The geometry of gauge fields
type: Item
instance of: scholarly article
title: The geometry of gauge fields
zbMATH Open document ID: 0679.53023
DOI: 10.1016/0393-0440(89)90002-8
author: G. Martucci
author: Kishore Marathe
published in: Journal of Geometry and Physics
publication date: 1989-01-01T00:00:00Z
review text: In this rather extensive article the authors endeavour to present a complete and systematic description of the mathematical background of gauge field theories. Some basic knowledge of the theory of differentiable manifolds is assumed, and accordingly the discussion of the exterior differential calculus and tensor bundles is quite terse. This is allowed by a brief description in Chapters 2 and 3 of principal and associated bundles, and the theory of connections. Chapter 4 is devoted to some aspects of algebraic topology, differential complexes, and cohomology theory. An outline of the basic concepts of gauge field theories is presented in Chapter 5, with special reference to the space of gauge connections and a variational approach to coupled field equations. In Chapter 6 special Yang-Mills fields and instantons are introduced, and a description is given of the moduli space of the complete (8k-3)-parameter family of instanton solutions over \(S^ 4\) with gauge group SU(2) and instanton number k. The final Chapter 7 deals with coupled Yang-Mills-Higgs fields, the notion of dimensional reduction, and monopole solutions. This brief summary does not reflect fully the unusually wide scope of the article under review. Clearly the omission in the latter of proofs and elaboration of many basic results is inevitable. However, a very detailed list of references serves to provide careful documentation.
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 81T08
zbMATH DE Number: 4113439
zbMATH Keywords: curvature
zbMATH Keywords: gauge field theories
zbMATH Keywords: connections
zbMATH Keywords: Yang-Mills fields
zbMATH Keywords: instantons
zbMATH Keywords: moduli space
zbMATH Keywords: dimensional reduction
zbMATH Keywords: monopole solutions
reviewed by: Hanno Rund
MaRDI profile type: Publication
full work available at URL: 0393-0440(89)90002-8
OpenAlex ID: W1989768663
cites work: Q4746527
cites work: Equivalent Norms for Sobolev Spaces
cites work: Q4721249
cites work: Q3994593
cites work: Cut points, conjugate points and Lorentzian comparison theorems
cites work: Q3926513
cites work: Q3938903
cites work: Topology and analysis. The Atiyah-Singer index formula and gauge- theoretic physics. Transl. from the German by D. D. Bleecker and A. Mader
cites work: Mathématique et physique. Seminaire de l'Ecole Normale Superieure (Paris) 1979-1982
cites work: Q5525829
cites work: Q4067705
cites work: Q3954541
cites work: Q4040290
cites work: Q4157686
cites work: Q3681575
cites work: Q3905390
cites work: Fiber bundle techniques in gauge theories. Lectures in mathematical physics at the University of Texas at Austin. Edited by A. Böhm and J. D. Dollard
cites work: Q5520521
cites work: Q3946763
cites work: Q3951266
cites work: Q5569065
cites work: Q5185713
cites work: Q4084975
cites work: Q4102494
cites work: Q4150360
cites work: Q3696183
cites work: Q3897665
cites work: Q4173935
cites work: Q4064722
cites work: Q3906754
cites work: Q5731354
cites work: Q5565587
cites work: Q5649693
cites work: Q3730551
cites work: Q4143915
cites work: Exact couples in algebraic topology. I.-V
cites work: Q3871594
cites work: Characteristic Classes. (AM-76)
cites work: Q3312869
cites work: Q5548503
cites work: Q5599559
cites work: Q5579608
cites work: General relativity and cosmology
cites work: Q3714308
cites work: The Atiyah-Singer index theorem
cites work: Q5522742
cites work: Q3881472
cites work: Q5825483
cites work: Q3729634
cites work: Q3938902
cites work: Smoothness of the action of the gauge transformation group on connections
cites work: The Lie group of automorphisms of a principal bundle
cites work: Significance of Electromagnetic Potentials in the Quantum Theory

The geometry of gauge fields (cites work)
cites work: Further Considerations on Electromagnetic Potentials in the Quantum Theory

The geometry of gauge fields (cites work)
cites work: The structure of the solution set for the Yang-Mills equations

