The Self-Duality Equations on a Riemann Surface

DOI10.1112/plms/s3-55.1.59zbMath0634.53045OpenAlexW1571072992MaRDI QIDQ3772878

Nigel J. Hitchin

Publication date: 1987

Published in: Proceedings of the London Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1112/plms/s3-55.1.59

zbMATH Keywords

Teichmüller spaceStein manifoldmoduli spacecompact Riemann surfacesymplectic manifoldcomplex structuresself dualityhyper-Kähler metricstable paircomplex connectionconnection on a principal G-bundle

Mathematics Subject Classification ID

Global differential geometry of Hermitian and Kählerian manifolds (53C55) Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces (32-02) Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) (32G15) Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems (37J99) Research exposition (monographs, survey articles) pertaining to differential geometry (53-02) Connections (general theory) (53C05) Differentials on Riemann surfaces (30F30) Generalized geometries (à la Hitchin) (53D18)

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II, On a topology property for the moduli space of Kapustin-Witten equations, Higgs bundles for the non-compact dual of the special orthogonal group, Wild quiver gauge theories, Higgs bundles over elliptic curves, Quaternionic Kähler metrics associated with special Kähler manifolds, Stable Higgs bundles with trivial Chern classes. Several examples, Moduli of rank 2 Higgs sheaves on elliptic surfaces, The Plancherel theorem for Fourier-Laplace-Nahm transform for connections on the projective line, Bohr-Sommerfeld Lagrangians of moduli spaces of Higgs bundles, Simpson filtration and oper stratum conjecture, Quadratic differentials as stability conditions, Higgs Bundles and Characteristic Classes, Spectral data for parabolic projective symplectic/orthogonal Higgs bundles, INTERSECTION COHOMOLOGY OF THE MODULI SPACE OF HIGGS BUNDLES ON A GENUS 2 CURVE, An elementary discussion of the representation and geometric invariant theory of equioriented quivers of type D with an application to quiver bundles, COHOMOLOGY RINGS OF TORIC HYPERKÄHLER MANIFOLDS, Notes on coherent systems, Relations in the cohomology ring of the moduli space of rank 2 Higgs bundles, Dimensional reduction and quiver bundles, From Hyperbolic Dehn Filling to Surgeries in Representation Varieties, Opers and the twisted Bogomolny equations, GEOMETRY OF THE MODULI SPACES OF HARMONIC MAPS INTO LIE GROUPS VIA GAUGE THEORY OVER RIEMANN SURFACES, GRADIENT FLOWS OF HIGHER ORDER YANG–MILLS–HIGGS FUNCTIONALS, Unnamed Item, SOLUTIONS OF VANISHING CURVATURE NON-ABELIAN MONOPOLE EQUATIONS, Spaces of Abelian Differentials and Hitchin’s Spectral Covers, On the Moyal deformation of Kapustin-Witten systems, Yang–Mills theories on geometric spaces, 25 open questions about vector bundles and their moduli, The reduction of five dimensional Chern–Simons theories, Vortices over Riemann surfaces and dominated splittings, Semi-stable holomorphic vector bundles over generalized Kähler manifolds, The quantum trace as a quantum non-abelianization map, Matrix divisors on Riemann surfaces and Lax operator algebras, Principal bundles on two-dimensional CW-complexes with disconnected structure group, Unnamed Item, Asymptotics for minimizers of a Donaldson functional and mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds, Representation variety of free or surface groups and Reidemeister torsion, AdS 3-manifolds and Higgs bundles, Perturbative and nonperturbative aspects of complex Chern–Simons theory, Constructions of Compact $$G_2$$-Holonomy Manifolds, The classification of real projective structures on compact surfaces, Fibrés de higgs et connexions intégrables : Le cas logarithmique (diviseur lisse), Higgs Bundles and Local Systems on Riemann Surfaces, Topological field theories and integrable models, Self-dual gravity via Hitchin’s equations, Symplectic resolutions for Higgs moduli spaces, Residue Mirror Symmetry for Grassmannians, Flat Rank Two Vector Bundles on Genus Two Curves, Supersymmetric field theories and geometric Langlands: The other side of the coin, A journey from the Hitchin section to the oper moduli, S-duality of boundary conditions and the Geometric Langlands program, Quantization of spectral curves for meromorphic Higgs bundles through topological recursion, METRICS AND SPECIAL KÄHLER GEOMETRY ON THE MODULI SPACES OF HIGGS BUNDLES AND HITCHIN SYSTEMS, The Seiberg-Witten equations and 4-manifold topology, Generalized quivers, orthogonal and symplectic