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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 studied the integrability conditions of a manifold with Spin(9)-structure and derived the corresponding differential equation for the unique self-dual 8-form differential equation assigned to any type of spin-9-structure.
Abstract: The aim of the present paper is the investigation of $Spin(9)$-structures on 16-dimensional manifolds from the point of view of topology as well as holonomy theory. First we construct several examples. Then we study the necessary topological conditions resulting from the existence of a $Spin(9)$-reduction of the frame bundle of a 16-dimensional compact manifold (Stiefel-Whitney and Pontrjagin classes). We compute the homotopy groups $\pi_i (X^{84})$ of the space $X^{84}= SO(16) / Spin(9)$ for $i \le 14$. Next we introduce different geometric types of $Spin(9)$-structures and derive the corresponding differential equation for the unique self-dual 8-form $\Omega^8$ assigned to any type of $Spin(9)$-structure. Finally we construct the twistor space of a 16-dimensional manifold with $Spin(9)$-structure and study the integrability conditions for its universal almost complex structure as well as the structure of the holomorphic normal bundle.
58 citations
TL;DR: In this article, a twistor version of the hyperkahler/quaternion Kahler correspondence is established and the corresponding holomorphic line bundle on twistor space is described and many examples computed, including monopole and Higgs bundle moduli spaces.
Abstract: A hyperkahler manifold with a circle action fixing just one complex structure admits a natural hyperholomorphic line bundle. This observation forms the basis for the construction of a corresponding quaternionic Kahler manifold in the work of A.Haydys. In this paper the corresponding holomorphic line bundle on twistor space is described and many examples computed, including monopole and Higgs bundle moduli spaces. Finally a twistor version of the hyperkahler/quaternion Kahler correspondence is established.
58 citations
TL;DR: In this article, it was shown that 4-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle.
Abstract: Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle. The case G = GL(4,R) is especially interesting because every 4-manifold is then naturally equipped with a principal G-bundle, namely its frame bundle. In this case, the partition function of a compact oriented 4-manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of 4-manifolds.
56 citations
TL;DR: In this article, the notion of translation maps in quantum principal bundles is introduced and a translation map is used to prove that the cross sections of a quantum fibre bundle associated to a quantum principal bundle P(B, V, A) are in bijective correspondence with equivariant maps V → P.
56 citations
56 citations