of the right or left actions of G on itself by arbitrary ones, followed by reduction (to G cross a Weyl‐chamber, e.g.). This method also works on principal bundles.

40 citations

Journal ArticleDOI
TL;DR: The twisted WZW model as discussed by the authors is a variant of the Wess-Zumino-Witten model which is associated to a certain Lie group bundle on a family of elliptic curves.
Abstract: Investigated is a variant of the Wess-Zumino-Witten model called a twisted WZW model, which is associated to a certain Lie group bundle on a family of elliptic curves. The Lie group bundle is a non-trivial bundle with flat connection and related to the classical elliptic r-matrix. (The usual (non-twisted) WZW model is associated to a trivial group bundle with trivial connection on a family of compact Riemann surfaces and a family of its principal bundles.) The twisted WZW model on a fixed elliptic curve at the critical level describes the XYZ Gaudin model. The elliptic Knizhnik-Zamolodchikov equations associated to the classical elliptic r-matrix appear as flat connections on the sheaves of conformal blocks in the twisted WZW model.

40 citations

Journal ArticleDOI
TL;DR: In this paper, a U1-valued latice gauge field u defined on a periodic, 2-dimensional lattice satisfies the generic continuity condition uuuu ≠ − 1, it can be used to construct a principal U 1-bundle over the torus and in that bundle a connection such that parallel transport along bonds is given by u. The characteristic classes and numbers of this bundle can then be calculated from u in a straightforward way.

40 citations

Journal ArticleDOI
TL;DR: In this paper, the authors make the category of bundle gerbes on a manifold $M$ into a 2$-category by providing $2$-cells in the form of transformations of the bundle gerbe morphisms.
Abstract: We make the category $\textbf{BGrb}_M$ of bundle gerbes on a manifold $M$ into a $2$-category by providing $2$-cells in the form of transformations of bundle gerbe morphisms. This description of $\textbf{BGrb}_M$ as a $2$-category is used to define the notion of a bundle $2$-gerbe. To every bundle $2$-gerbe on $M$ is associated a class in $H^4(M ; \mathbb{Z})$. We define the notion of a bundle $2$-gerbe connection and show how this leads to a closed, integral, differential $4$-form on $M$ which represents the image in real cohomology of the class in $H^4(M ; \mathbb{Z})$. Some examples of bundle $2$-gerbes are discussed, including the bundle $2$-gerbe associated to a principal $G$ bundle $P \to M$. It is shown that the class in $H^4(M ; \mathbb{Z})$ associated to this bundle $2$-gerbe coincides with the first Pontryagin class of $P$: this example was previously considered from the point of view of $2$-gerbes by Brylinski and McLaughlin.

40 citations

Posted Content
TL;DR: In this article, an algebro-geometric construction for the holomorphic Chern-Simons functional is presented giving the local analytic moduli scheme of a vector bundle and an analogous gradient scheme construction for Brill-Noether loci on ample divisors is also given.
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.

39 citations

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