Homogeneous spaces and Faddeev-Skyrme models

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dc.contributor.author Koshkin, Sergiy
dc.date.accessioned 2006-06-12T19:56:45Z
dc.date.available 2006-06-12T19:56:45Z
dc.date.issued 2006-06-12T19:56:45Z
dc.date.submitted August 2006 en
dc.identifier.uri http://hdl.handle.net/2097/171
dc.description.abstract We study geometric variational problems for a class of models in quantum field theory known as Faddeev-Skyrme models. Mathematically one considers minimizing an energy functional on homotopy classes of maps from closed 3-manifolds into homogeneous spaces of compact Lie groups. The energy minimizers known as Hopfions describe stable configurations of subatomic particles such as protons and their strong interactions. The Hopfions exhibit distinct localized knot-like structure and received a lot of attention lately in both mathematical and physical literature. High non-linearity of the energy functional presents both analytical and algebraic difficulties for studying it. In particular we introduce novel Sobolev spaces suitable for our variational problem and develop the notion of homotopy type for maps in such spaces that generalizes homotopy for smooth and continuous maps. As the spaces in question are neither linear nor even convex we take advantage of the algebraic structure on homogeneous spaces to represent maps by gauge potentials that form a linear space and reformulate the problem in terms of these potentials. However this representation of maps introduces some gauge ambiguity into the picture and we work out 'gauge calculus' for the principal bundles involved to apply the gauge-fixing techniques that eliminate the ambiguity. These bundles arise as pullbacks of the structure bundles H[arrow pointing right with hook on tail]G[arrow pointing right]G/H of homogeneous spaces and we study their topology and geometry that are of independent interest. Our main results include proving existence of Hopfions as finite energy Sobolev maps in each (generalized) homotopy class when the target space is a symmetric space. For more general spaces we obtain a weaker result on existence of minimizers only in each 2-homotopy class. en
dc.format.extent 540016 bytes
dc.format.mimetype application/pdf
dc.language.iso en_US en
dc.publisher Kansas State University en
dc.subject Skyrme energy en
dc.subject Hopfions en
dc.subject Knotted solitons en
dc.subject Coset models en
dc.subject Gauge-fixing en
dc.subject Weak wedge products en
dc.title Homogeneous spaces and Faddeev-Skyrme models en
dc.type Dissertation en
dc.description.degree Doctor of Philosophy en
dc.description.level Doctoral en
dc.description.department Department of Mathematics en
dc.description.advisor David R. Auckly en
dc.subject.umi Mathematics (0405) en
dc.date.published 2006 en
dc.date.graduationmonth August en

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