Hitchin systems and their quantization

This is an expanded version of the notes by the second author of the lectures on Hitchin systems and their quantization given by the first author at the Beijing Summer Workshop in Mathematics and Mathematical Physics ``Integrable Systems and Algebraic Geometry" (BIMSA-2024).

💡 Research Summary

The manuscript is an expanded set of lecture notes that give a comprehensive introduction to Hitchin systems and their quantization, aimed at graduate students and researchers interested in the interplay between algebraic geometry, integrable systems, and representation theory. The authors begin by recalling the notion of a principal (G)-bundle on an algebraic variety, emphasizing the role of (\acute{e}tale) charts, clutching functions, and the classification of bundles via (H^{1}_{\acute{e}t}(X,G)). They discuss associated and induced bundles, the classification of line bundles through the Picard group, and the Serre GAGA theorem that equates algebraic and analytic bundles on complex projective varieties.

The second part of the notes moves to the moduli problem. For a smooth projective curve (X) and a split reductive group (G) over a field (k), the stack (\operatorname{Bun}_G(X)) of principal (G)-bundles is introduced. The authors explain how this stack can be realized as a double quotient of the loop group (G(\mathcal{K})) by its positive loop subgroup (G(\mathcal{O})) and a space of adèles, drawing an analogy with the adelic description of automorphic forms. They then restrict to the open substack (\operatorname{Bun}_G^\circ(X)) of stable bundles, whose cotangent bundle (T^*\operatorname{Bun}_G^\circ(X)) is the space of Higgs pairs ((E,\phi)). The Hitchin map \