ICTP Course on Geometry and Quantization of the moduli of Higgs bundles (Shehryar SIKANDER) starts TOMORROW at 4PM

[ICTP Math Section]

Tomorrow at 4:00PM the first lecture of the course
"Geometry and Quantization of the moduli of Higgs bundles" by Shehryar 
Sikander (ICTP)

will take place at SISSA - room number 134


Below is an introduction to the course

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 completely 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 quantization leads to a 2+1 dimensional topological quantum 
eld 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 
hyperkahler 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 
associated 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 à la Beilinson-Drinfeld and 
formulation of the geometric Langlands correspondence
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