The geometry of gauge fields (cites work)
cites work: The structure of the space of solutions of Einstein's equations. II: Several Killing fields and the Einstein-Yang-Mills equations

The geometry of gauge fields (cites work)
cites work: Q3865184

The geometry of gauge fields (cites work)
cites work: Q3902147

The geometry of gauge fields (cites work)
cites work: Instantons in two and four dimensions

The geometry of gauge fields (cites work)
cites work: Q4207337

The geometry of gauge fields (cites work)
cites work: The Yang-Mills equations over Riemann surfaces

The geometry of gauge fields (cites work)
cites work: Construction of instantons

The geometry of gauge fields (cites work)
cites work: Deformations of instantons

The geometry of gauge fields (cites work)
cites work: Self-duality in four-dimensional Riemannian geometry

The geometry of gauge fields (cites work)
cites work: Topological aspects of Yang-Mills theory

The geometry of gauge fields (cites work)
cites work: Dirac operators coupled to vector potentials

The geometry of gauge fields (cites work)
cites work: Instantons and algebraic geometry

The geometry of gauge fields (cites work)
cites work: Some results for SU(N) gauge-fields on the hypertorus

The geometry of gauge fields (cites work)
cites work: The Riemannian geometry of the configuration space of gauge theories

The geometry of gauge fields (cites work)
cites work: Q3954548

The geometry of gauge fields (cites work)
cites work: Q3957592

The geometry of gauge fields (cites work)
cites work: Q3756926

The geometry of gauge fields (cites work)
cites work: Stability and isolation phenomena for Yang-Mills fields

The geometry of gauge fields (cites work)
cites work: Q3940515

The geometry of gauge fields (cites work)
cites work: Homology operations on instantons

The geometry of gauge fields (cites work)
cites work: Monopoles, non-linear \(\sigma\) models, and two-fold loop spaces

The geometry of gauge fields (cites work)
cites work: Magnetic monopoles on three-manifolds

The geometry of gauge fields (cites work)
cites work: Instantons on \({\mathbb{C}}{\mathbb{P}}_ 2\)

The geometry of gauge fields (cites work)
cites work: Geometrical gauge conditions in Yang–Mills theory: Some nonexistence results

The geometry of gauge fields (cites work)
cites work: Geometry of multidimensional universes

The geometry of gauge fields (cites work)
cites work: Q3326397

The geometry of gauge fields (cites work)
cites work: The action of the group of bundle-automorphisms on the space of connections and the geometry of gauge theories

The geometry of gauge fields (cites work)
cites work: Q3806388

The geometry of gauge fields (cites work)
cites work: A new proof of a theorem of Narasimhan and Seshadri

The geometry of gauge fields (cites work)
cites work: Self-dual connections and the topology of smooth 4-manifolds

The geometry of gauge fields (cites work)
cites work: An application of gauge theory to four dimensional topology

The geometry of gauge fields (cites work)
cites work: Q3676792

The geometry of gauge fields (cites work)
cites work: Instantons and geometric invariant theory

The geometry of gauge fields (cites work)
cites work: Nahm's equations and the classification of monopoles

The geometry of gauge fields (cites work)
cites work: Anti Self-Dual Yang-Mills Connections Over Complex Algebraic Surfaces and Stable Vector Bundles

The geometry of gauge fields (cites work)
cites work: Connections, cohomology and the intersection forms of 4-manifolds

The geometry of gauge fields (cites work)
cites work: Irrationality and the h-cobordism conjecture

The geometry of gauge fields (cites work)
cites work: Q4186962

The geometry of gauge fields (cites work)
cites work: Instantons and sheaves on \(C\mathbb{P}^3\)

The geometry of gauge fields (cites work)
cites work: Missed opportunities

The geometry of gauge fields (cites work)
cites work: Q5825485

The geometry of gauge fields (cites work)
cites work: Natural tensors on Riemannian manifolds

The geometry of gauge fields (cites work)
cites work: SO(3)-connections and the topology of 4-manifolds

The geometry of gauge fields (cites work)
cites work: An instanton-invariant for 3-manifolds