representations, and Hitchin–Kobayashi correspondences, Twistor geometry and gauge fields, Cayley and Langlands type correspondences for orthogonal Higgs bundles, Convergence of the Yang–Mills–Higgs flow on Gauged Holomorphic maps and applications, Homogeneous Higgs and co-Higgs bundles on Hermitian symmetric spaces, Monodromy of rank 2 twisted Hitchin systems and real character varieties, Uniqueness of minimal diffeomorphisms between surfaces, Atiyah’s work on holomorphic vector bundles and gauge theories, Harmonic maps and wild Teichmüller spaces, Moduli of wild Higgs bundles on with -actions, Twistor theory at fifty: from contour integrals to twistor strings, PARABOLIC HIGGS BUNDLES AND -HIGGS BUNDLES, Minimal n -noids in hyperbolic and anti-de Sitter 3-space, Moduli problem of Hitchin pairs over Deligne-Mumford stacks, Langlands Program, trace formulas, and their geometrization, Anti-self-dual Yang-Mills equations on noncommutative space-time, Canonical transformations for hyperkahler structures and hyperhamiltonian dynamics, Cohomologies of unipotent harmonic bundles over noncompact curves, MONODROMY OF THE SL₂ HITCHIN FIBRATION, Hodge polynomials of the SL(2, ℂ)-character variety of an elliptic curve with two marked points, Variations along the Fuchsian locus, Higgs bundles and four manifolds, INTEGRABLE SYSTEMS IN ONE AND TWO DIMENSIONS AND REDUCED TOPOLOGICAL GAUGE THEORIES, Deformation of Locally Free Sheaves and Hitchin Pairs over Nodal Curve, Locality in the Fukaya category of a hyperkähler manifold, Tau functions, Hodge classes, and discriminant loci on moduli spaces of Hitchin’s spectral covers, A Higgs bundle on a Hermitian symmetric space. II, Indecomposable vector bundles and stable Higgs bundles over smooth projective curves, Character varieties of virtually nilpotent Kähler groups and \(G\)-Higgs bundles, Hecke cycles associated to rank 2 twisted Higgs bundles on a curve, Unnamed Item, Model Higgs bundles in exceptional components of the Sp(4, ℝ)-character variety, A fully nonlinear equation in relativistic Teichmüller theory, Pseudo-Riemannian almost hypercomplex homogeneous spaces with irreducible isotropy, Orbifold Riemann Surfaces and the Yang-Mills-Higgs Equations, Metrics on semistable and numerically effective Higgs bundles, Nonabelian Hodge theory and vector valued modular forms, Quivers from Higgs bundles over P1 and quiver gauge theory, Counting Higgs bundles and type quiver bundles, Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces, Représentations linéaires des groupes kählériens : factorisations et conjecture de Shafarevich linéaire, Łojasiewicz–Simon gradient inequalities for coupled Yang–Mills energy functions, On Shimura’s isomorphism and (Γ,𝐺)-bundles on the upper-half plane, HIGGS BUNDLES OVER ELLIPTIC CURVES FOR COMPLEX REDUCTIVE GROUPS, Limits of harmonic maps and crowned hyperbolic surfaces, An Application of Spherical Geometry to Hyperkähler Slices, Multi-Hamiltonian structures for r-matrix systems, Yang–Mills connections on compact complex tori, Equivariant Morse Theory for the Norm-Square of a Moment Map on a Variety, Higgs bundles over non-compact Gauduchon manifolds, Strict plurisubharmonicity of the energy on Teichmüller space associated to Hitchin representations, A Note on Representation Variety of Abelian Groups and Reidemeister Torsion, Higher Teichmüller Theory for Surface Groups and Shifts of Finite Type, Uniformization of branched surfaces and Higgs bundles, 3d TQFTs from Argyres–Douglas theories, Geometric prequantization of the moduli space of the vortex equations on a Riemann surface, Aspects of Calabi-Yau integrable and Hitchin systems, Studying deformations of Fuchsian representations with Higgs bundles, Perspectives on the asymptotic geometry of the Hitchin moduli space, On Higgs bundles on nodal curves, Subspaces of moduli spaces of rank one local systems, Topological mirror symmetry for parabolic Higgs bundles, Classification of the automorphism and isometry groups of Higgs bundle moduli spaces, Maximal \(\mathsf{Sp}(4,\mathbb R)\) surface group representations, minimal immersions and cyclic surfaces, On a canonical quantization of 3D anti de Sitter pure gravity, The universal connection for principal bundles over homogeneous spaces and twistor space of coadjoint orbits, Quantization of Chern-Simons gauge theory with complex gauge group, Introduction to Nonabelian Hodge Theory, Flat connections and geometric quantization, Approximate Hitchin–Kobayashi correspondence for Higgs G-bundles, Fibrés