scholarly journals A BRIEF SURVEY OF HIGGS BUNDLES

2019 ◽  
Vol 26 (2) ◽  
pp. 197-214
Author(s):  
RONALD A. ZÚÑIGA ROJAS
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Mirror Symmetry ◽  
Higgs Field ◽  
Higgs Bundle ◽  
Higgs Bundles ◽  
Self Duality ◽  
Holomorphic Bundle ◽  
The Subject ◽  
Higgs Fields

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 space of Higgs bundles.

Mirror Symmetry with Branes by Equivariant Verlinde Formulas

2018 ◽  
pp. 189-218
Author(s):  
Tamás Hausel ◽  
Anton Mellit ◽  
Du Pei
Keyword(s):  
Functional Equation ◽  
Riemann Surface ◽  
Moduli Space ◽  
Mirror Symmetry ◽  
Higgs Bundles ◽  
Exterior Powers

This chapter finds an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the GL2 Higgs moduli space on a Riemann surface. On one side of the agreement, components of the Lagrangian brane of U(1,1) Higgs bundles, whose mirror was proposed by Hitchin to be certain even exterior powers of the hyperholomorphic Dirac bundle on the SL2 Higgs moduli space, are present. The agreement arises from a mysterious functional equation. This gives strong computational evidence for Hitchin’s proposal.

Model Higgs bundles in exceptional components of the Sp(4, ℝ)-character variety

2021 ◽  
pp. 2150067
Author(s):  
Georgios Kydonakis
Keyword(s):  
Elliptic Operator ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Higgs Bundle ◽  
Character Variety ◽  
Topological Invariants ◽  
Higgs Bundles ◽  
Connected Sum ◽  
Anosov Representations

We establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the [Formula: see text]-Hitchin equations using the linearization of a relevant elliptic operator. The construction can be used to provide model Higgs bundles in all the [Formula: see text] exceptional components of the maximal [Formula: see text]-Higgs bundle moduli space, which correspond to components solely consisting of Zariski dense representations. This also allows a comparison between the invariants for maximal Higgs bundles and the topological invariants for Anosov representations constructed by Guichard and Wienhard.

ON OVALS OF NON-CONJUGATE SYMMETRIES OF RIEMANN SURFACES

2009 ◽  
Vol 20 (01) ◽  
pp. 1-13 ◽  
Author(s):  
GRZEGORZ GROMADZKI ◽  
EWA KOZŁOWSKA-WALANIA
Keyword(s):  
Riemann Surface ◽  
Algebraic Structure ◽  
Upper Bound ◽  
Riemann Surfaces ◽  
Additional Assumption ◽  
The Subject

We find the upper bound for the total number of ovals of k (5 ≤ k ≤ 8) non-conjugate symmetries of a Riemann surface of genus g, without an additional assumption concerning their separabilities, filling this way the gap existing in known literature of the subject. We also prove that our bound is attained for infinitely many values of g. Finally, as a byproduct of our considerations, we obtain the algebraic structure of the group generated by the extremal configurations of the symmetries, provided it is a 2-group.

scholarly journals Uniformization of p-adic curves via Higgs–de Rham flows

2019 ◽  
Vol 2019 (747) ◽  
pp. 63-108
Author(s):  
Guitang Lan ◽  
Mao Sheng ◽  
Yanhong Yang ◽  
Kang Zuo
Keyword(s):  
Riemann Surfaces ◽  
Algebraic Closure ◽  
Higgs Field ◽  
Close Relation ◽  
Higgs Bundle ◽  
Generic Fiber ◽  
Hyperbolic Curve ◽  
De Rham ◽  
Associated Representation ◽  
Hyperbolic Riemann Surfaces

Abstract Let k be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve {X_{1}} defined over k, there exists a lifting X of the curve to the ring {W(k)} of Witt vectors as well as a lifting of the Higgs bundle to a periodic Higgs bundle over {X/W(k)} . These liftings give rise to a two-dimensional absolutely irreducible representation of the arithmetic fundamental group {\pi_{1}(X_{K})} of the generic fiber of X. This curve X and its associated representation is in close relation to the canonical curve and its associated canonical crystalline representation in the p-adic Teichmüller theory for curves due to S. Mochizuki. Our result may be viewed as an analogue of the Hitchin–Simpson’s uniformization theory of hyperbolic Riemann surfaces via Higgs bundles.

