Nigel Hitchin

Selfduality in Four-Dimensional Riemannian Geometry

We present a self-contained account of the ideas of R. Penrose connecting four-dimensional Riemannian geometry with three-dimensional complex analysis. In particular we apply this to the self-dual Yang-Mills equations in Euclidean 4-space and compute the number of moduli for any compact gauge group. Results previously announced are treated with full detail and extended in a number of directions.

Construction of instantons

A complete construction, involving only linear algebra, is given for all self-dual euclidean Yang-Mills fields.

Hyperkähler metrics and supersymmetry

We describe two constructions of hyperkähler manifolds, one based on a Legendre transform, and one on a sympletic quotient. These constructions arose in the context of supersymmetric nonlinear σ-models, but can be described entirely geometrically. In this general setting, we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry, along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience.

Stable bundles and integrable systems

On considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de Riemann. On montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

Monopoles and geodesics

AbstractUsing the holomorphic geometry of the space of straight lines in Euclidean 3-space, it is shown that every static monopole of chargek may be constructed canonically from an algebraic curve by means of the Atiyah-Ward Ansatz

Flat connections and geometric quantization

A_k

Low energy scattering of non-abelian monopoles

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Instantons, Poisson Structures and Generalized Kähler Geometry

We show that any self-dual SU (2) monopole may be constructed either by Wards twistor method, or Nahms use of the ADHM construction. The common factor in both approaches is an algebraic curve whose Jacobian is used to linearize the non-linear ordinary differential equations which arise in Nahms method. We derive the non-singularity condition for the monopole in terms of this curve and apply the result to prove the regularity of axially symmetric solutions.