Simpson, C.T. Higgs bundles and local systems. Publications Mathématiques de l'Institut des Hautes Scientifiques 75, 5-95 (1992). https://doi.org/10.1007/BF02699491. Download citation. Received: 04 May 1990. Revised: 09 April 1991. Issue Date: December 1992. DOI: https://doi.org/10.1007/BF0269949 HIGGS BUNDLES AND LOCAL SYSTEMS 5 Remark 2.1. We will mostly be dealing with connections on bundles that induce a xed connection on the determinant bundle. These will correspond, for example, to representations into SL n as opposed to GL n. In this case, the bundles g E and gC E should be taken to consist of traceless endomorphisms Higgs bundles and local systems on Riemann surfaces Richard A. Wentworth Department of Mathematics, University of Maryland, College Park, MD 20742, USA November 28, 202 Title: Higgs bundles and local systems on Riemann surfaces. Authors: Richard A. Wentworth. Download PDF Abstract: Lecture notes from the Third International School on Geometry and Physics at the Centre de Recerca Matematica in Barcelona, March 26--30, 2012. Comments Higgs bundles and local systems Carlos T. Simpson. Publications Mathématiques de l'IHÉS (1992) Volume: 75, page 5-95; ISSN: 0073-8301; Access Full Article top Access to full text Full (PDF) How to cite to
In mathematics, a Higgs bundle is a pair (,) consisting of a holomorphic vector bundle E and a Higgs field, a holomorphic 1-form taking values in End(E) such that =. Such pairs were introduced by Nigel Hitchin ( 1987 ), who named the field φ {\displaystyle \varphi } after Peter Higgs because of an analogy with Higgs bosons COHOMOLOGY SUPPORT LOCI FOR LOCAL SYSTEMS AND HIGGS BUNDLES 3 Theorem 1.5. When ˆis unitary, this degenerates at E 1 i.e. Hi(X;C ˆ) ˘= M p+q=i Hq(X; p X L ˆ) Proof. The connection rextends to C1forms ( E L ˆ) and its (0;1) part is @ . The left and right sides are given by H (( E L ˆ);r) and H (( E L ˆ);@ ) respectively
Download PDF: Sorry, we are unable to provide the full text but you may find it at the following location(s): http://arxiv.org/pdf/1402.4203... (external link G-Local Systems as Principal Bundles A G-Local System is the same thing as a Principal G-bundle described by transition functions that are locally constant, i.e. dg = 0 De nition (Flat bundles) For a bundle E, a choice of local trivializations for which dg = 0 is called a at structure on the bundle. A bundle together with a at structure is called a at bundle Filtered local systems and parabolic Higgs bundles. Moduli space of (parabolic) Higgs bundles over (noncompact) Riemann surfaces are well studied in the past twenty years. Based on Mochizuki's work on Hichin-Kobayashi correspondence for quasi-projective varieties, we study the moduli spac algebra. There exists a notion of stability for Higgs bundles, extending the notion of stability for bundles. For the Higgs vector bundle case this notion coincides with the usual notion of stability, but measured with respect to subbundles, invariant for the Higgs eld. A local system is a locally constant sheaf of vector spaces, or, equivalently, ON THE COHOMOLOGY GROUPS OF LOCAL SYSTEMS OVER HILBERT MODULAR VARIETIES VIA HIGGS BUNDLES By STEFANMULLER¨-STACH,MAO SHENG,XUANMING YE,andKANG ZUO Abstract. Let X be a Hilbert modular variety and Va non-trivial local system over X with inﬁnite monodromy. In this paper we study Saito's mixed Hodge structure (MHS) on the cohomology grou
