ON MODULI SPACE OF HIGGS Gp(2n, C)-BUNDLES OVER A RIEMANN SURFACE (original) (raw)
Related papers
Communications in Mathematical Physics, 2001
The moduli space of triples of the form (E, θ, s) are considered, where (E, θ) is a Higgs bundle on a fixed Riemann surface X, and s is a nonzero holomorphic section of E. Such a moduli space admits a natural map to the moduli space of Higgs bundles simply by forgetting s. If (Y, L) is the spectral data for the Higgs bundle (E, θ), then s defines a section of the line bundle L over Y. The divisor of this section gives a point of a Hilbert scheme, parametrizing 0-dimensional subschemes of the total space of the canonical bundle K X , since Y is a curve on K X. The main result says that the pullback of the symplectic form on the moduli space of Higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the Hilbert scheme using the map that sends any triple (E, θ, s) to the divisor of the corresponding section of the line bundle on the spectral curve.
Symplectic Structures on Moduli Spaces of Parabolic Higgs Bundles and Hilbert Scheme
Communications in Mathematical Physics, 2003
Parabolic triples of the form (E * , θ, σ) are considered, where (E * , θ) is a parabolic Higgs bundle on a given compact Riemann surface X with parabolic structure on a fixed divisor S, and σ is a nonzero section of the underlying vector bundle. Sending such a triple to the Higgs bundle (E * , θ) a map from the moduli space of stable parabolic triples to the moduli space of stable parabolic Higgs bundles is obtained. The pull back, by this map, of the symplectic form on the moduli space of stable parabolic Higgs bundles will be denoted by dΩ ′. On the other hand, there is a map from the moduli space of stable parabolic triples to a Hilbert scheme Hilb δ (Z), where Z denotes the total space of the line bundle K X ⊗ O X (S), that sends a triple (E * , θ, σ) to the divisor defined by the section σ on the spectral curve corresponding to the parabolic Higgs bundle (E * , θ). Using this map and a meromorphic one-form on Hilb δ (Z), a natural two-form on the moduli space of stable parabolic triples is constructed. It is shown here that this form coincides with the above mentioned form dΩ ′ .
Symplectic structures on moduli spaces of parabolic Higgs and Hilbert scheme
2001
Parabolic triples of the form (E * , θ, σ) are considered, where (E * , θ) is a parabolic Higgs bundle on a given compact Riemann surface X with parabolic structure on a fixed divisor S, and σ is a nonzero section of the underlying vector bundle. Sending such a triple to the Higgs bundle (E * , θ) a map from the moduli space of stable parabolic triples to the moduli space of stable parabolic Higgs bundles is obtained. The pull back, by this map, of the symplectic form on the moduli space of stable parabolic Higgs bundles will be denoted by dΩ ′. On the other hand, there is a map from the moduli space of stable parabolic triples to a Hilbert scheme Hilb δ (Z), where Z denotes the total space of the line bundle K X ⊗ O X (S), that sends a triple (E * , θ, σ) to the divisor defined by the section σ on the spectral curve corresponding to the parabolic Higgs bundle (E * , θ). Using this map and a meromorphic one-form on Hilb δ (Z), a natural two-form on the moduli space of stable parabolic triples is constructed. It is shown here that this form coincides with the above mentioned form dΩ ′ .
Quantization of a Moduli Space of Parabolic Higgs Bundles
International Journal of Mathematics, 2004
Let [Formula: see text] be a moduli space of stable parabolic Higgs bundles of rank two over a Riemann surface X. It is a smooth variety defined over [Formula: see text] equipped with a holomorphic symplectic form. Fix a projective structure [Formula: see text] on X. Using [Formula: see text], we construct a quantization of a certain Zariski open dense subset of the symplectic variety [Formula: see text].
Geometry of moduli spaces of Higgs bundles
Communications in Analysis and Geometry, 2006
We construct a Petersson-Weil type Kähler form on the moduli spaces of Higgs bundles over a compact Kähler manifold. A fiber integral formula for this form is proved, from which it follows that the Petersson-Weil form is the curvature of a certain determinant line bundle, equipped with a Quillen metric, on the moduli space of Higgs bundles over a projective manifold. The curvature of the Petersson-Weil Kähler form is computed. We also show that, under certain assumptions, a moduli space of Higgs bundles supports of natural hyper-Kähler structure.
Archiv der Mathematik, 2003
Fix a smooth very ample curve C on a K3 or abelian surface X. Let M denote the moduli space of pairs of the form (F, s), where F is a stable sheaf over X whose Hilbert polynomial coincides with that of the direct image, by the inclusion map of C in X, of a line bundle of degree d over C, and s is a nonzero section of F. Assume d to be sufficiently large such that F has a nonzero section. The pullback of the Mukai symplectic form on moduli spaces of stable sheaves over X is a holomorphic 2-form on M. On the other hand, M has a map to a Hilbert scheme parametrizing 0-dimensional subschemes of X that sends (F, s) to the divisor, defined by s, on the curve defined by the support of F. We prove that the above 2-form on M coincides with the pullback of the symplectic form on Hilbert scheme.
A Brief Survey of Higgs Bundles
Revista de Matemática: Teoría y Aplicaciones, 2019
Considering a compact Riemann surface of genus greater or equal than two, a Higgs bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin’s work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as Higgs fields because in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief but complete construction of the moduli spa...
A Brief Survey of Higgs Bundles Un Estudio Conciso De Fibrados De Higgs
2019
Considering a compact Riemann surface of genus greater or equal than two, a Higgs bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin’s work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as Higgs fields because in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief but complete construction of the moduli spa...
Anti-holomorphic involutions of the moduli spaces of Higgs bundles
Journal de l’École polytechnique — Mathématiques, 2015
We study anti-holomorphic involutions of the moduli space of G-Higgs bundles over a compact Riemann surface X, where G is a complex semisimple Lie group. These involutions are defined by fixing anti-holomorphic involutions on both X and G. We analyze the fixed point locus in the moduli space and their relation with representations of the orbifold fundamental group of X equipped with the antiholomorphic involution. We also study the relation with branes. This generalizes work by Biswas-García-Prada-Hurtubise and Baraglia-Schaposnik.
On Higgs Bundles on Elliptic Surfaces
The Quarterly Journal of Mathematics, 2015
Let π : X → C be a relatively minimal non-isotrivial elliptic surface over the field of complex numbers, where g(C) ≥ 2. In this article, we demonstrate an equivalence between the category of semistable parabolic Higgs bundles on C, and the category of semistable Higgs bundles on X with vanishing second Chern class, and determinant a vertical divisor. Contents 1. Introduction 1 Motivation and statement of results 1 Strategy 2 Related work and further comments 3 Acknowledgement 3 2. Preliminaries 3 2.1. Elliptic surfaces 4 2.2. Vertical Bundles 5 3. Main Theorem 8 3.1. Proof of Theorem 1 in the case of no multiple fibers 9 3.2. Proof of Theorem 1 in the case of multiple fibers 13 References 15