A Torelli Theorem for the Moduli Space of Higgs Bundles on a Curve (original) (raw)

Branes and moduli spaces of Higgs bundles on smooth projective varieties

Research in the Mathematical Sciences, 2021

Given a smooth complex projective variety M and a smooth closed curve X ⊂ M such that the homomorphism of fundamental groups π 1 (X) −→ π 1 (M) is surjective, we study the restriction map of Higgs bundles, namely from the Higgs bundles on M to those on X. In particular, we investigate the interplay between this restriction map and various types of branes contained in the moduli spaces of Higgs bundles on M and X. We also consider the setup where a finite group is acting on M via holomorphic automorphisms or anti-holomorphic involutions, and the curve X is preserved by this action. Branes are studied in this context.

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.

Higgs bundles over elliptic curves

2013

In this paper we study GGG-Higgs bundles over an elliptic curve when the structure group GGG is a classical complex reductive Lie group. Modifying the notion of family, we define a new moduli problem for the classification of semistable GGG-Higgs bundles of a given topological type over an elliptic curve and we give an explicit description of the associated moduli space as a finite quotient of a product of copies of the cotangent bundle of the elliptic curve. We construct a bijective morphism from this new moduli space to the usual moduli space of semistable GGG-Higgs bundles, proving that the former is the normalization of the latter. We also obtain an explicit description of the Hitchin fibration for our (new) moduli space of GGG-Higgs bundles and we study the generic and non-generic fibres.

ON MODULI SPACE OF HIGGS Gp(2n, C)-BUNDLES OVER A RIEMANN SURFACE

Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denoted by K X. Let M H denote the moduli space of semistable Higgs Gp(2n, C)-bundles over X of fixed topological type. The complex variety M H has a natural holomorphic symplectic structure. On the other hand, for any ≥ 1, the Liouville symplectic from on the total space of K X defines a holomorphic symplectic structure on the Hilbert scheme Hilb (K X) parametrizing the zero-dimensional subschemes of K X. We relate the symplectic form on Hilb (K X) with the symplectic form on M H .

A compactification of the moduli space of principal Higgs bundles over singular curves

2011

A principal Higgs bundle (P,phi)(P,\phi)(P,phi) over a singular curve XXX is a pair consisting of a principal bundle PPP and a morphism phi:XtotextAdPotimesOmega1X\phi:X\to\text{Ad}P \otimes \Omega^1_Xphi:XtotextAdPotimesOmega1X. We construct the moduli space of principal Higgs G-bundles over an irreducible singular curve XXX using the theory of decorated vector bundles. More precisely, given a faithful representation rho:GtoSl(V)\rho:G\to Sl(V)rho:GtoSl(V) of GGG, we consider principal Higgs bundles as triples (E,q,phi)(E,q,\phi)(E,q,phi) where EEE is a vector bundle with rkE=dimV\rk{E}=\dim VrkE=dimV over the normalization xtilde\xtildextilde of XXX, qqq is a parabolic structure on EEE and phi:EabtoL\phi:E\ab{}\to Lphi:EabtoL is a morphism of bundles, being LLL a line bundle and Eabdoteqdot(Eotimesa)oplusbE\ab{}\doteqdot (E^{\otimes a})^{\oplus b}Eabdoteqdot(Eotimesa)oplusb a vector bundle depending on the Higgs field phi\phiphi and on the principal bundle structure. Moreover we show that this moduli space for suitable integers a,ba,ba,b is related to the space of framed modules.

On the rational homotopy type of a moduli space of vector bundles over a curve

Communications in Analysis and Geometry, 2008

We study the rational homotopy of the moduli space N X of stable vector bundles of rank two and fixed determinant of odd degree over a compact connected Riemann surface X of genus g, with g ≥ 2. The symplectic group Aut(H 1 (X, Z)) ∼ = Sp(2g, Z) has a natural action on the rational homotopy groups π n (N X)⊗ Z Q. We prove that this action extends to an action of Sp(2g, C) on π n (N X)⊗ Z C. We also show that π n (N X)⊗ Z C is a non-trivial representation of Sp(2g, C) ∼ = Aut(H 1 (X, C)) for all n ≥ 2g −1. In particular, N X is a rationally hyperbolic space. In the special case where g = 2, for each n ∈ N, we compute the leading Sp(2g, C)-representation occurring in π n (N X) ⊗ Z C.

On rational maps between moduli spaces of curves and of vector bundles

arXiv: Algebraic Geometry, 2009

Let SU_C(2) be the moduli space of rank 2 semistable vector bundles with trivial de terminant on a smooth complex algebraic curve C of genus g > 1, we assume C non-hyperellptic if g > 2. In this paper we construct large families of pointed rational normal curves over certain linear sections of SU_C(2). This allows us to give an interpretation of these subvarieties of SUC(2) in terms of the moduli space of curves M_{0,2g}. In fact, there exists a natural linear map SU_C(2) -> P^g with modular meaning, whose fibers are birational to M_{0,2g}, the moduli space of 2g-pointed genus zero curves. If g < 4, these modular fibers are even isomorphic to the GIT compactification M^{GIT}_{0,2g}. The families of pointed rational normal curves are recovered as the fibers of the maps that classify extensions of line bundles associated to some effective divisors.

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...

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...