Topological mirror symmetry for parabolic Higgs bundles

We prove the topological mirror symmetry conjecture of Hausel-Thaddeus (Hausel and Thaddeus, 2001, 2003) for the moduli space of strongly parabolic Higgs bundles of rank two and three, with full flags, for any generic weights. Although the main theorem is proved only for rank at most three, most of the results are proved for any prime rank

Quiver bundles and wall crossing for chains

Holomorphic chains on a Riemann surface arise naturally as fixed points of the natural C-action on the moduli space of Higgs bundles. In this paper we associate a new quiver bundle to the Hom-complex of two chains, and prove that stability of the chains implies stability of this new quiver bundle. Our approach uses the Hitchin-Kobayashi correspondence for quiver bundles. Moreover, we use our result to give a new proof of a key lemma on chains (due to alvarez-Consul-Garcia-Prada-Schmitt), which has been important in the study of Higgs bundle moduli; this proof relies on stability and thus avoids the direct use of the chain vortex equations

Stratifications on the nilpotent cone of the moduli space of Hitchin pairs

We consider the problem of finding the limit at infinity (corresponding to the downward Morse flow) of a Higgs bundle in the nilpotent cone under the natural C*-action on the moduli space. For general rank we provide an answer for Higgs bundles with regular nilpotent Higgs field, while in rank three we give the complete answer. Our results show that the limit can be described in terms of data defined by the Higgs field, via a filtration of the underlying vector bundle

On Eulers Rotation Theorem

Summary: It is well known that a rigid motion of the Euclidean plane can be written as the composition of at most three reflections. It is perhaps not so widely known that a rigid motion of n-dimensional Euclidean space can be written as the composition of at most n + 1 reflections. The purpose of the present article is, firstly, to present a natural proof of this result in dimension 3 by explicitly constructing a suitable sequence of reflections, and, secondly, to show how a careful analysis of this construction provides a quick and pleasant geometric path to Eulers rotation theorem, and to the complete classification of rigid motions of space, whether orientation preserving or not. We believe that our presentation will highlight the elementary nature of the results and hope that readers, perhaps especially those more familiar with the usual linear algebra approach, will appreciate the simplicity and geometric flavor of the arguments

Homotopy type of moduli spaces of G-Higgs bundles and reducibility of the nilpotent cone

Let G be a real reductive Lie group, and H-C the complexification of its maximal compact subgroup H subset of G. We consider classes of semistable G-Higgs bundles over a Riemann surface X of genus g >= 2 whose underlying H-C-principal bundle is unstable. This allows us to find obstructions to a deformation retract from the moduli space of G-Higgs bundles over X to the moduli space of H-C-bundles over X, in contrast with the situation when g = 1, and to show reducibility of the nilpotent cone of the moduli space of G-Higgs bundles, for G complex

Birationality of moduli spaces of twisted U(p, q)-Higgs bundles

A U(p, q)-Higgs bundle on a Riemann surface (twisted by a line bundle) consists of a pair of holomorphic vector bundles, together with a pair of (twisted) maps between them. Their moduli spaces depend on a real parameter alpha. In this paper we study wall crossing for the moduli spaces of alpha-polystable twisted U(p, q)-Higgs bundles. Our main result is that the moduli spaces are birational for a certain range of the parameter and we deduce irreducibility results using known results on Higgs bundles. Quiver bundles and the Hitchin-Kobayashi correspondence play an essential role

Unramified covers and branes on the Hitchin system

We study the locus of the moduli space of GL(n, C)-Higgs bundles on a curve given by those Higgs bundles obtained by pushforward under a connected unramified cover. We equip these loci with a hyperholomorphic bundle so that they can be viewed as BBB-branes, and we introduce corresponding BAA-branes which can be described via Hecke modifications. We then show how these branes are naturally dual via explicit Fourier-Mukai transform (recall that GL(n, C) is Langlands self dual). It is noteworthy that these branes lie over the singular locus of the Hitchin fibration. As a particular case, our construction describes the behavior under mirror symmetry of the fixed loci for the action of tensorization by a line bundle of order n. These loci play a key role in the work of Hausel and Thaddeus on topological mirror symmetry for Higgs moduli spaces. (C) 2020 Published by Elsevier Inc

Stratifications on the moduli space of Higgs bundles

The moduli space of Higgs bundles has two stratifications. The Bialynicki-Birula stratification comes from the action of the non-zero complex numbers by multiplication on the Higgs field, and the Shatz stratification arises from the Harder-Narasimhan type of the vector bundle underlying a Higgs bundle. While these two stratifications coincide in the case of rank two Higgs bundles, this is not the case in higher rank. In this paper we analyze the relation between the two stratifications for the moduli space of rank three Higgs bundles

Higgs bundles for the non-compact dual of the special orthogonal group

Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group . In this paper we examine the case . We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this and Morse theory in the moduli space of Higgs bundles, we show that the moduli space is connected in this maximal Toledo case. The Morse theory also allows us to show connectedness when the Toledo invariant is zero. The correspondence between Higgs bundles and surface group representations thus allows us to count the connected components with zero and maximal Toledo invariant in the moduli space of representations of the fundamental group of the surface in SO* (2n)