I am a theoretical physics major student working on string theory. I want to understand the work of MF Atiyah and R Bott, "The Yang-Mills equations over riemann surfaces" . What kinds of mathematical background does it need? books or papers? (I only learned Nakahara's book on geometry)
Thanks in advance.
Atiyah and Bott’s 1983 Paper – Key Insights
dg.differential-geometrygeometryreference-request
Best Answer
This is one very beautiful, influential and very challenging paper. You need to know differential geometry, some topology (cohomology, characteristic classes a bit of Morse theory), some algebraic geometry, and a bit of analysis (elliptic operators and complexes).
There is no one book that contains all of these, though Frankel's book The Geometry of Physics would be a good book to have by your side. Perhaps the best way to go is start reading the paper and whenever you get stuck get help from books and, better yet, an experienced mathematician. Early on, you'll get stuck quite often, so you need to stay cool and persevere. It beats spending a year reading all the required material. In any case, I want to emphasize that having adult supervision would make this experience more bearable and much more enriching.