Branes and moduli spaces of Higgs bundles on smooth projective varieties (Q2050852)

Let \(M\) be a smooth complex projective variety and \(X\subset M\) a smooth closed curve such that the homomorphism of fundamental groups \(\pi_1(X)\longrightarrow\pi_1(M)\) is surjective. The aim of this paper is to study the restriction map of Higgs bundles, namely from the Higgs bundles on \(M\) to those on \(X\). In particular, the authors investigate the interplay between this restriction map and various types of branes contained in the moduli spaces of Higgs bundles on \(M\) and \(X\). They also consider the setup where a finite group is acting on \(M\) via holomorphic automorphisms or anti-holomorphic involutions, and the curve \(X\) is preserved by this action. Branes are studied in this context. This paper is organized as follows: Section 1, is an introduction to the subject and a description of the results obtained. Section 2 deals with nonabelian Hodge theory. Here the authors introduce Higgs bundles from a differential geometric perspective, and also the Betti moduli space of representations, and then give a brief description of nonabelian Hodge theory. Section 3 deals with restriction to curves. The authors dedicate this section to understanding the restriction of the ideas of Section 1 to hypersurfaces, which will become useful when studying Lagrangians within the moduli space of Higgs bundles. Section 4 deals with Deligne-Hitchin moduli space and twistor space. In this section the authors first recall the notion of \(\lambda\)-connections and the Deligne-Hitchin moduli space (whose details can be found in [\textit{C. Simpson}, Proc. Symp. Pure Math. 62, 217--281 (1997; Zbl 0914.14003)]). Then, they look into the relationship of these spaces with the twistor spaces of certain hyper-Kähler manifolds. Section 5 concerns real structures. Here the authors dedicate this section to the study of real structures and their fixed point sets, and in particular, their appearance in the context of Higgs bundles. Section 6 is devoted to holomorphic action of a finite group.