ترغب بنشر مسار تعليمي؟ اضغط هنا

Surface group representations to $SL(2,{mathbb C})$ and Higgs bundles with smooth spectral data

214   0   0.0 ( 0 )
 نشر من قبل Richard Wentworth
 تاريخ النشر 2015
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

We show that for every nonelementary representation of a surface group into $SL(2,{mathbb C})$ there is a Riemann surface structure such that the Higgs bundle associated to the representation lies outside the discriminant locus of the Hitchin fibration.



قيم البحث

اقرأ أيضاً

179 - Brian Collier 2017
Recall that the group $PSL(2,mathbb R)$ is isomorphic to $PSp(2,mathbb R), SO_0(1,2)$ and $PU(1,1).$ The goal of this paper is to examine the various ways in which Fuchsian representations of the fundamental group of a closed surface of genus $g$ int o $PSL(2,mathbb R)$ and their associated Higgs bundles generalize to the higher rank groups $PSL(n,mathbb R), PSp(2n,mathbb R), SO_0(2,n), SO_0(n,n+1)$ and $PU(n,n)$. For the $SO_0(n,n+1)$-character variety, we parameterize $n(2g-2)$ new connected components as the total space of vector bundles over appropriate symmetric powers of the surface and study how these components deform in the $SO_0(n,n+2)$-character variety. This generalizes results of Hitchin for $PSL(2,mathbb R)$.
155 - Brian Collier 2017
We study the character variety of representations of the fundamental group of a closed surface of genus $ggeq2$ into the Lie group SO(n,n+1) using Higgs bundles. For each integer $0<dleq n(2g-2),$ we show there is a smooth connected component of the character variety which is diffeomorphic to the product of a certain vector bundle over a symmetric product of a Riemann surface with the vector space of holomorphic differentials of degree 2,4,...,2n-2. In particular, when d=n(2g-2), this recovers Hitchins parameterization of the Hitchin component. We also exhibit $2^{2g+1}-1$ additional connected components of the SO(n,n+1)-character variety and compute their topology. Moreover, representations in all of these new components cannot be continuously deformed to representations with compact Zariski closure. Using recent work of Guichard and Wienhard on positivity, it is shown that each of the representations which define singularities (i.e. those which are not irreducible) in these $2^{2g+1}-1$ connected components are positive Anosov representations.
Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental grou p of the surface. For surfaces of genus at least two, such orbits correspond to homomorphisms with finite image. For genus one, they correspond to the finite or special dihedral representations. We also obtain an analogous result for bounded orbits in the moduli space.
We develop a complete Hitchin-Kobayashi correspondence for twisted pairs on a compact Riemann surface X. The main novelty lies in a careful study of the the notion of polystability for pairs, required for having a bijective correspondence between sol utions to the Hermite-Einstein equations, on one hand, and polystable pairs, on the other. Our results allow us to establish rigorously the homemomorphism between the moduli space of polystable G-Higgs bundles on X and the character variety for representations of the fundamental group of X in G. We also study in detail several interesting examples of the correspondence for particular groups and show how to significantly simplify the general stability condition in these cases.
154 - Peter B. Gothen 2011
We give an overview of the work of Corlette, Donaldson, Hitchin and Simpson leading to the non-abelian Hodge theory correspondence between representations of the fundamental group of a surface and the moduli space of Higgs bundles. We then explain ho w this can be generalized to a correspondence between character varieties for representations of surface groups in real Lie groups G and the moduli space of G-Higgs bundles. Finally we survey recent joint work with Bradlow, Garcia-Prada and Mundet i Riera on the moduli space of maximal Sp(2n,R)-Higgs bundles.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا