The Dirac-Higgs bundle

Thesis

The Dirac–Higgs bundle is a vector bundle with a natural connection on the moduli space of stable Higgs bundles on a compact Riemann surface. It is a vector bundle of null-spaces of a Dirac-operator coupled to stable Higgs bundles. In this thesis, we study various aspects of this bundle and its natural connection.

The Dirac–Higgs bundle is hyperholomorphic on the smooth hyperkähler moduli space of stable Higgs bundles. This property is a generalisation of the four-dimensional anti- self-duality equations to hyperkähler manifolds. One use of the Dirac–Higgs bundle is the construction of a Nahm transform for Higgs bundles. This transform produces hyperholo- morphic bundles on the moduli space of rank one Higgs bundles.

The Higgs bundle moduli space is non-compact and we study the asymptotics of the connection in the Nahm transform of a Higgs bundle. We show that elements of the null- spaces concentrate at a finite number of points on the Riemann surface. This asymptotical behaviour naturally defines a frame for the Nahm transform, which is conjectured to be asymptotically unitary.

By considering only the holomorphic structure, the Nahm transform of a Higgs bundle extends to a holomorphic bundle on the natural compactification of the rank one Higgs bundle moduli space. We discuss various aspects of this extended holomorphic bundle. Most importantly, it is a sheaf extension in which the constituent sheaves and the extension class have natural interpretations in terms of the original Higgs bundle. Furthermore, the extended bundle is not fixed at the divisor at infinity; explicit examples show that it depends on the type of Riemann surface, for example.

The Dirac–Higgs bundle has a parabolic cousin. In the parabolic case the rank depends on the number of marked points and the total multiplicity of the zero weights in the parabolic structure. The moduli space of stable rank two parabolic Higgs bundles on the Riemann sphere with four marked points has complex dimension two. Furthermore, there is a combination of parabolic weights such that the Dirac–Higgs bundle is a line bundle with an instanton connection. We study the topology of this line bundle and find that the instanton does not have finite energy. As in the non-parabolic case we define a Nahm transform for parabolic Higgs bundles, and in the case of genus one Riemann surfaces use it to produce doubly-periodic instantons of finite energy.

Authors: Blaavand, JL

• DOI: 10.5287/ora-00my9j2qy
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: self-duality equations; Dirac operators; instantons; Higgs bundles; Fourier-Mukai transform; Nahm transform
• Subjects: Geometry, Algebraic; Mathematics; Geometry, Differential