99 research outputs found
Higgs bundles and higher Teichm\"uller spaces
This paper is a survey on the role of Higgs bundle theory in the study of
higher Teichm\"uller spaces. Recall that the Teichm\"uller space of a compact
surface can be identified with a certain connected component of the moduli
space of representations of the fundamental group of the surface into
. Higher Teichm\"uller spaces correspond to
special components of the moduli space of representations when one replaces
by a real non-compact semisimple Lie group of
higher rank. Examples of these spaces are provided by the Hitchin components
for split real groups, and maximal Toledo invariant components for groups of
Hermitian type. More recently, the existence of such components has been proved
for , in agreement with the conjecture of Guichard and
Wienhard relating the existence of higher Teichm\"uller spaces to a certain
notion of positivity on a Lie group that they have introduced. We review these
three different situations, and end up explaining briefly the conjectural
general picture from the point of view of Higgs bundle theory.Comment: arXiv admin note: substantial text overlap with arXiv:1511.0775
Connectedness of Higgs bundle moduli for complex reductive Lie groups
We carry an intrinsic approach to the study of the connectedness of the
moduli space of -Higgs bundles, over a compact Riemann
surface, when is a complex reductive (not necessarily connected) Lie group.
We prove that the number of connected components of is indexed
by the corresponding topological invariants. In particular, this gives an
alternative proof of the counting by J. Li of the number of connected
components of the moduli space of flat -connections in the case in which
is connected and semisimple.Comment: Due to some mistake the authors did not appear in the previous
version. Fixed this. Final version; to appear in the Asian Journal of
Mathematics. 19 page
The y-genus of the moduli space of PGL_n-Higgs bundles on a curve (for degree coprime to n)
Building on our previous joint work with A. Schmitt [7] we explain a
recursive algorithm to determine the cohomology of moduli spaces of Higgs
bundles on any given curve (in the coprime situation). As an application of the
method we compute the y-genus of the space of PGL_n-Higgs bundles for any rank
n, confirming a conjecture of T. Hausel.Comment: 13 page
Anti-holomorphic involutions of the moduli spaces of Higgs bundles
We study anti-holomorphic involutions of the moduli space of principal
-Higgs bundles over a compact Riemann surface , where is a complex
semisimple Lie group. These involutions are defined by fixing anti-holomorphic
involutions on both and . We analyze the fixed point locus in the moduli
space and their relation with representations of the orbifold fundamental group
of equipped with the anti-holomorphic involution. We also study the
relation with branes. This generalizes work by
Biswas--Garc\'{\i}a-Prada--Hurtubise and Baraglia--Schaposnik.Comment: Final version; to appear in Journal de l'\'Ecole
polytechnique--Math\'ematique
Higgs bundles for the non-compact dual of the unitary group
Using Morse-theoretic techniques, we show that the moduli space of
U*(2n)-Higgs bundles over a compact Riemann surface is connected.Comment: 20 pages; v2: several improvements and corrections; main results are
unchange
Hitchin-Kobayashi correspondence, quivers, and vortices
A twisted quiver bundle is a set of holomorphic vector bundles over a complex
manifold, labelled by the vertices of a quiver, linked by a set of morphisms
twisted by a fixed collection of holomorphic vector bundles, labelled by the
arrows. When the manifold is Kaelher, quiver bundles admit natural
gauge-theoretic equations, which unify many known equations for bundles with
extra structure. In this paper we prove a Hitchin--Kobayashi correspondence for
twisted quiver bundles over a compact Kaehler manifold, relating the existence
of solutions to the gauge equations to a stability criterion, and consider its
application to a number of situations related to Higgs bundles and dimensional
reductions of the Hermitian--Einstein equations.Comment: 28 pages; larger introduction, added references for the introduction,
added a short comment in Section 1, typos corrected, accepted in Comm. Math.
Phy
Higgs bundles, abelian gerbes and cameral data
We study the Hitchin map for -Higgs bundles on a smooth
curve, where is a quasi-split real form of a complex reductive
algebraic group . By looking at the moduli stack of regular
-Higgs bundles, we prove it induces a banded gerbe structure on
a slightly larger stack, whose band is given by sheaves of tori. This
characterization yields a cocyclic description of the fibres of the
corresponding Hitchin map by means of cameral data. According to this, fibres
of the Hitchin map are categories of principal torus bundles on the cameral
cover. The corresponding points inside the stack of -Higgs bundles are
contained in the substack of points fixed by an involution induced by the
Cartan involution of . We determine this substack of fixed
points and prove that stable points are in correspondence with stable
-Higgs bundles.Comment: 34 page
K\"ahler-Yang-Mills Equations and Vortices
The K\"ahler-Yang-Mills equations are coupled equations for a K\"ahler metric
on a compact complex manifold and a connection on a complex vector bundle over
it. After briefly reviewing the main aspects of the geometry of the
K\"ahler-Yang-Mills equations, we consider dimensional reductions of the
equations related to vortices - solutions to certain Yang-Mills-Higgs
equations.Comment: Paper in honour of Jean-Pierre Bourguigno
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