The Self-Duality Equations on a Riemann Surface

Hitchin, Nigel

doi:10.1112/plms/s3-55.1.59

Cited by 1,444 publications

(2,548 citation statements)

References 33 publications

Supporting

Mentioning

2,482

Contrasting

Unclassified

Order By: Relevance

“…In view of such a connection, our main existence theorem for calorons may be stated as follows. In the subsequent sections, we aim at solving the coupled equations (2.13) and (2.14), which belong to a category of gauge field equations over Riemann surfaces known as Hitchin's equations [21].…”

Section: Hyperbolic Calorons and Vorticesmentioning

confidence: 99%

Existence of hyperbolic calorons

Sibner

Yang

2015

Proc. R. Soc. A.

View full text Add to dashboard Cite

Recent work of Harland shows that the SO(3)-symmetric, dimensionally reduced, charge-N selfdual Yang-Mills calorons on the hyperbolic space H 3 × S 1 may be obtained through constructing N-vortex solutions of an Abelian Higgs model as in the study of Witten on multiple instantons. In this paper, we establish the existence of such minimal action charge-N calorons by constructing arbitrarily prescribed N-vortex solutions of the Witten type equations.

show abstract

Section: Hyperbolic Calorons and Vorticesmentioning

confidence: 99%

Existence of hyperbolic calorons

Sibner

Yang

2015

Proc. R. Soc. A.

View full text Add to dashboard Cite

show abstract

“…Even once this question is asked, it is difficult to answer it without some additional structure. The additional structure that comes in handy is provided by Hitchin's equations, see Hitchin (1987a). Until this point, C has simply been an oriented two-manifold (compact and without boundary).…”

Section: Mirror Symmetry and Hitchin's Equationsmentioning

confidence: 99%

“…As an aside, one may ask how closely related φ, known in the present context as the Higgs field, is to the Higgs fields of particle physics. Thus, to what extent is the terminology that was introduced in Hitchin (1987a) actually justified? The main difference is that Higgs fields in particle physics are scalar fields, while φ is a one-form on C (valued in each case in some representation of the gauge group).…”

Section: Mirror Symmetry and Hitchin's Equationsmentioning

confidence: 99%

“…The analog of what we explained for M(G, C) is to realize M H (G, C) as the hyper-Kahler quotient of a smooth space Y by a group W . It may be impossible to do this with finitedimensional Y and W , but in the infinite-dimensional world, this problem has a natural solution described in Hitchin's original paper on the Hitchin equations, Hitchin (1987a). (Y is the space of pairs (A, φ) on C, and W = Maps(C, G).)…”

Section: Four-dimensional Gauge Theory and Stacksmentioning

confidence: 99%

See 1 more Smart Citation

Mirror Symmetry, Hitchin's Equations, and Langlands Duality

Witten¹

2010

The Many Facets of Geometry

View full text Add to dashboard Cite

Abstract. Geometric Langlands duality can be understood from statements of mirror symmetry that can be formulated in purely topological terms for an oriented two-manifold C. But understanding these statements is extremely difficult without picking a complex structure on C and using Hitchin's equations. We sketch the essential statements both for the "unramified" case that C is a compact oriented two-manifold without boundary, and the "ramified" case that one allows punctures. We also give a few indications of why a more precise description requires a starting point in four-dimensional gauge theory.

show abstract

“…The powerful theory developed by Hitchin ([27], [26]) gives precise topological information concerning the deformation space X(S). In particular Hitchin [27] shows that X(S) has exactly three connected components:…”

Section: Representations Of the Fundamental Groupmentioning

confidence: 99%

The classification of real projective structures on compact surfaces

Choi

Goldman

1997

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Real projective structures (RP 2 -structures) on compact surfaces are classified. The space of projective equivalence classes of real projective structures on a closed orientable surface of genus g > 1 is a countable disjoint union of open cells of dimension 16g − 16. A key idea is Choi's admissible decomposition of a real projective structure into convex subsurfaces along closed geodesics. The deformation space of convex structures forms a connected component in the moduli space of representations of the fundamental group in PGL(3, R), establishing a conjecture of Hitchin. Real projective structuresProjective differential geometry began in the early twentieth and late nineteenth century as an attempt to apply infinitesimal methods on manifolds to concepts from projective geometry. Most of the work, culminating in the 1930's, concentrated on local questions. Global questions became more prominent with Chern's work on the Gauss-Bonnet theorem and characteristic classes. Thurston's work [43] in the late 1970's on geometrization of 3-manifolds underscored the importance of geometric structures in low-dimensional topology, renewing interest in global projective differential geometry. In this note we summarize some recent advances in two-dimensional projective differential geometry. Although many of these ideas can be expressed in terms of affine connections and projective connections, we prefer

show abstract

The Self-Duality Equations on a Riemann Surface

Cited by 1,444 publications

References 33 publications

Existence of hyperbolic calorons

Existence of hyperbolic calorons

Mirror Symmetry, Hitchin's Equations, and Langlands Duality

The classification of real projective structures on compact surfaces

Contact Info

Product

Resources

About