THE YANG–MILLS FUNCTIONAL AND BOGOMOLOV INEQUALITY FOR ARBITRARY PRINCIPAL BUNDLES OVER KÄHLER MANIFOLDS

2013 ◽  
Vol 10 (08) ◽  
pp. 1360015
Author(s):  
CARLOS TEJERO PRIETO
Keyword(s):  
Type Inequality ◽  
Structure Group ◽  
Fiber Bundles ◽  
Characteristic Classes ◽  
Compact Lie Group ◽  
Absolute Minimum ◽  
Yang Mills ◽  
Group A ◽  
Absolute Minimizers ◽  
Principal Fiber Bundles

We study the Yang–Mills functional for principal fiber bundles with structure group a compact Lie group K over a Kähler manifold. In particular, we analyze the absolute minimizers for this functional and prove that they are exactly the Einstein K-connections. By means of the structure of the Yang–Mills functional at an absolute minimum, we prove that the characteristic classes of a principal K-bundle which admits an Einstein connection satisfy two inequalities. One of them is a generalization of the Bogomolov inequality whereas the other is an inequality related to the center of the structure group. Therefore, this way we offer a new and natural proof of the Bogomolov inequality that helps understanding its origin. Finally, in view of the Hitchin–Kobayashi correspondence we prove that every (poly-)stable principal Kℂ-bundle has to satisfy this generalized Bogomolov type inequality.

CLASSIFICATION OF GAUGE-RELATED INVARIANT CONNECTIONS

1993 ◽  
Vol 05 (01) ◽  
pp. 69-103 ◽  
Author(s):  
R. BAUTISTA ◽  
J. MUCIÑO ◽  
E. NAHMAD-ACHAR ◽  
M. ROSENBAUM
Keyword(s):  
Symmetry Group ◽  
Moduli Space ◽  
Structure Group ◽  
Fiber Bundles ◽  
Explicit Solutions ◽  
Arbitrary Structure ◽  
Holonomy Groups ◽  
Principal Fiber Bundles

Connection 1-forms on principal fiber bundles with arbitrary structure groups are considered, and a characterization of gauge-equivalent connections in terms of their associated holonomy groups is given. These results are then applied to invariant connections in the case where the symmetry group acts transitively on fibers, and both local and global conditions are derived which lead to an algebraic procedure for classifying orbits in the moduli space of these connections. As an application of the developed techniques, explicit solutions for SU (2) × SU (2)-symmetric connections over S2 × S2, with SU(2) structure group, are derived and classified into non-gauge-related families, and multi-instanton solutions are identified.

The uniqueness of bundle transfers

1983 ◽  
Vol 93 (1) ◽  
pp. 87-111 ◽  
Author(s):  
L. G. Lewis
Keyword(s):  
Lie Group ◽  
Cohomology Theory ◽  
Structure Group ◽  
Compact Lie Group ◽  
Group A

AbstractLet K be a cohomology theory. Axioms are given which uniquely characterize the transfer in K-cohomology for bundles with structure group a compact Lie group and fibre a finite C.W. complex. Also, a family of generalized transfers is defined which includes both the standard transfer and Atiyah's holomorphic transfer.

GEOMETRIC STRUCTURE OF THE HILBERT–YANG–MILLS FUNCTIONAL

2008 ◽  
Vol 05 (03) ◽  
pp. 387-405 ◽  
Author(s):  
A. PATÁK ◽  
D. KRUPKA
Keyword(s):  
Einstein Equations ◽  
Fiber Bundles ◽  
Variational Theory ◽  
Geometric Structures ◽  
Variation Formula ◽  
Yang Mills ◽  
Variational Functional ◽  
Fibered Manifolds ◽  
Prolongation Theory ◽  
Principal Fiber Bundles

The global variational functional, defined by the Hilbert–Yang–Mills Lagrangian over a smooth manifold, is investigated within the framework of prolongation theory of principal fiber bundles, and global variational theory on fibered manifolds. The principal Lepage equivalent of this Lagrangian is constructed, and the corresponding infinitesimal first variation formula is obtained. It is shown, in particular, that the Noether currents, associated with isomorphisms of the underlying geometric structures, split naturally into several terms, one of which is the exterior derivative of the Komar–Yang–Mills superpotential. Consequences of invariance of the Hilbert–Yang–Mills Lagrangian under isomorphisms of underlying geometric structures such as Noether's conservation laws for global currents are then established. As an example, a general formula for the Komar–Yang–Mills superpotential of the Reissner–Nordström solution of the Einstein equations is found.

Approximate Hitchin–Kobayashi correspondence for Higgs G-bundles

2014 ◽  
Vol 11 (07) ◽  
pp. 1460015 ◽  
Author(s):  
Ugo Bruzzo ◽  
Beatriz Graña Otero
Keyword(s):  
Reductive Group ◽  
Kähler Manifold ◽  
Higgs Bundle ◽  
Structure Group ◽  
Higgs Bundles ◽  
Yang Mills ◽  
Kahler Manifold ◽  
Group A ◽  
Compact Kähler Manifold

We announce a result about the extension of the Hitchin–Kobayashi correspondence to principal Higgs bundles. A principal Higgs bundle on a compact Kähler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian–Yang–Mills structure.

scholarly journals A consistent measure for lattice Yang–Mills

2017 ◽  
Vol 32 (02n03) ◽  
pp. 1750016
Author(s):  
R. Vilela Mendes
Keyword(s):  
Structure Group ◽  
Cartesian Products ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Past Work ◽  
Mass Gap ◽  
Hypercubic Lattice ◽  
Group A ◽  
Interaction Measure ◽  
Projective Limits

