# The moduli space of Higgs bundles

## Course information

Code: MAT1305S

Instructor: Marco Gualtieri

Class schedule: W1:30-3 & F1:30-3 in BA6180, starting Jan 11, 2017

Evaluation: Class participation and reading project
(presentations if class size allows)

This course is an introduction to the moduli space of Higgs bundles, introduced by Hitchin in 1987 and sometimes called the Hitchin moduli space. It is a central object of study in the geo- metric Langlands programme, and it is an archetypal example of a completely integrable system. It is endowed with an incredibly elaborate structure, one major part of which is a Hyperkahler structure.

Main topics include:

- Hyperkahler geometry and hyperkahler reduction
- The twistor space of a hyperkahler manifold
- Integrable systems
- The spectral correspondence
- Singularities
- Non-abelian Hodge
- Dirac operator associated to a Higgs bundle
- Fourier transform for Higgs bundles
- Opers and brane

## Notes

## Resources

### Articles

- N. Hitchin. Stable bundles and integrable systems
- N. Hitchin. The self-duality equations on a Riemann surface
- N. Hitchin. Lie groups and Teichmuller space
- N. Hitchin. The Dirac operator
- N. Hitchin. Hyperkahler manifolds
- N. Hitchin. Polygons and gravitons
- D. Gaiotto, G. W. Moore, & A. Neitzke. Wall-crossing, Hitchin systems, and the WKB approximation
- O. Biquard & P. Boalch. Wild nonabelian Hodge theory on curves
- P. Boalch. Habilitation memoir

### Theses

- Geometry of the moduli space of Higgs bundles, T. Hausel
- A Fourier transform for Higgs bundles, J. Bonsdorff
- The Dirac-Higgs bundle, Jakob Blaavand
- Higgs bundles and opers, P. Dalakov
- Nahm transform for integrable connections on the Riemann sphere, S. Szabo

## Research project

The research project will consist of an expansion on one of the topics touched on in class, written up and possibly presented in class or in an external seminar.