Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces

@inproceedings{Goldman2004RankOH,
  title={Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces},
  author={William M. Goldman and Eugene Z. Xia},
  year={2004}
}
Introduction Equivalences of deformation theories The Betti and de Rham deformation theories and their moduli spaces The Dolbeault groupoid Equivalence of de Rham and Dolbeault groupoids Hyperkahler geometry on the moduli space The twistor space The moduli space and the Riemann period matrix Bibliography. 

A weight two phenomenon for the moduli of rank one local systems on open varieties

The twistor space of representations on an open variety maps to a weight two space of local monodromy transformations around a divisor com- ponent at infinty. The space of �-invariant sections of

Moduli Spaces of Higgs Bundles – Old and New

  • J. Swoboda
  • Mathematics
    Jahresbericht der Deutschen Mathematiker-Vereinigung
  • 2021
We give an overview of the differential geometric and analytic aspects of Higgs bundles and their moduli spaces and highlight some of their interrelations with neighboring fields. We review various

Abelian and non-abelian cohomology

We place the representation variety in the broader context of abelian and nonabelian cohomology. We outline the equivalent constructions of the moduli spaces of flat bundles, of smooth integrable

Non-Abelian Hodge theory gives a real analytic isomorphism between two algebraically quite di erent varieties associated to a Riemann surface X : The Betti moduli

Non-Abelian Hodge theory gives a real analytic isomorphism between two algebraically quite di erent varieties associated to a Riemann surface X: The Betti moduli space, the moduli space of

Recent results and conjectures on the non abelian Hodge theory of curves

We give an introduction to non-abelian Hodge theory for curves with the aim of stating the $$P = W$$P=W conjecture both in its original cohomological version and in the more recent geometric one, and

On the geometric P=W conjecture

We formulate the geometric P=W conjecture for singular character varieties. We establish it for compact Riemann surfaces of genus one, and obtain partial results in arbitrary genus. To this end, we

Hitchin's equations and Fourier transform on curves

These are informal notes to my talks at the Lie algebras and moduli spaces seminar of the Eotvos Lorand University of Budapest in March 2007. First, we review some standard facts and constructions

Deligne pairings and families of rank one local systems on algebraic curves

For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting,

METRICS AND SPECIAL KÄHLER GEOMETRY ON THE MODULI SPACES OF HIGGS BUNDLES AND HITCHIN SYSTEMS

  • Zhenxiao Huang
  • Mathematics
    Bulletin of the Australian Mathematical Society
  • 2019
The notions of Hitchin systems and Higgs bundles (also called Higgs pairs) were introduced by N. Hitchin in 1987. They rapidly formed a subject lying on the crossroads of representation theory,

References

SHOWING 1-10 OF 57 REFERENCES

Fundamental Groups of Compact Kähler Manifolds

Introduction Fibering Kahler manifolds and Kahler groups The de Rham fundamental group $L^2$-cohomology of Kahler groups Existence theorems for harmonic maps Applications of harmonic maps Non-Abelian

Topology of U(2, 1) Representation Spaces

The Betti numbers of moduli spaces of representations of a universal central extension of a surface group in the groups U(2, 1) and SU(2, 1) are calculated. The results are obtained using the

The Yang-Mills equations over Riemann surfaces

  • M. AtiyahR. Bott
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1983
The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect' functional provided due account is taken of its gauge

THE BETTI NUMBERS OF THE MODULI SPACE OF STABLE RANK 3 HIGGS BUNDLES ON A RIEMANN SURFACE

In this paper we calculate the Poincaré polynomial of the moduli space of stable Higgs bundles of rank 3 on a Riemann surface Σ of genus g ≥ 2. This space was introduced by Hitchin in [10] and we

Surface group representations and U(p, q)-Higgs bundles

Using the L2 norm of the Higgs field as a Morse function, we study the moduli spaces of U(p, q)-Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but

Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces

1. Introduction 2. Riemann surfaces and integrable systems 3. Integrable systems and inverse scattering 4. Integrable systems and twistors Index

Lectures on Symplectic Manifolds

Introduction Symplectic manifolds and lagrangian submanifolds, examples Lagrangian splittings, real and complex polarizations, Kahler manifolds Reduction, the calculus of canonical relations,

Twisted harmonic maps and the self-duality equations

Si M est une surface de Riemann compacte, on utilise des resultats de la theorie des applications harmoniques pour trouver une solution aux equations d'auto-dualite associees a une representation

Hyperkähler metrics and supersymmetry

We describe two constructions of hyperkähler manifolds, one based on a Legendre transform, and one on a sympletic quotient. These constructions arose in the context of supersymmetric nonlinear
...