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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, it was shown that the nonperturbative superpotential generated by a heterotic superstring wrapped once around a genus-zero holomorphic curve is proportional to the Pfaffian involving the determinant of a Dirac operator.
Abstract: The non-perturbative superpotential generated by a heterotic superstring wrapped once around a genus-zero holomorphic curve is proportional to the Pfaffian involving the determinant of a Dirac operator on this curve. We show that the space of zero modes of this Dirac operator is the kernel of a linear mapping that is dependent on the associated vector bundle moduli. By explicitly computing the determinant of this map, one can deduce whether or not the dimension of the space of zero modes vanishes. It is shown that this information is sufficient to completely determine the Pfaffian and, hence, the non-perturbative superpotential as explicit holomorphic functions of the vector bundle moduli. This method is illustrated by a number of non-trivial examples.
81 citations
TL;DR: In this paper, a generalized Higgs field is defined as a map from P to V, equivariant under the action of G in P. If the image of α is an orbitW ⊂V of G, then a breaks (spontaneously) the symmetry: the isotropy (little) group ofw 0 eW is the “unbroken” groupH.
Abstract: Principal fibre bundles with connections provide geometrical models of gauge theories. Bundles allow for a global formulation of gauge theories: the potentials used in physics are pull-backs, by means of local sections, of the connection form defined on the total spaceP of the bundle. Given a representationP of the structure (gauge) groupG in a vector spaceV, one defines a (generalized) Higgs field α as a map fromP toV, equivariant under the action ofG inP. If the image of α is an orbitW ⊂V ofG, then a breaks (spontaneously) the symmetry: the isotropy (little) group ofw
0 eW is the “unbroken” groupH. The principal bundleP is then reduced to a subbundleQ with structure groupH. Gravitation corresponds to a linear connection, i.e. to a connection on the bundle of frames. This bundle has more structure than an abstract principal bundle: it is soldered to the base. Soldering results in the occurrence of torsion. The metric tensor is a Higgs field breaking the symmetry fromGL (4,R) to the Lorentz group.
78 citations
TL;DR: In this article, the case of a matter bundle E whose standard fiber admits action only of an exact symmetry subgroup H of G is examined, where the bundle E fails to be associated with a principal bundle and the canonical jet bundle morphism J1EH×J1Σ→J1E is used.
Abstract: Given a principal bundle P→X with a structure group G and an associated Higgs bundle Σ with a standard fiber G/H, the case of a matter bundle E whose standard fiber admits action only of an exact symmetry subgroup H of G is examined. In the presence of a fixed Higgs field σ: X→Σ, matter fields are represented by sections of a matter bundle Eh associated with the corresponding reduced subbundle Ph of P. The totality of matter fields and Higgs fields is described by sections of the bundle E which is the composite bundle EH→Σ→X where EH→Σ is the bundle associated with the principal H‐bundle P→Σ. The bundle E fails to be associated with a principal bundle. To construct a connection Γ: E→J1E on E, the canonical jet bundle morphism J1EH×J1Σ→J1E is used.
78 citations
TL;DR: In this article, the Yang-Mills heat flow in a vector bundle over a compact Riemannian four-manifold for given initial connection of finite energy is established.
Abstract: Global existence and uniqueness is established for the Yang-Mills heat flow in a vector bundle over a compact Riemannian four-manifold for given initial connection of finite energy. Our results are analogous to those valid for the evolution of harmonic maps of Riemannian surfaces.
78 citations
TL;DR: In this paper, the Atiyah-Patodi-Singer index theory construction of the bundle of fermionic Fock spaces parameterized by vector potentials in odd space dimensions is presented.
Abstract: This paper reviews recent work on a new geometric object called a bundle gerbe and discusses some new examples arising in quantum field theory. One application is to an Atiyah–Patodi–Singer index theory construction of the bundle of fermionic Fock spaces parameterized by vector potentials in odd space dimensions and a proof that this leads in a simple manner to the known Schwinger terms (Mickelsson–Faddeev cocycle) for the gauge group action. This gives an explicit computation of the Dixmier–Douady class of the associated bundle gerbe. The method also works in other cases of fermions in external fields (external gravitational field, for example) provided that the APS theorem can be applied; however, we have worked out the details only in the case of vector potentials. Another example, in which the bundle gerbe curvature plays a role, arises from the WZW model on Riemann surfaces. A further example is the "existence of string structures" question. We conclude by showing how global Hamiltonian anomalies fit within this framework.
78 citations