Higgs Bundles—Recent Applications (original) (raw)

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

Advanced topics in gauge theory: Mathematics and Physics of Higgs bundles

Quantum Field Theory and Manifold Invariants, 2021

These notes have been prepared as reading material for the mini-course given by the author at the 2019 Graduate Summer School at Park City Mathematics Institute-Institute for Advanced Study. We begin by introducing Higgs bundles and their main properties (Lecture 1), and then we discuss the Hitchin fibration and its different uses (Lecture 2). The second half of the course is dedicated to studying different types of subspaces (branes) of the moduli space of complex Higgs bundles, their appearances in terms of flat connections and representations (Lecture 3), as well as correspondences between them (Lecture 4). Contents 1 Introduction 1 2 The geometry of the moduli space of Higgs bundles 3 3 The geometry of the Hitchin fibration 15 4 Branes in the moduli space of Higgs bundles 26 5 Higgs bundles and correspondences 39

Spectral data for G-Higgs bundles

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

Higgs bundles without geometry

arXiv: Algebraic Geometry, 2020

Higgs bundles appeared a few decades ago as solutions to certain equations from physics and have attracted much attention in geometry as well as other areas of mathematics and physics. Here, we take a very informal stroll through some aspects of linear algebra that anticipate the deeper structure in the moduli space of Higgs bundles. (This note was produced for the MFO Snapshots of Modern Mathematics series, which is "designed to promote the understanding and appreciation of modern mathematics and mathematical research in the interested public world-wide.")

Cayley and Langlands type correspondences for orthogonal Higgs bundles

Transactions of the American Mathematical Society, 2018

Through Cayley and Langlands type correspondences, we give a geometric description of the moduli spaces of real orthogonal and symplectic Higgs bundles of any signature in the regular fibres of the Hitchin fibration. As applications of our methods, we complete the concrete abelianization of real slices corresponding to all quasi-split real forms, and describe how extra components emerge naturally from the spectral data point of view.

A geometric approach to orthogonal Higgs bundles

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.

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.

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 .

Higgs bundles and applications

arXiv: Algebraic Geometry, 2016

This short note gives an overview of how a few conjectures and theorems of the author and collaborators fit together. It was prepared for Oberwolfach's workshop Differentialgeometrie im Gro{\ss}en, 28 June - 4 July 2015, and contains no new results.

Higgs Bundles—Recent Applications (original) (raw)

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