On Spectral Cover Equations in Simpson Integrable Systems

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...

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...

1995

This is the expanded text of a series of CIME lectures. We present an algebro-geometric approach to integrable systems, starting with those which can be described in terms of spectral curves. The prototype is Hitchin's system on the cotangent bundle of the moduli space of stable bundles on a curve. A variant involving meromorphic Higgs bundles specializes to many familiar

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.

Canadian Journal of Mathematics, 2015

We prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight systemis generic. When the genus is at least two, using this result we also prove a Torelli theoremfor the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise.

Notices of the American Mathematical Society

arXiv (Cornell University), 2013

Overview and statement of results Since Higgs bundles were introduced in 1987 [Hit87], they have found applications in many areas of mathematics and mathematical physics. In particular, Hitchin showed in [Hit87] that their moduli spaces give examples of Hyper-Kähler manifolds and that they provide an interesting example of integrable systems [Hit87a]. More recently, Hausel and Thaddeus [HT03] related Higgs bundles to mirror symmetry, and in the work of Kapustin and Witten [KW07] Higgs bundles were used to give a physical derivation of the geometric Langlands correspondence. The moduli space G of polystable G-Higgs bundles over a compact Riemann surface Σ, for G a real form of a complex semisimple Lie group G c , may be identified through non-abelian Hodge theory with the moduli space of representations of the fundamental group of Σ (or certain central extension of the fundamental group) into G (see [G-PGM09] for the Hitchin-Kobayashi correspondence for G-Higgs bundles). Motivated partially by this identification, the moduli space of G-Higgs bundles has been studied by various researchers, mainly through a Morse theoretic approach (see, for example, [BW12] for an expository article on applications of Morse theory to moduli spaces of Higgs bundles). Real forms of S L(n,) and G L(n,) were initially considered in [Hit87] and [Hit92], where Hitchin studied S L(2,)-Higgs bundles, and later on extended those results to

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.

2016

This paper uses Morse-theoretic techniques to compute the equivariant Betti numbers of the space of semistable rank two degree zero Higgs bundles over a compact Riemann surface, a method in the spirit of Atiyah and Bott's original approach for semistable holomorphic bundles. This leads to a natural proof that the hyperkähler Kirwan map is surjective for the non-fixed determinant case.

European Journal of Mathematics, 2017

We give a geometric characterisation of the topological invariants associated to SO(m, m + 1)-Higgs bundles through KO-theory and the Langlands correspondence between orthogonal and symplectic Hitchin systems. By defining the split orthogonal spectral data, we obtain a natural grading of the moduli space of SO(m, m + 1)-Higgs bundles.