Reading seminar on Higgs bundles
Reading seminar on Higgs bundles
Talk Schedule
...
Tentative plan
To properly motivate Higgs bundles from the differential geometric standpoint, one should start by learning about harmonic maps (from surfaces). I can suggest the following program.
As a starting point, see Chapter 2 of my thesis and this survey. One could also see the early chapters of Schoen-Yau's book.
If one wants to learn about the foundational existence theorem of Eells-Sampson (paper here) and more about the heat flow, I like this book.
Wolf's thesis on harmonic maps and Teichmüller theory is an essential read. it proves a special case of non-abelian Hodge (without the fancy language), and relates harmonic maps to the Thurston compactification of Teichmüller space.
After the items above, you're probably ready to dive into Higgs bundles, but it might be nice to look at some applications of harmonic maps. One could read one or any of...
1) Siu's groundbreaking paper on harmonic maps between Kähler manifolds and supperigidity for certain Kähler manifolds (including complex hyperbolic manifolds). This work has been generalized in many contexts. Notably, by Gromov and Schoen.
2) Wolf's very digestible proof of the Hubbard-Masur theorem here.
3) Schoen-Yau's foundational existence theorem for minimal surfaces, with geometric applications here.
4) A nice and short paper by Deroin-Tholozan that uses equivariant harmonic maps to solve a problem about anti-de Sitter geometry.
5) A recent paper that uses harmonic maps to study problems about the mapping class group.
Moving onto Higgs bundles and the non-abelian Hodge correspondence, there are a lot of nice surveys. After seeing harmonic maps, the more approachable ones would be
Chapter 3 of my survey here.
Donaldson's very short proof of existence of equivariant harmonic maps (i.e., the passage from representations to Higgs bundles in non-abelian Hodge), which is really just a small twist on the Eells-Sampson method, is at the very end of this paper.
To understand more about non-abelian Hodge, in particular the passage from Higgs bundles to representations,
I would start with Wentworth's survey and Hitchin's paper.
If you're feeling ambitious, you can look at Simpson's paper.
And if you need a reference for gauge theory, Chapter 2 of Donaldson and Kronheimer's book is probably enough.
From here, one could start looking at any paper that uses Higgs bundles. We can decide what to do. Some initial suggestions:
The construction of the Hitchin section in this paper, or, in the same paper, the use of Higgs bundles to count connected components of character varieties.
This (rather challenging) paper that relates harmonic maps and Higgs bundles to pleated surfaces (building on Minksy's thesis and the work of many others).
Something about pseudo-Riemannian geometry or higher Teichmüller theory, like this paper of Collier-Tholozan-Toulisse.
Of course we could also read papers with a much more algebraic flavour, or papers related to physics, like that of Gaiotto-Moore-Neitzke.