A Higgs bundle is a holomorphic vector bundle E E together with a 1-form Φ \Phi with values in the endomorphisms of (the fibers of) E E, such that Φ ∧ Φ = 0 \Phi \wedge \Phi = 0. Higgs bundles play a central role in nonabelian Hodge theory The converse problem of going from a local system to a vector bundle or Higgs bundle is trivial in the unitary case, but in the case of nonunitary local systems it involves the notion of equivariant harmonic maps. This definition and the. HARMONIC BUNDLES ON NONCOMPACT CURVES 71 We construct five families of two-dimensional moduli spaces of parabolic Higgs bundles (respectively local systems) by taking the equivariant Hilbert scheme of a certain finite group acting on the cotangent bundle of an elliptic curve. We show that the Hilbert scheme of m points of these surfaces is again a moduli space of parabolic Higgs bundles (respectively local systems), confirming a. Higgs bundles and local systems, Preprint, Princeton University. | MR 1179076 [31] C. T. Simpson. Moduli of representations of the fundamental group of a smooth variety, Preprint, Princeton University. [32] C. T. Simpson In very vague terms a structure of Higgs bundle is kind of holomorphic shadow of a flat connection. The last statement can be might rigorous, but this is a very non-trivial theorem (or rather a collection of hard theorems by Simpson, Hitchin, Corlette, Uhlenbeck and Yau), also known as non-abelian Hodge-de Rham theorem
space of stable bundle in [31, 32, 33] and, more recently, for the moduli space of Higgs bundles (coprime case) in [3], see also [4, 5]. The extension to Higgs bundles uses the Hitchin system and should be the main focus of the talk. It would however be good (and time should permit this) to also explain the original argument of Mumford and. Kostenlose Lieferung möglic
parabolic Higgs bundles with trivial characteristic numbers. We also show the Bogomolov-Gieseker type inequality forμ L-stable parabolic Higgs bundles. Then we show that any local system on a smooth quasi projective variety can be deformed to a variation of polarized Hodge structure. As a consequence, we can con surface' (1986), and in this generality by Simpson in 'Higgs bundles and local systems' (1992). MGF (ICMAT) AMS-EMS-SPM Porto, 12 June 2 / 17. Twisted Higgs bundles Let X be a compact complex manifold and L an ample line bundle over X. Deﬁnition: A twisted higgs bundle over X is a pair (E,) consisting of Finally, a Higgs bundle is said to be H-numerically flat if E and its dual Higgs bundle E ∗ are H-numerically effective. H-numerically flat Higgs bundles make up again a neutral Tannakian category; the corresponding group scheme is denoted by π 1 H (X, x)
Steven Rayan (University of Saskatchewan) talks on Higgs bundles and the Hitchin system at the Center of Mathematical Sciences and Applications. Abstract: I. We study Higgs bundles over an elliptic curve with complex reductive structure group, describing the (normalisation of) its moduli spaces and the associated Hitchin fibration. The case of trivial degree is covered by the work of Thaddeus in 2001. Our arguments are different from those of Thaddeus and cover arbitrary degree HIGGS BUNDLES — EXISTENCE OF SOLUTIONS 3 De˙nition(Stability). A Higgs bundle (E,Φ)is said to be stableif for every Remark. (E,0)is stable if and only if Eis stable in the usual sense. Exercise A. There are no stable Higgs bundles on P1. [Hints below1] Lemma 1. For every E∈Vec 2(M)there is a short exact sequence 0→O →End(E)→End 0(E)→ Such Higgs bundles are referred to as virtually basic (Definition 4.2); see Section 5.3 for an example showing the necessity of this hypothesis. The following is the main result proved here (see Theorem 5.1). Higgs bundles and local systems..
