Abstract
The paper studies the locus in the moduli space of rank 2 Higgs bundles over a curve of genus g corresponding to points which are critical for d of the Poisson commuting functions defining the integrable system. These correspond to the Higgs field vanishing on a divisor D of degree d. The degree d critical locus is shown to have an induced integrable system related to K(−D)-twisted Higgs bundles. It is embedded in the singular part of the fibration and a description of these singular fibres using Hecke curves is given. The methods used are also applied to give information about the cohomology classes of components of the nilpotent cone and of the first critical locus. It is shown that whereas in the extreme case d = 2g − 2 the locus is a hyperkähler submanifold this does not hold in general. The example of genus 2 is studied concretely and the d = 1 integrable system is seen to be described by a pencil of Kummer surfaces.
Highlights
The moduli space of Higgs bundles over a curve has played a fundamental role in recent work relating Langlands duality and mirror symmetry [17]
Recall that an integrable system is a symplectic manifold M2m with a proper map F : M → B to a manifold Bm, the base of the system, which over an open set in B is a fibration by Lagrangian submanifolds
We show that the critical locus Cd ⊂ M occurs where the Higgs field vanishes on a divisor D of degree d, and the nondegenerate points correspond to having simple zeros and semisimple derivative at those zeros
Summary
The moduli space of Higgs bundles over a curve has played a fundamental role in recent work relating Langlands duality and mirror symmetry [17]. For an analytic integrable system there is a notion of nondegeneracy of a point on the critical locus. In this paper we shall apply this criterion and study the consequences in the case of the moduli space of rank 2 Higgs bundles over a curve of genus g ≥ 2. This is a holomorphic integrable system and so analyticity automatically holds. It follows that we can identify the torus fibres of the subintegrable system as Prym varieties of spectral curves for the K (−D)twisted Higgs bundle (V, /s) where s is the canonical section of O(D) vanishing on D. Using the explicit analytical expressions for the integrable system obtained in [18] we have equations for these
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