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SUMMARY:Abelian and non-Abelian Hopfions in all odd dimensions - Tchrakian
 \, T (DIAS)
DTSTART:20121207T114000Z
DTEND:20121207T120000Z
UID:TALK41930@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Hopfions are field configurations of scalar matter systems cha
 racterised prominently by the fact that they describe knots in configurati
 on space. Like the 'usual solitons'\, e.g. Skyrmions\, monopoles\, vortice
 s and instantons\, Hopfions are static and finite energy solutions that ar
 e stabilised by a topological charge\, which supplies the energy lower bou
 nd. \n\nIn contrast to the 'usual solitons' however\, the topological char
 ge of Hopfions is not the volume integral of a total divergence. While the
  topological charge densities of the 'usual solitons'\, namely the Chern-P
 ontryagin (CP) densities or their descendants\, are total divergence\, the
  corresponding quantities for Hopfions are the Chern-Simons (CS) densities
  which are not total divergence. Subject to the appropriate symmetries how
 ever\, these CS densities do reduce to total divergence and become candida
 tes for topological charges. Thus\, Hopfion field are necessarily subject 
 to the appropriate symmetry to decsribe knots\, excluding spherically symm
 etry\, in contrast to the 'usual solitons'. \n\nThe construction of these 
 CS densities is enabled by employing complex nonlinear sigma models\, whic
 h feature composite connections. The CS densities are defined in terms of 
 these connections and their curvatures. (In some dimensions the complex si
 gma model can be equivalent to a real sigma model\, e.g. in D=3 Skyrme-Fad
 de'ev O(3) model and the corresponding CP^1 model.) It is natural to propo
 se Hopfion fields in all odd space dimensions where a CS density can be de
 fined. This covers both Abelian and non-Abelian theories\, namely empolyin
 g projective-complex and Grassmannian models\, respectively. It is in this
  sense that we have used the terminology of Abelian and non-Abelian Hopfio
 ns. \n\nExplicit field configurations displaying the appropriate symmetrie
 s and specific asymptotic behaviours in several (higher) dimensions are pr
 oposed\, and it is verified that for these configurations the CS densities
  do indeed become total divergence.\n\n
LOCATION:Seminar Room 1\, Newton Institute
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