Simpson constructs his nonabelian Hodge correspondence first in the case of polystable Higgs bundles and semisimple local systems. In this setup, the correspondence is a consequence of existence theorems for pluri-harmonic metrics on the underlying bundles R. Wentworth , Higgs bundles and local systems on Riemann surfaces, in Geometry and Quantization of Moduli Spaces, Advanced Courses in Mathematics, CRM Barcelona (Birkhäuser, Basel-Boston, 2016), pp. 165-219. Google Scholar; 21. U. Bruzzo and B. Graña Otero , Metrics on semistable and numerically effective Higgs bundles, J. Reine Angew. Higgs bundles have been introduced by Hitchin as solutions of the self-duality equations on a Riemann surface.The subsequent foundational works due to him, Donaldson, Corlette, Simpson and others extended the scope of this concept to more general Kähler manifolds and revealed deep connections to neighboring subjects such as complex geometry, nonlinear PDEs on manifolds, representations of. on Higgs Bundles and Local Systems on Riemann Surfaces, which have been a very useful tool to better understand this amazing topic. Thank you! I want to thank to those people who shared their time with me, listening to my work. Of course, is impossible for me to mention all of them here, but I am truly grate
We prove the topological mirror symmetry conjecture of Hausel-Thaddeus (Hausel and Thaddeus, 2001, 2003) for the moduli space of strongly parabolic Higgs bundles of rank two and three, with full flags, for any generic weights. Although the main theorem is proved only for rank at most three, most of the results are proved for any prime rank Higgs Bundles and UV Completion in F-Theory Higgs Bundles and UV Completion in F-Theory Donagi, Ron; Wijnholt, Martijn 2014-01-31 00:00:00 F-theory admits 7-branes with exceptional gauge symmetries, which can be compactified to give phenomenological four-dimensional GUT models. Here we study general supersymmetric compactifications of eight-dimensional Yang-Mills theory We construct five families of 2D moduli spaces of parabolic Higgs bundles (respectively, local systems) by taking the equivariant Hilbert scheme of a certain finite group acting on the cotangent bundle of an elliptic curve (respectively, twisted cotangent bundle). We show that the Hilbert scheme of m points of these surfaces is again a moduli space of parabolic Higgs bundles (respectively. Higher genus Yangians and Higgs bundles: 13:30-15:00: Lunch: 15:00-16:00: Graeme Wilkin Representations of the Heisenberg algebra on a singular Morse complex: 16:20-17:20: Thomas Sutherland From Higgs bundles to local systems: a (non)abelian perspective: 17:30-18:30: Samson Shatashvili Higgs Bundles in Physic local systems is clarified by this correspondence. A representation underlies a variation of Hodge structure if and only if the associated Higgs bundle can be given a structure of system of Hodge bundles. The points in M Hi which have structures of systems of Hodge bundles may be characterized as the fixed points of a natural C* action
respondence between representations of surface groups and Higgs bundles is the existence of an equivariant harmonic map from the universal cover of the surface to the symmetric space associated to a reductive Lie group G: The purpose of this note is to introduce the theory of Higgs bundles, with a strong emphasis put on the role of harmonic maps 2 Outline 1 Introduction to Higgs bundles and connections 2 Higgs bundles and quantum curves 3 Quantization of Airy function 4 The methamorphosis of quantum curves into opers 5 General theory of Hitchin systems for the Lie group G = SLr(C) 6 Oper Fock-Goncharov on moduli spaces of local systems over surfaces [8], as well as the foundational work of Hitchin [9] and Corlette, Donaldson, Simpson [10,11,12] on Higgs bundles without singularities, Simpson's extension to Higgs bundles with regular singularities [13], and Biquard-Boalch for Higgs bundles with wild ramiﬁ-cation [14] A Higgs bundle (H, θ) is said to be k-nilpotent if the associated bundle map satisfies η k + 1 ≡ 0. Remark: relation with a system of Hodge bundles . If ( H , θ ) is a system of Hodge bundles (see page 44 in [ 12 ] ) then it is k -nilpotent for some k Higgs bundles with rational weights and trivial weight ltration. In this paper we investigate parabolic Higgs bundles with varying weights and study the depen-dence of the harmonic metric on the weight and the Higgs bundle. The following theorem is the main result, which uses the Hitchin-Kobayashi correspondence for parabolic Higgs bundles ([18
Higgs bundlesURL: http://www.icts.res.in/program/hb2016DATES: Monday 21 Mar, 2016 - Friday 01 Apr, 2016 VENUE : Madhava Lecture Hall, ICTS BangaloreDESCRIPTI.. It gives a canonical diffeomorphism between the moduli space nof rank n stable Higgs bundle M Dol and the character variety of rank n stable local systems M B on this curve. Both moduli spaces are complex manifolds, however the canonical diffeomorphism does not preserve the complex structures Higgs bundles are a central tool used to study a range of intersecting brane systems in string compactifications. Solutions to the internal gauge theory equations of motion for the corresponding worldvolume theories of branes give rise to different low energy effective field theories. This has been heavily used in the study of M-theory on local ${G}_{2}$ spaces and F-theory on local. Sponsored by the Department of Mathematics
Carlos Simpson, Higgs bundles and local systems. Publications Mathématiques de l'IHES,75, 1992, p. 5--95. This course is meant to progressively introduce the theory of Anosov representations, developped during the past fifteen years. We will be guided by the following questions CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): We construct five families of 2D moduli spaces of parabolic Higgs bundles (respectively, local systems) by taking the equivariant Hilbert scheme of a certain finite group acting on the cotangent bundle of an elliptic curve (respectively, twisted cotangent bundle). We show that the Hilbert scheme of m points of these.
