Higgs Bundles and Opers. by Peter Dalakov at Bookstore UAE
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Higgs Bundles and Opers.

Higgs Bundles and Opers.


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About the Book

In this thesis we address the question of determining the Higgs bundles on a Riemann surface which correspond to opers by the non-abelian Hodge theorem. We study one specific Higgs bundle (P, theta) obtained by extension of structure group from a particular rank two Higgs bundle on K1/2 (+) K -1/2. For a simple complex group we identify the tangent space to the Dolbeault moduli space at that particular point as the Hitchin base plus its dual and exhibit harmonic representatives for the hypercohomology of the corresponding deformation complex. We study the Maurer-Cartan equation for (P, theta) and give an explicit, recursive (terminating after finitely many steps) method for writing a family of deformations parametrised by the tangent space which can be interpreted as a holomorphic exponential map. Next, we identify opers with their tangent space at the uniformisation oper. By quaternionic linear algebra we determine the subspace of infinitesimal deformations of (P, theta) corresponding to opers and apply the exponential map to those. In this way we obtain the germ of the family of Higgs bundles corresponding to opers. In the final chapter we sketch a plan for obtaining the complete solution, as well as certain questions for further investigation.


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Product Details
  • ISBN-13: 9781243497093
  • Publisher: Proquest, Umi Dissertation Publishing
  • Publisher Imprint: Proquest, Umi Dissertation Publishing
  • Height: 246 mm
  • Weight: 195 gr
  • ISBN-10: 1243497092
  • Publisher Date: 01 Sep 2011
  • Binding: Paperback
  • Spine Width: 5 mm
  • Width: 189 mm


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