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.

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.

scholarly journals GRADED GEOMETRIC STRUCTURES UNDERLYING F-THEORY RELATED DEFECT THEORIES

2013 ◽  
Vol 28 (21) ◽  
pp. 1350106
Author(s):  
V. K. OIKONOMOU
Keyword(s):  
Vector Space ◽  
Dimensional Space ◽  
Composite Fiber ◽  
Fiber Bundles ◽  
Canonical Bundle ◽  
Geometric Structures ◽  
Yang Mills ◽  
Zero Modes ◽  
Fermionic Fields ◽  
Related Defect

In the context of F-theory, we study the related eight-dimensional super-Yang–Mills theory and reveal the underlying supersymmetric quantum mechanics algebra that the fermionic fields localized on the corresponding defect theory are related to. Particularly, the localized fermionic fields constitute a graded vector space, and in turn this graded space enriches the geometric structures that can be built on the initial eight-dimensional space. We construct the implied composite fiber bundles, which include the graded affine vector space and demonstrate that the composite sections of this fiber bundle are in one-to-one correspondence to the sections of the square root of the canonical bundle corresponding to the submanifold on which the zero modes are localized.

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.

scholarly journals THE SEIBERG–WITTEN MAP FOR NON-COMMUTATIVE PURE GRAVITY AND VACUUM MAXWELL THEORY

2013 ◽  
Vol 10 (06) ◽  
pp. 1350023 ◽  
Author(s):  
ELISABETTA DI GREZIA ◽  
GIAMPIERO ESPOSITO ◽  
MARCO FIGLIOLIA ◽  
PATRIZIA VITALE
Keyword(s):  
Explicit Construction ◽  
Einstein Equations ◽  
Electromagnetic Potential ◽  
Field Equations ◽  
Maxwell Theory ◽  
Commutative Field ◽  
Yang Mills ◽  
First Order ◽  
Pure Gravity ◽  
Vacuum Solutions

In this paper the Seiberg–Witten map is first analyzed for non-commutative Yang–Mills theories with the related methods, developed in the literature, for its explicit construction, that hold for any gauge group. These are exploited to write down the second-order Seiberg–Witten map for pure gravity with a constant non-commutativity tensor. In the analysis of pure gravity when the classical space–time solves the vacuum Einstein equations, we find for three distinct vacuum solutions that the corresponding non-commutative field equations do not have solution to first order in non-commutativity, when the Seiberg–Witten map is eventually inserted. In the attempt of understanding whether or not this is a peculiar property of gravity, in the second part of the paper, the Seiberg–Witten map is considered in the simpler case of Maxwell theory in vacuum in the absence of charges and currents. Once more, no obvious solution of the non-commutative field equations is found, unless the electromagnetic potential depends in a very special way on the wave vector.

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