MODULI OF STABLE PAIRS FOR HOLOMORPHIC BUNDLES OVER RIEMANN SURFACES

1991 ◽  
Vol 02 (05) ◽  
pp. 477-513 ◽  
Author(s):  
STEVEN B. BRADLOW ◽  
GEORGIOS D. DASKALOPOULOS
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Holomorphic Section ◽  
Stable Bundles ◽  
Space Problem ◽  
Yang Mills ◽  
Closed Riemann Surface ◽  
Holomorphic Bundles ◽  
Stable Pairs ◽  
Compact Kähler Manifold

It this paper we study the space of gauge equivalence classes of pairs [Formula: see text] where [Formula: see text] represents a holomorphic structure on a complex bundle, E, over a closed Riemann Surface, and ϕ is a holomorphic section. We define a space of stable pairs and consider the moduli space problem for this space. The space of stable pairs, [Formula: see text], is related to the space of solution to the Vortex (Hermitian-Yang-Mills-Higgs) equation. Using the parameter, τ, which appears in this equation we can define subspaces [Formula: see text] within [Formula: see text]. We show that under suitable restrictions on τ and the degree of E, the space [Formula: see text] is naturally a finite dimensional, Hausdorff, compact Kähler manifold. We show further that there is a natural holomorphic map from this space onto the Seshadri compactification of the moduli space of stable bundles and that this map is generically a fibration.

MODULI OF STABLE PAIRS FOR HOLOMORPHIC BUNDLES OVER RIEMANN SURFACES II

1993 ◽  
Vol 04 (06) ◽  
pp. 903-925 ◽  
Author(s):  
STEVEN BRADLOW ◽  
GEORGIOS D. DASKALOPOULOS
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Projective Variety ◽  
Algebraic Varieties ◽  
Stable Bundles ◽  
Topological Information ◽  
Holomorphic Bundles ◽  
Stable Pairs

In this paper we continue our investigation of the moduli space of stable pairs introduced in Part I. We obtain certain topological information, and we give a proof that this moduli space admits the structure of a nonsingular projective variety. We show that the natural map from the moduli space of stable pairs onto the Seshadri compactification of stable bundles is a morphism of algebraic varieties.

scholarly journals Convergence of the Yang–Mills–Higgs flow on Gauged Holomorphic maps and applications

2018 ◽  
Vol 29 (04) ◽  
pp. 1850024
Author(s):  
Samuel Trautwein
Keyword(s):  
Uniqueness Theorem ◽  
Gradient Flow ◽  
Holomorphic Section ◽  
Holomorphic Maps ◽  
Yang Mills ◽  
Stable Case ◽  
Finite Dimensional ◽  
Closed Riemann Surface ◽  
The Moment ◽  
Negative Gradient

The symplectic vortex equations admit a variational description as global minimum of the Yang–Mills–Higgs functional. We study its negative gradient flow on holomorphic pairs [Formula: see text] where [Formula: see text] is a connection on a principal [Formula: see text]-bundle [Formula: see text] over a closed Riemann surface [Formula: see text] and [Formula: see text] is an equivariant map into a Kähler Hamiltonian [Formula: see text]-manifold. The connection [Formula: see text] induces a holomorphic structure on the Kähler fibration [Formula: see text] and we require that [Formula: see text] descends to a holomorphic section of this fibration. We prove a Łojasiewicz type gradient inequality and show uniform convergence of the negative gradient flow in the [Formula: see text]-topology when [Formula: see text] is equivariantly convex at infinity with proper moment map, [Formula: see text] is holomorphically aspherical and its Kähler metric is analytic. As applications we establish several results inspired by finite dimensional GIT: First, we prove a certain uniqueness property for the critical points of the Yang–Mills–Higgs functional which is the analogue of the Ness uniqueness theorem. Second, we extend Mundet’s Kobayashi–Hitchin correspondence to the polystable and semistable case. The arguments for the polystable case lead to a new proof in the stable case. Third, in proving the semistable correspondence, we establish the moment–weight inequality for the vortex equation and prove the analogue of the Kempf existence and uniqueness theorem.