paraboliques stables et connexions singulières plates, On the Voevodsky motive of the moduli space of Higgs bundles on a curve, The Hitchin-Kobayashi correspondence for quiver bundles over generalized Kähler manifolds, Irreducible flat \(\operatorname{SL}(2, \mathbb{R})\)-connections on the trivial holomorphic bundle, Index of rigidity of differential equations and Euler characteristic of their spectral curves, Automorphism group of a moduli space of framed bundles over a curve, Semi-stability for bi-holomorphic pairs over compact bi-Hermitian Gauduchon manifolds, Equivariant Verlinde formula from fivebranes and vortices, Degenerate, strong and stable Yang-Mills-Higgs pairs, Yang–Mills mass gap at large-N, noncommutative YM theory, topological quantum field theory and hyperfiniteness, Flat Higgs bundles over non-compact affine Gauduchon manifolds, Maximal representations of cocompact complex hyperbolic lattices, a uniform approach, Harmonic maps for Hitchin representations, Geometric models and variation of weights on moduli of parabolic Higgs bundles over the Riemann sphere: a case study, Motive of the \(\mathrm{SL}_4\)-character variety of torus knots, A Liouville theorem on complete non-Kähler manifolds, Intersection cohomology of character varieties for punctured Riemann surfaces, Conformal limits and the Białynicki-Birula stratification of the space of \(\lambda\)-connections, T-branes and defects, Involutions of Higgs moduli spaces over elliptic curves and pseudo-real Higgs bundles, Twisted hyperkähler symmetries and hyperholomorphic line bundles, Nonabelian Hodge theory for stacks and a stacky P=W conjecture, Comparison of Poisson structures on moduli spaces, Cluster-like coordinates in supersymmetric quantum field theory, Abelianization of Higgs bundles for quasi-split real groups, Quaternionic Kähler detour complexes and \({\mathcal{N} = 2}\) supersymmetric black holes, Partial differential equations. Abstracts from the workshop held July 25--31, 2021 (hybrid meeting), Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. II: Higgs sheaves and admissible structures, Geometry and braiding of Stokes data; fission and wild character varieties, Harmonic maps and hypersymplectic geometry, Hyperpolygon spaces and moduli spaces of parabolic Higgs bundles, Length spectrum compactification of the \(\mathrm{SO}_0(2, 3)\)-Hitchin component, Commuting elements in reductive groups and Higgs bundles on abelian varieties, Real projective structures on Riemann surfaces and new hyper-Kähler manifolds, Higgs bundles and UV completion in \(F\)-theory, Geometric quantization of the Hitchin system, Polygons in Minkowski three space and parabolic Higgs bundles of rank 2 on \(\mathbb{CP}^1\), Joyce-Song wall crossing as an asymptotic expansion, Affine representations of the fundamental group, Einstein manifolds and contact geometry, A Higgs bundle on a Hermitian symmetric space, EXISTENCE OF HERMITIAN–EINSTEIN METRICS ON STABLE HIGGS BUNDLES OVER OPEN KÄHLER MANIFOLDS, DIMENSIONAL REDUCTION, ${\rm SL} (2, {\mathbb C})$-EQUIVARIANT BUNDLES AND STABLE HOLOMORPHIC CHAINS, PROJECTIVE STRUCTURES AND HIGGS BUNDLES ON SURFACES, The G-generalized Hitchin systems and Prym varieties, Y-system for scattering amplitudes, Invariant forms, associated bundles and Calabi-Yau metrics, Connections and Higgs fields on a principal bundle, Hyper-Kähler and quaternionic Kähler manifolds with \(S^{1}\)-symmetries, Moduli spaces of local systems and higher Teichmüller theory, On length and product of harmonic forms in Kähler geometry, Geometric quantization of the moduli space of the self-duality equations on a Riemann surface, Orbifold projective structures, differential operators, and logarithmic connections on a pointed Riemann surface, Moduli of parabolic Higgs bundles and Atiyah algebroids, Density of Zariski density for surface groups, Exact Results on $${\mathcal N}=2$$ Supersymmetric Gauge Theories, Supersymmetric Gauge Theories, Quantization of $${\mathcal M}_\mathrm{flat}$$ , and Conformal Field Theory, Character varieties with Zariski closures of \(\mathrm{GL}_n\)-conjugacy classes at punctures, Harmonic Bundles on Noncompact Curves, Toledo invariants of 2-orbifolds and Higgs bundles on elliptic surfaces, A Fourier transformation for Higgs bundles, Endoscopic decompositions and the Hausel–Thaddeus conjecture, Noncommutative Homological Mirror Functor, Gauge Theory, Mirror Symmetry, and the Geometric Langlands Program, Quasi-parabolic Higgs bundles and null hyperpolygon spaces, Two-Dimensional