On Higgs Bundles on Nodal Curves

Abstract

This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and L-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic bundles and generalised parabolic Ltwisted Hitchin pairs on its normalisation as well as on an analogue of the Hitchin map for generalised parabolic L-twisted Hitchin pairs.

Families of Hitchin systems and N=2 theories

“…There has been considerable prior work on studying the moduli of bundles and parabolic bundles on nodal curves (see, for example, [20][21][22][23]). There has been some recent progress on extending some of these results to Higgs bundles on nodal curves [24][25][26]. See also [27,28] for some earlier work in this direction.…”

“…So, we develop this from the basics. Relating our work to the framework of [24][25][26] is an interesting direction for future work.…”

“…subject to the relation (25). Restricting to any given C λ , we get the additional relation (26), which cuts the dimension of the space of φ 3 s down to 3. Again, we'll choose a trivialization of π * L 3 which is good everywhere but at λ = ∞ and write…”

On character varieties of singular manifolds

“…Nevertheless, there is still some hope in recovering a sort of non-abelian Hodge correspondence in the nodal setting if appropriate framings for the flat vector bundle on the singular points are fixed, as sketched to some extent in [2] but the problem is still open. For a survey regarding this interplay on nodal curves, see [25].…”