scholarly journals EQUISYMMETRIC STRATA OF THE MODULI SPACE OF CYCLIC TRIGONAL RIEMANN SURFACES OF GENUS 4

2009 ◽  
Vol 51 (1) ◽  
pp. 19-29 ◽  
Author(s):  
MILAGROS IZQUIERDO ◽  
DANIEL YING
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Fuchsian Groups ◽  
Regular Covering ◽  
Closed Riemann Surface

AbstractA closed Riemann surface which can be realized as a three-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. If the trigonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic trigonal Riemann surface. Using the characterization of cyclic trigonality by Fuchsian groups, we find the structure of the space of cyclic trigonal Riemann surfaces of genus 4.

scholarly journals Moduli Spaces of Vector Bundles over a Real Curve: ℤ/2-Betti Numbers

2014 ◽  
Vol 66 (5) ◽  
pp. 961-992 ◽  
Author(s):  
Thomas Baird
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Lagrangian Submanifolds ◽  
Betti Numbers ◽  
Stable Bundles ◽  
Complex Curve ◽  
Yang Mills ◽  
Real Curve

AbstractModuli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi–stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah–Bott's “Yang–Mills over a Riemann Surface” to compute ℤ/2–Betti numbers of these spaces.

SINGULARITY STRUCTURE OF ${\mathcal N} = 2$ SUPERSYMMETRIC YANG–MILLS THEORIES

2013 ◽  
Vol 28 (20) ◽  
pp. 1330029 ◽  
Author(s):  
JIHYE SOFIA SEO
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Magnetic Monopoles ◽  
Gauge Groups ◽  
Singularity Structure ◽  
Yang Mills ◽  
Geometric Singularity ◽  
Physical Singularity ◽  
Monopoles And Dyons ◽  
Nontrivial Topology

In this paper, we consider the case where electrons, magnetic monopoles and dyons become massless. Here, we consider the [Formula: see text] supersymmetric Yang–Mills (SYM) theories with classical gauge groups with a rank r, SU(r+1), SO(2r), Sp(2r) and SO(2r+1) which are studied by Riemann surfaces called Seiberg–Witten curves. We discuss physical singularity associated with massless particles, which can be studied by geometric singularity of vanishing 1-cycles in Riemann surfaces in hyperelliptic form. We pay particular attention to the cases where mutually nonlocal states become massless (Argyres–Douglas theories), which corresponds to Riemann surfaces degenerating into cusps. We discuss nontrivial topology on the moduli space of the theory, which is reflected as monodromy associated to vanishing 1-cycles. We observe how dyon charges get changed as we move around and through singularity in moduli space.

The Yang-Mills equations over Riemann surfaces

1983 ◽  
Vol 308 (1505) ◽  
pp. 523-615 ◽  
Keyword(s):  
Riemann Surface ◽  
Gauge Symmetry ◽  
Moduli Space ◽  
Morse Theory ◽  
Riemann Surfaces ◽  
Point Of View ◽  
Yang Mills ◽  
Critical Sets

The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect' functional provided due account is taken of its gauge symmetry. This enables topological conclusions to be drawn about the critical sets and leads eventually to information about the moduli space of algebraic bundles over the Riemann surface. This in turn depends on the interplay between the holomorphic and unitary structures, which is analysed in detail.

A Primer on Mapping Class Groups (PMS-49)

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Graduate Students ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Surface Bundles ◽  
Surface Homeomorphisms ◽  
Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

scholarly journals The Volume of the Quiver Vortex Moduli Space

2021 ◽  
Author(s):  
Kazutoshi Ohta ◽  
Norisuke Sakai
Keyword(s):  
Gauge Theory ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Gauge Theories ◽  
Target Space ◽  
Contour Integral ◽  
Quiver Gauge Theory ◽  
Volume Formula ◽  
Volume Operator ◽  
Space Volume

Abstract We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vev of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to derive the volume formula. The contour integral formula of the volume (generalization of the Jeffrey-Kirwan residue formula) leads to the Bradlow bounds (upper bounds on the vorticity by the area of the Riemann surface divided by the intrinsic size of the vortex). We give some examples of various quiver gauge theories and discuss properties of the moduli space volume in these theories. Our formula are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with CPN target space, which is obtained by a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abelian generalization of the quiver gauge theory and “Abelianization” of the volume formula.

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