Higgs bundles over elliptic curves Franco, Emilio, Garcia-Prada, Oscar, and Newstead, P. E., Illinois Journal of Mathematics, 2014; Extensions of Higgs bundles Bradlow, Steven B. and Gómez, Tomás L., Illinois Journal of Mathematics, 2002; A Desingularization of the Moduli Space of Rank 2 Higgs Bundles over a Curve Yoo, Sang-Bum, Taiwanese Journal of Mathematics, 202 Nonabelianization of Higgs bundles Hitchin, Nigel and Schaposnik, Laura P., Journal of Differential Geometry, 2014; A Note on co-Higgs Bundles Ballico, Edoardo and Huh, Sukmoon, Taiwanese Journal of Mathematics, 2017; The reverse Yang-Mills-Higgs flow in a neighbourhood of a critical point Wilkin, Graeme, Journal of Differential Geometry, 2020.
MULTIPLICATIVE HITCHIN SYSTEMS AND SUPERSYMMETRIC GAUGE THEORY CHRIS ELLIOTT AND VASILY PESTUN Abstract. Multiplicative Hitchin systems are analogues of Hitchin's integrable system based on moduli spaces of G-Higgs bundles on a curve C where the Higgs eld is group-valued, rather than Lie algebra valued 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 anti. This entails a closed formula for the Poincaré polynomial of the moduli spaces of stable Higgs bundles over a compact Riemann surface, and hence also for the Poincaré polynomials of the twisted character varieties {Counting local systems with principal unipotent local monodromy}, journal = {Ann. of Math.}, fjournal = {Annals of. We define the quasi-compact Higgs G C-bundles over singular curves introduced in our previous paper for the Lie group SL(N).The quasi-compact structure means that the automorphism groups of the bundles are reduced to the maximal compact subgroups of G C at marked points of the curves. We demonstrate that in particular cases, this construction leads to the classical integrable systems of the. arXiv:2005.06855v1 [math.AG] 14 May 2020 Local systems on diamonds and p-adic vector bundles Lucas Mann and Annette Werner May 15, 2020 We use Scholze's framework of diamonds t
Later Simpson generalized Hitchin's construction to higher dimensional Kaehler manifolds and established the so called Hitchin-Simpson correspondence between Higgs bundles and local systems. In this talk I will give a brief introduction to the theory of Hitchin and Simpson, as well as some applications in complex geometry Stefan Müller-Stach, Mao Sheng, Xuanming Ye, Kang Zuo, On the cohomology groups of local systems over Hilbert modular varieties via Higgs bundles, Amer. J. Math. 137 (2015), no. 1, 1-35. Mao Sheng, He Xin, Kang Zuo, A note on the characteristic p nonabelian Hodge theory in the geometric case, Internat. J. Math. 26 (2015), no. 1, 1550011, 18 pp Moduli Space of Parabolic Higgs Bundles dles of degree 0 and the category of stable parabolic local systems of degree 0, or equivalently the category of tame harmonic bundles. I Later studied and generalized by Mochizuki [Moc06, Moc09] to higher dimensional case and ob The characterization is parallel to the one that R. Hain gave of those unipotent representations of π1(X, x) that can be realized as the monodromy of a flat connection on the holomorphically trivial vector bundle. We see that Hain's result and ours follow from a careful study of Simpson's correspondence between Higgs bundles and local systems