The construction of a consistent measure for Yang–Mills is a precondition for an accurate formulation of nonperturbative approaches to QCD, both analytical and numerical. Using projective limits as subsets of Cartesian products of homomorphisms from a lattice to the structure group, a consistent interaction measure and an infinite-dimensional calculus have been constructed for a theory of non-Abelian generalized connections on a hypercubic lattice. Here, after reviewing and clarifying past work, new results are obtained for the mass gap when the structure group is compact.

scholarly journals Yang–Mills connections on compact complex tori

2015 ◽  
Vol 07 (02) ◽  
pp. 293-307
Author(s):  
Indranil Biswas
Keyword(s):  
Algebraic Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Structure Group ◽  
Kähler Structure ◽  
Yang Mills ◽  
Complex Torus ◽  
Complex Tori

Let G be a connected reductive complex affine algebraic group and K ⊂ G a maximal compact subgroup. Let M be a compact complex torus equipped with a flat Kähler structure and (EG, θ) a polystable Higgs G-bundle on M. Take any C∞ reduction of structure group EK ⊂ EG to the subgroup K that solves the Yang–Mills equation for (EG, θ). We prove that the principal G-bundle EG is polystable and the above reduction EK solves the Einstein–Hermitian equation for EG. We also prove that for a semistable (respectively, polystable) Higgs G-bundle (EG, θ) on a compact connected Calabi–Yau manifold, the underlying principal G-bundle EG is semistable (respectively, polystable).

scholarly journals A MANIFESTLY GAUGE INVARIANT AND UNIVERSAL CALCULUS FORSU(N) YANG–MILLS

2006 ◽  
Vol 21 (23n24) ◽  
pp. 4627-4761 ◽  
Author(s):  
OLIVER J. ROSTEN
Keyword(s):  
Perturbation Theory ◽  
Renormalization Group ◽  
Explicit Dependence ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Group A ◽  
Β Function ◽  
Exact Renormalization Group

Within the framework of the Exact Renormalization Group, a manifestly gauge invariant calculus is constructed for SU (N) Yang–Mills. The methodology is comprehensively illustrated with a proof, to all orders in perturbation theory, that the β function has no explicit dependence on either the seed action or details of the covariantization of the cutoff. The cancellation of these nonuniversal contributions is done in an entirely diagrammatic fashion.

Holonomies of Yang–Mills connections for bundles on a surface with disconnected structure group

1994 ◽  
Vol 116 (2) ◽  
pp. 375-384 ◽  
Author(s):  
Johannes Huebschmann
Keyword(s):  
Moduli Space ◽  
Scalar Product ◽  
Additive Group ◽  
Closed Surface ◽  
Riemannian Metric ◽  
Compact Lie Group ◽  
Universal Central Extension ◽  
Yang Mills ◽  
Disconnected Structure ◽  
Image Position

AbstractLet Σ be a closed surface of genus ≥ 1, G a compact Lie group, not necessarily connected with Lie algebra g, ξ,: P → Σ a principal G-bundle, and suppose Σ equipped with a Riemannian metric and g with an invariant scalar product so that the Yang—Mills equations on ξ are defined. Further, letbe the universal central extension of the fundamental group π of Σ and ΓR the group obtained from Γ when its centre Z is extended to the additive group R of the reals. We show that there are bijective correspondences between various spaces of classes of Yang—Mills connections on ξ and spaces of representations of Γ and ΓR (as appropriate) in G. In particular, we show that the holonomy establishes a homeomorphism between the moduli space N(ξ) of central Yang–Mills connections on ξ and the space Repξ(Γ, G) of representations of Γ in G determined by ξ. Our results rely on a detailed study of the holonomy of a central Yang–Mills connection and extend corresponding ones of Atiyah and Bott for the case where G is connected.

CQI Tools Sentinel Events, Warning, and Action Limits

1993 ◽  
Vol 14 (9) ◽  
pp. 537-539
Author(s):  
David Birnbaum
Keyword(s):  
Management Control ◽  
Detailed Knowledge ◽  
Specific Situation ◽  
Absolute Minimum ◽  
Continuous Quality ◽  
Surveillance Programs ◽  
Group A ◽  
Sentinel Events ◽  
To Come ◽  
Systematic Response

Sentinel events, that is, events whose single occurrence is of sufficient concern to trigger systematic response, recently were advocated as an important component in managing quality under Continuous Quality Improvement (CQI). Ideally, sentinel events are exceedingly rare and invariably indicate preventable deficiencies (eg, operation on the wrong patient, which should never occur, denotes a lack of management control requiring corrective action). However, although single cases of nosocomial group A streptococcal surgical wound infection or of tuberculosis are harbingers of worse to come unless prompt intervention is initiated, sentinel events generally have not served infection surveillance programs adequately. This is because nosocomial infection is the result of multifactorial chains of events that produce a probability, not a certainty, of infection, and our knowledge is incomplete. Without detailed knowledge of these probabilistic chains, the absolute minimum rates of infection that may be attained, and the optimal statistical approaches to define predictive outbreak “warning” or “action” levels, we will not be able to define meaningful sentinel events. How, then, can we best assure health services quality?Nearly 30 years ago, Drucker suggested that three elements comprise effective management decisions. They involve determining 1) whether a specific situation is generic or an exception, 2) clear specification of what the decision has to accomplish, and 3) what is right rather than simply what is acceptable.

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