scholarly journals Uniformization of branched surfaces and Higgs bundles

2021 ◽  
pp. 2150096
Author(s):  
Indranil Biswas ◽  
Steven Bradlow ◽  
Sorin Dumitrescu ◽  
Sebastian Heller
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Cone Angle ◽  
Higgs Bundles ◽  
Constant Negative Curvature ◽  
Conical Singularities ◽  
Genus Formula ◽  
Conical Metric ◽  
Branched Surfaces ◽  
Cone Metric

Given a compact connected Riemann surface [Formula: see text] of genus [Formula: see text], and an effective divisor [Formula: see text] on [Formula: see text] with [Formula: see text], there is a unique cone metric on [Formula: see text] of constant negative curvature [Formula: see text] such that the cone angle at each point [Formula: see text] is [Formula: see text] [R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988) 222–224; M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821]. We describe the Higgs bundle on [Formula: see text] corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on [Formula: see text] parametrized by a nonempty open subset of [Formula: see text] that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin’s results in [N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126] for the case [Formula: see text].

scholarly journals Model Higgs Bundles in Exceptional Components of the Sp(4,R)-Character Variety

2018 ◽  
Vol 14 (2) ◽  
pp. 7744-7786 ◽  
Author(s):  
Georgios Kydonakis
Keyword(s):  
Elliptic Operator ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Higgs Bundle ◽  
Character Variety ◽  
Topological Invariants ◽  
Higgs Bundles ◽  
Connected Sum ◽  
Anosov Representations

We establish a gluing construction for Higgs bundles over a connected sum of Riemann  surfaces in terms of  solutions to the Sp(4,R)-Hitchin equations using the linearization of a relevant elliptic operator. The construction can be used to provide model Higgs bundles in all the 2g-3 exceptional components of the maximal Sp(4,R)-Higgs bundle moduli space, which correspond to components solely consisted of Zariski dense representations. This alsoallows a comparison between the invariants for maximal Higgs bundles and the topological invariants for Anosov representations constructed by O. Guichard and A. Wienhard.

scholarly journals BIRATIONAL EQUIVALENCE OF HIGGS MODULI

2005 ◽  
Vol 16 (04) ◽  
pp. 365-386
Author(s):  
MRIDUL MEHTA
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Compact Riemann Surface ◽  
Holomorphic Section ◽  
Higgs Bundle ◽  
Higgs Bundles ◽  
Birational Equivalence

In this paper, we study triples of the form (E, θ, ϕ) over a compact Riemann surface, where (E, θ) is a Higgs bundle and ϕ is a global holomorphic section of the Higgs bundle. Our main result is an description of a birational equivalence which relates geometrically the moduli space of Higgs bundles of rank r and degree d to the moduli space of Higgs bundles of rank r-1 and degree d.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

scholarly journals Local G2-manifolds, Higgs bundles and a colored quantum mechanics

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Max Hübner
Keyword(s):  
Quantum Mechanics ◽  
Gauge Theories ◽  
Supersymmetric Quantum Mechanics ◽  
Higgs Bundle ◽  
Higgs Bundles ◽  
View Point ◽  
Yang Mills ◽  
G2 Manifolds ◽  
Supersymmetric Gauge ◽  
M Theory

Abstract M-theory on local G2-manifolds engineers 4d minimally supersymmetric gauge theories. We consider ALE-fibered G2-manifolds and study the 4d physics from the view point of a partially twisted 7d supersymmetric Yang-Mills theory and its Higgs bundle. Euclidean M2-brane instantons descend to non-perturbative effects of the 7d supersymmetric Yang-Mills theory, which are found to be in one to one correspondence with the instantons of a colored supersymmetric quantum mechanics. We compute the contributions of M2-brane instantons to the 4d superpotential in the effective 7d description via localization in the colored quantum mechanics. Further we consider non-split Higgs bundles and analyze their 4d spectrum.

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