The geometry of gauge fields (cites work)
cites work: The topology of four-dimensional manifolds

The geometry of gauge fields (cites work)
cites work: Q3324665

The geometry of gauge fields (cites work)
cites work: Yang-Mills equations on the two-dimensional sphere

The geometry of gauge fields (cites work)
cites work: Splitting of the connection in gauge theories with broken symmetry

The geometry of gauge fields (cites work)
cites work: Q4059578

The geometry of gauge fields (cites work)
cites work: Gauge algebras, curvature and symplectic structure

The geometry of gauge fields (cites work)
cites work: Q3855860

The geometry of gauge fields (cites work)
cites work: Why instantons are monopoles

The geometry of gauge fields (cites work)
cites work: The Hamilton-Cartan formalism in the calculus of variations

The geometry of gauge fields (cites work)
cites work: Three exotic \(R^ 4\)'s and other anomalies

The geometry of gauge fields (cites work)
cites work: An infinite set of exotic \({\mathbb{R}}^ 4\)'s.

The geometry of gauge fields (cites work)
cites work: Q4119756

The geometry of gauge fields (cites work)
cites work: Q3672638

The geometry of gauge fields (cites work)
cites work: Integrality of the monopole number in SU(2) Yang-Mills-Higgs theory on \({\mathbb{R}}^ 3\)

The geometry of gauge fields (cites work)
cites work: The geometry of the moduli space of \(CP^ 2\) instantons

The geometry of gauge fields (cites work)
cites work: The Riemannian geometry of the Yang-Mills moduli space

The geometry of gauge fields (cites work)
cites work: The geometry of the Yang-Mills moduli space for definite manifolds

The geometry of gauge fields (cites work)
cites work: Solutions to Yang-Mills field equations in eight dimensions and the last Hopf map

The geometry of gauge fields (cites work)
cites work: On the geometry of Sp(1)-instantons with Pontrjagin index 1 on the 4- sphere

The geometry of gauge fields (cites work)
cites work: Group actions on principal bundles and invariance conditions for gauge fields

The geometry of gauge fields (cites work)
cites work: Group actions on principal bundles and dimensional reduction

The geometry of gauge fields (cites work)
cites work: Canonical connections on Riemannian symmetric spaces and solutions to the Einstein–Yang–Mills equations

The geometry of gauge fields (cites work)
cites work: Stable vector bundles and instantons

The geometry of gauge fields (cites work)
cites work: On the construction of monopoles

The geometry of gauge fields (cites work)
cites work: Monopole charges for arbitrary compact gauge groups and Higgs fields in any representation

The geometry of gauge fields (cites work)
cites work: On the moduli space of anti-self-dual Yang-Mills connections on Kähler surfaces

The geometry of gauge fields (cites work)
cites work: The moduli space of Yang-Mills connections over a Kähler surface is a complex manifold

The geometry of gauge fields (cites work)
cites work: Geometry of anti-self-dual connections and Kuranishi map

The geometry of gauge fields (cites work)
cites work: Symmetry of Einstein-Yang-Mills systems and dimensional reduction

The geometry of gauge fields (cites work)
cites work: Gauge-invariant description of several \((2+1)\)-dimensional integrable nonlinear evolution equations

The geometry of gauge fields (cites work)
cites work: Spontaneous symmetry breaking and principal fibre bundles

The geometry of gauge fields (cites work)
cites work: Geometric structure on the orbit space of gauge connections

The geometry of gauge fields (cites work)
cites work: Q3881604

The geometry of gauge fields (cites work)
cites work: The natural spinor connection on \(S_ 8\) is a gauge field

The geometry of gauge fields (cites work)
cites work: Graded Lie algebras and connections on a fibered space

The geometry of gauge fields (cites work)
cites work: Q3326383

The geometry of gauge fields (cites work)
cites work: On the geometric structure of gauge theories

The geometry of gauge fields (cites work)
cites work: Q3217319

The geometry of gauge fields (cites work)
cites work: A condition for paracompactness of a manifold

The geometry of gauge fields (cites work)
cites work: Spaces admitting gravitational fields

The geometry of gauge fields (cites work)
cites work: The mean curvature of gravitational fields

