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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 present nonabelian bundle gerbes as a higher version of principal bundles and study connection, curving, curvature and gauge transformations both in a global coordinate independent formalism and in local coordinates.
Abstract: Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge fields needed for the construction of Yang-Mills theories with 2-form gauge potential.
131 citations
TL;DR: In this paper, a compact Kahler manifold M and a connected reductive algebraic group G over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] admits an Einstein Hermitian connection if and only if the principal bundle is polystable.
Abstract: Given a compact Kahler manifold M and a connected reductive algebraic group G over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], a prinlcipal G -bundle over M admits an Einstein-Hermitian connection if and only if the principal bundle is polystable. If M is a projective manifold, a Higgs G -bundle over M admits an Einstein-Hermitian connection if and only if the Higgs bundle is polystable.
125 citations
TL;DR: In this paper, the authors give an explicit description of the vector bundle of WZW conformal blocks on elliptic curves with marked points as a subbundle of Weyl group invariant vector valued theta functions on a Cartan subalgebra.
Abstract: We give an explicit description of the vector bundle of WZW conformal blocks on elliptic curves with marked points as a subbundle of a vector bundle of Weyl group invariant vector valued theta functions on a Cartan subalgebra. We give a partly conjectural characterization of this subbundle in terms of certain vanishing conditions on affine hyperplanes. In some cases, explicit calculations are possible and confirm the conjecture. The Friedan-Shenker flat connection is calculated, and it is shown that horizontal sections are solutions of Bernard's generalization of the Knizhnik-Zamolodchikov equation.
123 citations
TL;DR: The theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in H4(BG, ℤ) for a compact semi-simple Lie group was developed in this paper.
Abstract: We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in H4(BG, ℤ) for a compact semi-simple Lie group G. The Chern-Simons bundle 2-gerbe realises differential geometrically the Cheeger-Simons invariant. We apply these notions to refine the Dijkgraaf-Witten correspondence between three dimensional Chern-Simons functionals and Wess-Zumino-Witten models associated to the group G. We do this by introducing a lifting to the level of bundle gerbes of the natural map from H4(BG, ℤ) to H3(G, ℤ). The notion of a multiplicative bundle gerbe accounts geometrically for the subtleties in this correspondence for non-simply connected Lie groups. The implications for Wess-Zumino-Witten models are also discussed.
123 citations
TL;DR: In this paper, a direct method is employed to minimize the Yang-Mills functional over a 4-dimensional manifold, and it is shown that a topological invariant of the bundle is preserved by the minimizing process.
Abstract: A direct method is employed to minimize the Yang-Mills functional over a 4-dimensional manifold. The limiting connection is shown to be Yang-Mills, but in a possibly new bundle. We show that a topological invariant of the bundle is preserved by the minimizing process. This implies the existence of an absolute minimum of the Yang-Mills functional in a wide class of bundles.
120 citations