Removable Singularities for Monopoles on a Sasakian 3-manifold (PhD project)

Project title: Removable Singularities for Monopoles on a Sasakian 3-manifold (PhD project)

Supervisor: Dr Adam Harris (UNE)

PhD student: Mr Kumbu Dorji

Funding: UNE International Postgraduate Scholarship  

Project description:
A classic problem of higher-dimensional complex analysis concerns the extension of complex-analytic structures, such as holomorphic functions or vector bundles, across a gap locus, which in the first instance may simply be a point. Theorems of this kind for functions are more than a century old, whereas "removable singularities" theorems for vector bundles have a more recent origin. A famous theorem of Uhlenbeck shows that point singularities of unitary vector bundles over a punctured ball in real four-dimensional space are removable if certain conditions are imposed on the curvature form associated with a choice of hermitian metric. Such conditions are typically representable as differential equations governing gauge-fields which arise in theoretical physics.  One such is the so-called anti self-dual form of the Yang-Mills equation, in terms of which a unitary bundle can be endowed with a natural holomorphic structure. Techniques available for the study of removable singularities in this setting are applied to this project by means of a two-fold reduction from four dimensions to three. On one hand, a particular form of the magnetic monopole equation, first studied by Dirac, can be realised as a time-independent reduction of the anti self-dual equation. On the other, every Sasakian three-manifold is the canonical section of a certain complex Kahler surface.

Related publications:

  1. I. Biswas and J. Hurtubise, Monopoles on Sasakian three-folds, arXiv:1412.4050v1 (2014)
  2. C.P. Boyer and K. Galicki, Sasakian Geometry, Oxford Math. Monographs, OUP (2008)
  3. N.P. Buchdahl and A. Harris , Holomorphic connections and extension of complex vector bundles. Math. Nachr. 204 (1999) 29-39
  4. A. Harris and Y. Tonegawa, Analytic continuation of vector bundles with Lp-curvature. Int. J. Math. 11, No. 1 (2000) 29-40