Higgs Bundles: An Introduction Through Gauge Theory

This is a three-part mini-course on Higgs bundles that I thaught at the University of Lisbon in January of 2025.

Lecture Notes

Here are my handwritten notes of the lectures:

Lecture 1 - Holomorphic Vector Bundles & Higgs Bundles. [PDF]

Lecture 2 - Connections and Metrics. [PDF]

Lecture 3 - Non-abelian Hodge Correspondence. [PDF]

Main References

Lecture 1 - Mostly follows the book of R. C. Gunning [2] and part of the appendix by García-Prada of the book by R.O. Wells [4]. The treatment of the theorem of Riemann-Roch is in chapter 7 of [2], with the statement on page 111 and the definitions of point bundles just after.

Lecture 2 - This is mostly standard so again the introductory chapter of [4] and also Griffiths and Harris [1], chapter 0, part 5. The calculations with the language of connections are very well-explained in the book by Tu [3], in particular, the relation between linear differential operators and bundle maps is explained in chapter 1, part 7.

Lecture 3 - Again we mostly follow the appendix by García-Prada in [4].

[1] P. Griffiths & J. Harris; Principles of Algebraic Geometry, John Wiley & Sons, Ltd, 1994. [Link]

[2] R. C. Gunning; Lectures on Riemann Surfaces, Princeton University Press, Princeton, N.J., 1966. [PDF]

[3] L.W. Tu; Differential Geometry, Graduate Texts in Mathematics, vol 275, Springer, Cham., 2017. [Link]

[4] R.O. Wells; Differential Analysis on Complex Manifolds, Graduate Texts in Mathematics, vol 65. Springer, N.Y., 2008. [Link]

Other References

There are currently a lot of lecture notes on Higgs bundles. I also used these to prepare the mini-course.

Very nice overview of the theory:

Peter B. Gothen; Higgs bundles and the real symplectic group. AIP Conf. Proc. 5 July 2011; 1360 (1): 39–50. ArXiv

Details:

Olivier Guichard; An Introduction to the Differential Geometry of Flat Bundles and of Higgs Bundles, in The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles, World Sci. Publ., Hackensack, N.J., 2016. [PDF]

Qiongling Li; An Introduction to Higgs Bundles via Harmonic Maps, SIGMA 15 - 2019, 035, Contribution to the Special Issue on Geometry and Physics of Hitchin Systems. ArXiv

Our colleague Jean Douçot also found the following lecture notes to be pedagogical and helpful.

A. García-Raboso, S. Rayan.; Introduction to Nonabelian Hodge Theory. In: Laza, R., Schütt, M., Yui, N. (eds) Calabi-Yau Varieties: Arithmetic, Geometry and Physics. Fields Institute Monographs, vol 34. Springer, New York, 2015. [ArXiv]

You can also check this compilation of resources by Peter B. Gothen for the IST Courses on Algebraic Geometry - 2013. [PDF]

Or the publication page of Laura Schaposnik where you can find more recent writings about Higgs bundles. [Link]