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Yang-Mills Flows on Nearly Kähler Manifolds and G 2-Instantons

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Abstract

We consider Lie(G)-valued G-invariant connections on bundles over spaces \({G/H,\, \mathbb{R}\times G/H\, {\rm and}\, \mathbb{R}^2\times G/H}\), where G/H is a compact nearly Kähler six-dimensional homogeneous space, and the manifolds \({\mathbb{R}\times G/H}\) and \({\mathbb{R}^2\times G/H}\) carry G 2- and Spin(7)-structures, respectively. By making a G-invariant ansatz, Yang-Mills theory with torsion on \({\mathbb{R}\times G/H}\) is reduced to Newtonian mechanics of a particle moving in a plane with a quartic potential. For particular values of the torsion, we find explicit particle trajectories, which obey first-order gradient or hamiltonian flow equations. In two cases, these solutions correspond to anti-self-dual instantons associated with one of two G 2-structures on \({\mathbb{R}\times G/H}\). It is shown that both G 2-instanton equations can be obtained from a single Spin(7)-instanton equation on \({\mathbb{R}^2\times G/H}\).

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Authors and Affiliations

  1. Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167, Hannover, Germany

    Derek Harland & Olaf Lechtenfeld

  2. Bogoliubov Laboratory of Theoretical Physics, JINR, 141980, Dubna, Moscow Region, Russia

    Tatiana A. Ivanova & Alexander D. Popov

Authors
  1. Derek Harland
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  2. Tatiana A. Ivanova
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  3. Olaf Lechtenfeld
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  4. Alexander D. Popov
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Correspondence to Derek Harland.

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Communicated by P.T. Chruściel

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Harland, D., Ivanova, T.A., Lechtenfeld, O. et al. Yang-Mills Flows on Nearly Kähler Manifolds and G 2-Instantons. Commun. Math. Phys. 300, 185–204 (2010). https://doi.org/10.1007/s00220-010-1115-7

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  • DOI: https://doi.org/10.1007/s00220-010-1115-7

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