The ubiquity of moduli spaces of semi-stable higgs bundles on a smooth projective curve both in mathematics and physics is rather impressive. These moduli spaces have proven to be grounds of extremely fruitful interaction between the two disciplines. As an example, the techniques developed by physicists to quantize a symplectic manifold and to quantize a com- pletely integrable Hamiltonian system when applied to these moduli spaces yield remarkable mathematical results. E. Witten showed that one can quantize moduli spaces of bundles, and that this quanti- zation leads to a 2+1 dimensional topological quantum field theory. This TQFT not only provides new topological invariants of three manifolds and knots but has also proven to be extremely successful in attacking long standing problems in low dimensional topology. One of the most remarkable fact about the moduli spaces of higgs bundles, discovered by N. Hitchin, is that they admit the structure of a completely integrable Hamiltonian system. A. Beilinson and V. Drinfeld gave an explicit quantization of this Hamiltonian system and showed that this quantization naturally gives formulation of the geometric Langlands correspondence. Interestingly, again relying on these moduli spaces, E. Witten and A. Kapustin showed that all the mathematical objects that appear in the Beilinson-Drinfeld formulation of the geometric Langlands have natural interpretation in terms of N “ 4 super Yang-Mills theory in four dimensions, and that the Langlands correspondence can be interpreted as the electric-magnetic duality in Yang-Mills theory. We will first go through the construction of relevant geometric structures on the moduli spaces of higgs bundles, namely the hyperk ̈ahler structure and the structure of a completely integrable Hamiltonian system. The main objective is to go through the Beilinson-Drinfeld quantization procedure, and if time permits to show how this quantization relates to the geometric Langlands correspondence. Following topics are planned to be covered.
- First example: Jacobians as moduli spaces of line bundles
- Moduli spaces of vector bundles, the Narasimhan-Seshadri and Donaldson theorem
- Quantization of moduli space of vector bundles ́a la Hitchin and the asso- ciated three dimensional Topological Quantum Field Theory
- Moduli spaces of Higgs bundles, their relation to the character variety and non-abelian Hodge theory
- The Hitchin system
- Quantization of the Hitchin system ́a la Beilinson-Drinfeld and formulation of the geometric Langlands correspondence