The geometry of generalized Hitchin systems (Q2709052)

The original Hitchin systems [\textit{N. Hitchin}, Duke Math. J 54, 91-114 (1987; Zbl 0627.14024)] are completely integrable Hamiltonian systems on the cotangent bundle of the moduli space of stable principal \(G\)-bundles (of fixed degree, for a complex reductive group \(G\)) over a compact Riemann surface \(\Sigma\). An element of this cotangent bundle can be thought of as a principal \(G\)-bundle \(E\to\Sigma\) equipped with a ``Higgs field'' \(\varphi\), a holomorphic 1-form over \(\Sigma\) with values in the coadjoint bundle of \(E\). The pair \((E,\varphi)\) is called a Higgs bundle. The total space has the natural symplectic structure of a cotangent bundle and the Hamiltonians which define the systems arise from the \(G\)-invariant polynomials of the coadjoint representation. The most important feature of these systems is that they are ``algebraically completely integrable'', meaning that their Lagrangian submanifolds are (open subsets) of abelian varieties. Hitchin used these systems for computing certain cohomology groups, but for some time it was unclear whether these systems had recognizable realisations as systems of differential equations. NEWLINENEWLINENEWLINEThe article under review is a relatively short, readable survey of the author's contributions towards understanding generalized Hitchin systems. These systems, introduced by \textit{F. Bottacin} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 28, No. 4, 391-433 (1995; Zbl 0864.14004)] and \textit{E. Markman} [Compos. Math. 93, 255-290 (1994; Zbl 0824.14013)], allow for the Higgs field to be meromorphic. The total space \(\mathcal{M}\) of these Higgs bundles is no longer a cotangent bundle, but it can be given a Poisson structure by realising it (over open sets) as the quotient of a cotangent bundle by the Hamiltonian action of a certain Lie group. The first section of this survey describes this in clear detail. This section shows that what we win from this generalization is the ability to recognize a number of familiar systems (for example, systems which arise from \(R\)-matrix structures on loop algebras) as generalized Hitchin systems. NEWLINENEWLINENEWLINEThe second section describes attempts by \textit{J. C. Hurtubise} and \textit{E. Markman} [Commun. Math. Phys. 223, 533-552 (2001; Zbl 0999.37048)] to use generalized Hitchin systems to give geometry to the Calogero-Moser systems: There is one of these for each root system. The article discusses how this is done by using, instead of the natural semi-simple Lie group associated to each root system, a semi-direct product of the maximal torus and the sum of the root spaces. This approach produces a phase space for Calogero-Moser but does not give a clean explanation of why it is an algebraically completely integrable system. One gets the impression that the Calogero-Moser systems are yet to be fully understood. NEWLINENEWLINENEWLINEThe third section discusses the role of \(Gl(N)\)-Hitchin systems and ``abelianisation''. This is the process of associating to each Higgs bundle \((E,\varphi)\) (which we now think of as the associated rank \(n\) vector bundle equipped with the Higgs field) a line bundle over a covering curve \(\pi: S\to\Sigma\), called the spectral curve since it is the curve of eigenvalues of the Higgs field. This curve lives in the total space of the line bundle \(K_\Sigma(D)\) of holomorphic 1-forms with divisor of poles no worse than \(D\). The Jacobi variety of \(S\) (or, to be precise, a Zariski open subset of it) lives as a Lagrangian submanifold of the Poisson manifold \(\mathcal{M}\): This is how one sees that the Hitchin systems are algebraically completely integrable. The author then explains how on each symplectic leaf of \(\mathcal{M}\) the family of spectral curves is encoded by a surface \(Q\). Each spectral curve embeds in \(Q\) and there is a symplectic isomorphism from a Hilbert scheme of points on \(Q\) to the family of Jacobi varieties of the spectral curves: When restricted to any spectral curve this map is just the Abel map. This isomorphism provides a way of separating variables for the Hamiltonian system. NEWLINENEWLINENEWLINEThe final section is a brief review of what is known about abelianisation when the group is an arbitrary reductive group.NEWLINENEWLINEFor the entire collection see [Zbl 0952.00031].