Notes on Higgs bundles and D-branes L.Katzarkov D.Orlov T.Pantev Contents 1 A brief introduction to D-branes 1 2 Higgs bundles 4 Is there a deformation of the notion of a complex local system that captures the quantum corrected con gurations of nbranes wrapping ? (iii). are Higgs bundles on a complex projective variety that are numerically ﬂat in a suitable sense, and show how the Higgs fundamental group is related to a conjecture about semistable Higgs bundles. Keywords: Fundamental groups, Tannakian categories, Higgs bundles, curve semista-bility. MS Classiﬁcation 2010: 14F05, 14F35, 14H60, 14J60. Local G2-Manifolds and 7d twisted SYM Higgs Bundles and a Colored Supersymmetric QM Abelian Higgs Backgrounds Summary and Outlook Probing Higgs Bundles for Local G 2-Manifolds Max Hubner Max.Hubner@maths.ox.ac.uk Seminar Series on String Phenomenology 1 December 2020, arXiv:2009.07136 Max Hubner Probing Higgs Bundles for Local G2-Manifold
Vector bundles and local systems on non-Archimedean analytic spaces we plan to make use of Scholze's theory of diamonds to study bundles with numerically flat reduction after proper pullback, and finally we plan to attack the case of non-vanishing Higgs field II. HIGGS BUNDLE VACUA In this section we introduce the different Higgs bundles associated with local M- and F-theory models. We refer to the corresponding effective field theories generated by these compactifications as Higgs bundle vacua. As a warmup, we first discuss the case of 5D N ¼ 1 vacua as generatedbyM.
Higher Teichmuller¨ theory via Higgs bundles This seminar is intended to give an overview on recent developments in higher Teichmuller theory using Higgs bundles. Higgs bundles and local systems on Riemann surfaces. English. In: Geometry and quantization of moduli spaces. Based on 4 courses, Barcelona, Spain, March - June 2012 Hitchin-type system in G 2background • Gauge degrees in G 2background realized on three-dimensional manifold M 3associated with co-dimension four (ADE) singularities. •Described as a partial topological twist of a six-brane wrapped on three-manifold M 3 , whose supersymmetric gauge theory -Hitchin-type system- specified by (one-form Higgs field ϕ, vector bundle connection A This workshop, fourth in a series of Graduate workshops on Higgs bundles, is aimed at graduate students and young postdocs, will expose participants to some current research topics on the geometry and physics of Higgs bundles. The Lecture notes will apprear in the special issue of SIGMA (Symmetry, Integrability and Geometry: Methods and Applications
Higgs Bundles, Integrability, and holomorphic forms (dvi file) (published in Motives, Polylogarithms and Hodge theory, Int'l Press 2002) Intro to mixed Hodge modules (pdf file) (published in Cycles, Motives and Shimura varieties, TIFR 2010) Cohomology support loci for local systems and Higgs bundles Talks at U. Michiga The blue social bookmark and publication sharing system Thus local systems are \simpler in the sense that you only need the rst homotopy group to vanish to be trivial while vector bundles need all homotopy groups to vanish. Theorem 1 Assume Xis connected, locally connected, path-connected, paracompact, Hausdor , etc. Then the following categories are equivalent. (i) A local systems on X Higgs bundles appear in several guises including (a) as solutions to gauge-theoretic equations for connections and sections of a bundle (b) as holomorphic realizations of fundamental group representations or, equivalently, local systems and (c) as special cases of principal bundles with extra structure (principal pairs) 2020-11-08 15:03:55 -0500 Moduli of Higgs Bundles, preliminary and incomplete draft 35d043e Moduli of Higgs Bundles Andrew Neitzke Preliminary and incomplete draft These are the notes for a spring 2016 course at UT Austin. The lectures are now ﬁn-ished but the notes are not: they are extremely incomplete, unreliable, full of mistake