Frame bundle

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

The ring of global sections of multiples of a line bundle on a toric variety

Abstract: In this article we prove that for any complete toric variety, and for any Cartier divisor, the ring of global sections of multiples of the line bundle associated to the divisor is finitely generated.

Loop Groups, Higgs Fields and Generalised String Classes

Abstract: We consider various generalisations of the string class of a loop group bundle. The string class is the obstruction to lifting a bundle whose structure group is the loop group $LG$ to one whose structure group is the Kac-Moody central extension of the loop group. We develop a notion of higher string classes for bundles whose structure group is the group of based loops, $\Omega G$. In particular, we give a formula for characteristic classes in odd dimensions for such bundles which are associated to characteristic classes for $G$-bundles in the same way that the string class is related to the first Pontrjagyn class of a certain $G$-bundle associated to the loop group bundle in question. This provides us with a theory of characteristic classes for $\Omega G$-bundles analogous to Chern-Weil theory in finite dimensions. This also gives us a geometric interpretation of the well-known transgression map $H^{2k}(BG) \to H^{2k-1}(G).$ We also consider the obstruction to lifting a bundle whose structure group is not the loop group but the semi-direct product of the loop group with the circle, $LG \rtimes S^1$. We review the theory of bundle gerbes and their application to central extensions and lifting problems and use these methods to obtain an explicit expression for the de Rham representative of the obstruction to lifting such a bundle. We also relate this to a generalisation of the so-called `caloron correspondence' (which relates $LG$-bundles over $M$ to $G$-bundles over $M \times S^1$) to a correspondence which relates $LG \rtimes S^1$-bundles over $M$ to $G$-bundles over $S^1$-bundles over $M$.

A note on metric connections for chiral and Dirac spinors

Abstract: It is known that the bundle of Dirac spinors is produced as a direct sum of two bundles - the bundle of chiral spinors and its Hermitian conjugate bundle. In this paper some aspects of metric connections for chiral and Dirac spinors are resumed and their relation is studied.

Vanishing theorems for the kernel of a Dirac operator

Abstract: We obtain a vanishing theorem for the kernel of a Dirac operator on a Clifford module twisted by a sufficiently large power of a line bundle, whose curvature is non-degenerate at any point of the base manifold. In particular, if the base manifold is almost complex, we prove a vanishing theorem for the kernel of a $\spin^c$ Dirac operator twisted by a line bundle with curvature of a mixed sign. In this case we also relax the assumption of non-degeneracy of the curvature. These results are generalization of a vanishing theorem of Borthwick and Uribe. As an application we obtain a new proof of the classical Andreotti-Grauert vanishing theorem for the cohomology of a compact complex manifold with values in the sheaf of holomorphic sections of a holomorphic vector bundle, twisted by a large power of a holomorphic line bundle with curvature of a mixed sign. As another application we calculate the sign of the index of a signature operator twisted by a large power of a line bundle.