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.

ON THE CONNECTEDNESS OF THE LOCUS OF REAL ELLIPTIC-HYPERELLIPTIC RIEMANN SURFACES

2009 ◽  
Vol 20 (08) ◽  
pp. 1069-1080 ◽  
Author(s):  
JOSÉ A. BUJALANCE ◽  
ANTONIO F. COSTA ◽  
ANA M. PORTO
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Algebraic Curves ◽  
Orbit Space ◽  
Previous Article ◽  
Connected Components ◽  
Hyperelliptic Involution ◽  
Hyperelliptic Surfaces ◽  
Genus One

A Riemann surface X of genus g > 2 is elliptic-hyperelliptic if it admits a conformal involution h such that the orbit space X/〈h〉 has genus one. This elliptic-hyperelliptic involution h is unique for g > 5 [1]. In a previous article [3], we established the non-connectedness of the subspace [Formula: see text] of real elliptic-hyperelliptic algebraic curves in the moduli space [Formula: see text] of Riemann surfaces of genus g, when g is even and > 5. In this paper we improve this result and give a complete answer to the connectedness problem of the space [Formula: see text] of real elliptic-hyperelliptic surfaces of genus > 5: we show that [Formula: see text] is connected if g is odd and has exactly two connected components if g is even; in both cases the closure [Formula: see text] of [Formula: see text] in the compactified moduli space [Formula: see text] is connected.

scholarly journals COHERENT STATES AND THE QUANTIZATION OF (1+1)-DIMENSIONAL YANG–MILLS THEORY

2001 ◽  
Vol 13 (10) ◽  
pp. 1281-1305 ◽  
Author(s):  
BRIAN C. HALL
Keyword(s):  
Phase Space ◽  
Gauge Symmetry ◽  
Coherent States ◽  
Complex Structure ◽  
Canonical Quantization ◽  
Point Of View ◽  
Joint Work ◽  
Bargmann Transform ◽  
Yang Mills ◽  
New Family

This paper discusses the canonical quantization of (1+1)-dimensional Yang–Mills theory on a spacetime cylinder from the point of view of coherent states, or equivalently, the Segal–Bargmann transform. Before gauge symmetry is imposed, the coherent states are simply ordinary coherent states labeled by points in an infinite-dimensional linear phase space. Gauge symmetry is imposed by projecting the original coherent states onto the gauge-invariant subspace, using a suitable regularization procedure. We obtain in this way a new family of "reduced" coherent states labeled by points in the reduced phase space, which in this case is simply the cotangent bundle of the structure group K. The main result explained here, obtained originally in a joint work of the author with B. Driver, is this: The reduced coherent states are precisely those associated to the generalized Segal–Bargmann transform for K, as introduced by the author from a different point of view. This result agrees with that of K. Wren, who uses a different method of implementing the gauge symmetry. The coherent states also provide a rigorous way of making sense out of the quantum Hamiltonian for the unreduced system. Various related issues are discussed, including the complex structure on the reduced phase space and the question of whether quantization commutes with reduction.

PARABOLIC VECTOR BUNDLES AND HERMITIAN-YANG-MILLS CONNECTIONS OVER A RIEMANN SURFACE

1993 ◽  
Vol 04 (03) ◽  
pp. 467-501 ◽  
Author(s):  
JONATHAN A. PORITZ
Keyword(s):  
Riemann Surface ◽  
Vector Bundle ◽  
Sobolev Space ◽  
Moduli Space ◽  
Vector Bundles ◽  
Weighted Sobolev Space ◽  
Conjugacy Classes ◽  
Stable Bundle ◽  
Yang Mills ◽  
Parabolic Bundles

We study a certain moduli space of irreducible Hermitian-Yang-Mills connections on a unitary vector bundle over a punctured Riemann surface. The connections used have non-trivial holonomy around the punctures lying in fixed conjugacy classes of U (n) and differ from each other by elements of a weighted Sobolev space; these connections give rise to parabolic bundles in the sense of Mehta and Seshadri. We show in fact that the moduli space of stable parabolic bundles can be identified with our moduli space of HYM connections, by proving that every stable bundle admits a unique unitary gauge orbit of Hermitian-Yang-Mills connections.

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.

scholarly journals Truncation of lattice N = 4 super Yang-Mills

2018 ◽  
Vol 175 ◽  
pp. 11008 ◽  
Author(s):  
Joel Giedt ◽  
Simon Catterall ◽  
Raghav Govind Jha
Keyword(s):  
Gauge Theory ◽  
Gauge Symmetry ◽  
Moduli Space ◽  
Lattice Spacing ◽  
Continuum Limit ◽  
Power Counting ◽  
Yang Mills ◽  
Free Theory ◽  
Lattice Gauge ◽  
Two Sectors

In twisted and orbifold formulations of lattice N = 4 super Yang-Mills, the gauge group is necessarily U(1) × SU(N), in order to be consistent with the exact scalar supersymmetry Q. In the classical continuum limit of the theory, where one expands the link fields around a point in the moduli space and sends the lattice spacing to zero, the diagonal U(1) modes decouple from the SU(N) sector, and give an uninteresting free theory. However, lattice artifacts (described by irrelevant operators according to naive power-counting) couple the two sectors, so removing the U(1) modes is a delicate issue. We describe how this truncation to an SU(N) gauge theory can be obtained in a systematic way, with violations of Q that fall off as powers of 1=N2. We are able to achieve this while retaining exact SU(N) lattice gauge symmetry at all N, and provide both theoretical arguments and numerical evidence for the 1=N2 suppression of Q violation.

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.

CUTTING AND PASTING OF RIEMANN SURFACES WITH ABELIAN DIFFERENTIALS I

1999 ◽  
Vol 10 (05) ◽  
pp. 587-617
Author(s):  
YOSHITAKE HASHIMOTO ◽  
KIYOSHI OHBA
Keyword(s):  
Riemann Surface ◽  
Positive Integer ◽  
Elliptic Curves ◽  
Complex Plane ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Complex Structures ◽  
Line Segments ◽  
Abelian Differentials ◽  
The Family

We introduce a method of constructing once punctured Riemann surfaces by cutting the complex plane along "line segments" and pasting by "parallel transformations". The advantage of this construction is to give a good visualization of the deformation of complex structures of Riemann surfaces. In fact, given a positive integer g, there appears a family of once punctured Riemann surfaces of genus g which is complete and effectively parametrized at any point. Our construction naturally gives each of the resulting surfaces what we call a Lagrangian lattice Λ, a certain subgroup of the first homology. Furthermore Λ and the puncture determine an Abelian differential ωΛ of the second kind on the Riemann surface. Using Λ and ωΛ we consider the Kodaira–Spencer maps and some extension of the family to obtain any once punctured Riemann surface with a Lagrangian lattice. In particular we describe the moduli space of once punctured elliptic curves with Lagrangian lattices.

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