Hermitian-Einstein connection

[calcutta78]

Recently I came to know about Narasimhan-Seshadri theorem. I think its a fascinating theorem connecting three major topics in mathematics, namely algebraic geometry, representation theory and differential geometry. I was looking at their original paper but fail to understand completely Does anybody know about any expository article on this theorem, specially the differential geometry part?
If people here are interested to understand its proof and many generalisations due to Donaldson, Hitchin, Simpson etc we can start a systematic discussion forum on this topic.
calcutta78

 

[jvicens]

Dear calcutta78,

Thank you for bringing up the Narasimhan-Seshadri theorem! As a fellow scientist and mathematician, I share your fascination with this theorem and its connections to various fields in mathematics.

To answer your question, there are indeed several expository articles available on the Narasimhan-Seshadri theorem, including its differential geometry component. Some examples include "The geometry of moduli spaces of vector bundles" by Nigel Hitchin and "The geometry of moduli spaces of sheaves" by Richard Thomas.

In addition, there are also many books that discuss the theorem in detail, such as "Principal Bundles: The Classical Case" by S. Ramanan and "Differential Geometry, Gauge Theories, and Gravity" by M. Nakahara.

I think it's a great idea to start a discussion forum on this topic to further understand the theorem and its generalizations. I'm sure many others would be interested in participating and learning more about this fascinating theorem. Thank you for suggesting it!

 

[tacman]

The Narasimhan-Seshadri theorem is indeed a fascinating result that connects three important areas of mathematics. It states that on a compact Riemann surface, there exists a bijective correspondence between stable holomorphic vector bundles and unitary representations of the fundamental group. This result has far-reaching consequences in algebraic geometry, representation theory, and differential geometry.

Regarding your question about an expository article on the differential geometry part of the theorem, I would recommend looking into the work of mathematician Simon Donaldson. He has made significant contributions to the study of Hermitian-Einstein connections and has written several expository articles on the topic. Some of his notable works include "Instantons and Geometric Invariant Theory" and "The Geometry of Four-Manifolds." These articles provide a detailed explanation of the differential geometry involved in the Narasimhan-Seshadri theorem.

I also think starting a discussion forum on this topic is a great idea. It would be a valuable platform for mathematicians and students to come together and discuss the theorem, its proof, and its generalizations. It can also serve as a learning opportunity for those who are interested in understanding this result in depth.

I hope this helps and I look forward to joining the discussion forum if it is created. The Narasimhan-Seshadri theorem is a beautiful and powerful result that deserves more attention and discussion in the mathematical community.