Quantum Yang–Mills Theory and the Makeenko–Migdal Equations, The Beauville-Narasimhan-Ramanan correspondence for twisted Higgs 𝑉-bundles and components of parabolic 𝑆𝑝(2𝑛,ℝ)-Higgs moduli spaces, On moduli spaces of Hitchin pairs, Stable bundles and polyvector fields, The asymptotic behavior of the monodromy representation of the associated family of a compact CMC surface, On Some Elliptic Problems in the Study of Selfdual Chern-Simons Vortices, Character varieties, Hitchin's self-duality equations on complete Riemannian manifolds, On the universal coverings of algebraic surfaces, Dirichlet problem for Hermitian Yang-Mills-Higgs equations over Hermitian manifolds, Twisted quiver bundles over almost complex manifolds, Special almost Hermitian geometry, Gravity and unification: a review, NEW POLARIZATIONS ON THE MODULI SPACES AND THE THURSTON COMPACTIFICATION OF TEICHMÜLLER SPACE, PROJECTIVE MODULI FOR HITCHIN PAIRS, Parabolic Higgs bundles and Teichmüller spaces for punctured surfaces, A Torelli type theorem for the moduli space of rank two connections on a curve, Higgs bundles for real groups and the Hitchin–Kostant–Rallis section, Constructing Variations of Hodge Structure Using Yang-Mills Theory and Applications to Uniformization, Geometrization of trace formulas, Eisenstein metrics, Arakelov–Milnor inequalities and maximal variations of Hodge structure, Relative cluster categories and Higgs categories, Nilpotent Higgs bundles and families of flat connections, Branes on the singular locus of the Hitchin system via Borel and other parabolic subgroups, BPS cohomology for rank 2 degree 0 Higgs bundles (and more), E-polynomials of character varieties for real curves, Galois \(E_6\)-bundles over a hyperelliptic algebraic curve, A Teichmüller space for negatively curved surfaces, The non-abelian Hodge correspondence on some non-Kähler manifolds, Harmonic metrics of generically regular semisimple Higgs bundles on noncompact Riemann surfaces, Working session: Higher rank Teichmüller theory. Abstracts from the working session held October 9--14, 2022, On the intersection cohomology of the moduli of SLn$\mathrm{SL}_n$‐Higgs bundles on a curve, sl(2)$\mathfrak {sl}(2)$‐Type singular fibres of the symplectic and odd orthogonal Hitchin system, Quasi Poisson structures, weakly quasi Hamiltonian structures, and Poisson geometry of various moduli spaces, INTERSECTION COHOMOLOGY OF RANK 2 CHARACTER VARIETIES OF SURFACE GROUPS, Anti-quasi-Sasakian manifolds, Enhanced mirror symmetry for Langlands dual Hitchin systems, Zeta functions in higher Teichmüller theory, Fixed points of automorphisms of the vector bundle moduli space over a compact Riemann surface, Mirror symmetry for parabolic Higgs bundles via \(p\)-adic integration, The covariance metric in the Blaschke locus, Domains of discontinuity in oriented flag manifolds, Domination results in n$n$‐Fuchsian fibers in the moduli space of Higgs bundles, Semi-stable twisted holomorphic vector bundles over Gauduchon manifolds, Cyclic Higgs bundles and minimal surfaces in pseudo-hyperbolic spaces, On the existence of harmonic metrics on non‐Hermitian Yang–Mills bundles, Geodesic coordinates for the pressure metric at the Fuchsian locus, Cohomological \(\chi\)-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles, Hitchin fibration under ramified coverings, Hodge theory on \(\mathrm{ALG}^\ast\) manifolds, Infinite energy equivariant harmonic maps, domination, and anti-de Sitter \(3\)-manifolds, On character varieties of singular manifolds, Meromorphic parahoric Higgs torsors and filtered Stokes G-local systems on curves, Motives of moduli spaces of bundles on curves via variation of stability and flips, Resurgence analysis of quantum invariants of Seifert fibered homology spheres, From complex contact structures to real almost contact 3-structures, A proof of N. Takahashi's conjecture for \(( \mathbb{P}^2, E)\) and a refined sheaves/Gromov-Witten correspondence, Complex analysis -- differential and algebraic methods in Kähler spaces. Abstracts from the workshop held April 9--14, 2023, Szegő kernel and symplectic aspects of spectral transform for extended spaces of rational matrices, On 2k-Hitchin’s equations and Higgs bundles: A survey, Generalized Deligne-Hitchin twistor spaces: construction and properties, Gauge theory and the analytic form of the geometric Langlands program, Opers and Non-Abelian Hodge: Numerical Studies, Principal Higgs bundles and Schottky representations, Airy structures and deformations of curves in surfaces