The geometry of gauge fields (cites work)
cites work: Q3805417

The geometry of gauge fields (cites work)
cites work: The geometry of gauge fields

The geometry of gauge fields (cites work)
cites work: Spin structures on manifolds

The geometry of gauge fields (cites work)
cites work: On the bundle of connections and the gauge orbit manifold in Yang-Mills theory

The geometry of gauge fields (cites work)
cites work: Q4721200

The geometry of gauge fields (cites work)
cites work: Canonical formulations of a classical particle in a Yang-Mills field and Wong's equations

The geometry of gauge fields (cites work)
cites work: Geometry of SU(2) gauge fields

The geometry of gauge fields (cites work)
cites work: Natural connections on Stiefel bundles are sourceless gauge fields

The geometry of gauge fields (cites work)
cites work: The fundamental equations of a submersion

The geometry of gauge fields (cites work)
cites work: Gauge theories on four dimensional Riemannian manifolds

The geometry of gauge fields (cites work)
cites work: Q3346914

The geometry of gauge fields (cites work)
cites work: A new definition of singular points in general relativity

The geometry of gauge fields (cites work)
cites work: A direct method for minimizing the Yang-Mills functional over 4- manifolds

The geometry of gauge fields (cites work)
cites work: Q3345477

The geometry of gauge fields (cites work)
cites work: Singular Sobolev connections with holonomy

The geometry of gauge fields (cites work)
cites work: Classification of singular Sobolev connections by their holonomy

The geometry of gauge fields (cites work)
cites work: Solutions to Yang—Mills equations that are not self-dual

The geometry of gauge fields (cites work)
cites work: Some remarks on the Gribov ambiguity

The geometry of gauge fields (cites work)
cites work: The Geometry of the Orbit Space for Non-Abelian Gauge Theories

The geometry of gauge fields (cites work)
cites work: Q5597399

The geometry of gauge fields (cites work)
cites work: Q3221053

The geometry of gauge fields (cites work)
cites work: Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field

The geometry of gauge fields (cites work)
cites work: Natural differential operators on Riemannian manifolds and representations of the orthogonal and special orthogonal groups

The geometry of gauge fields (cites work)
cites work: Surface integrals and monopole charges in non-Abelian gauge theories

The geometry of gauge fields (cites work)
cites work: The existence of a non-minimal solution to the SU(2) Yang-Mills-Higgs equations on \(R^ 3.\) I

The geometry of gauge fields (cites work)
cites work: The existence of a non-minimal solution to the SU(2) Yang-Mills-Higgs equations on \(R^ 3.\) II

The geometry of gauge fields (cites work)
cites work: Self-dual Yang-Mills connections on non-self-dual 4-manifolds

The geometry of gauge fields (cites work)
cites work: Path-connected Yang-Mills moduli spaces

The geometry of gauge fields (cites work)
cites work: Self-dual connections on 4-manifolds with indefinite intersection matrix

The geometry of gauge fields (cites work)
cites work: Some twisted self-dual solutions for the Yang-Mills equations on a hypertorus

The geometry of gauge fields (cites work)
cites work: Removable singularities in Yang-Mills fields

The geometry of gauge fields (cites work)
cites work: Connections with \(L^ p \)bounds on curvature

The geometry of gauge fields (cites work)
cites work: Q3675144

The geometry of gauge fields (cites work)
cites work: The Chern classes of Sobolev connections

The geometry of gauge fields (cites work)
cites work: Invariant Theoretical Interpretation of Interaction

The geometry of gauge fields (cites work)
cites work: On self-dual gauge fields

The geometry of gauge fields (cites work)
cites work: A universal phase space for particles in Yang-Mills fields

The geometry of gauge fields (cites work)
cites work: The unreasonable effectiveness of mathematics in the natural sciences. Richard courant lecture in mathematical sciences delivered at New York University, May 11, 1959

The geometry of gauge fields (cites work)
cites work: Q4179492

The geometry of gauge fields (cites work)
cites work: Integral Formalism for Gauge Fields

The geometry of gauge fields (cites work)
cites work: Q3906162

The geometry of gauge fields (cites work)
cites work: Conservation of Isotopic Spin and Isotopic Gauge Invariance