The geometrical framework for Yang–Mills theories

In the gauge natural bundle framework a new space is introduced and a first-order purely frame-formulation of General Relativity is obtained. In some of our recent works [1, 2, 3] a new geometrical framework for YangMills field theories and General Relativity in the tetrad-affine formulation has been developed.The construction of the new geometrical setting has been obtained quotienting the first-jet bundles of the configuration spaces of the above theories in a suitable way, resulting into the introduction of a new family of fiber bundles.In this letter we show that these new spaces allow a (covariant) first-order purely frame-formulation of General Relativity.The whole geometrical construction will be developed within the gauge natural bundle framework [4], which provides the suitable mathematical setting for globally describing gravity in the tetrad formalism.To start with, let M be a space-time manifold, allowing a metric tensor g with signature η = (1, 3): the manifold M will be called a η-manifold and the metric tensor canonical representation will be η µν := diag (−1, 1, 1, 1). Moreover, let L(M ) be the frame-bundle over M and P → M a principal fiber bundle over M with structural group SO (1, 3).The configuration space of the theory (the tetrad space) is a GL(4, ℜ) bundle π : E → M , associated to P × M L(M ) through the left-action λ : (SO(1, 3) × GL(4, ℜ)) × GL(4, ℜ) → GL(4, ℜ), λ(Λ, J; X) = Λ · X · J −1 (1) 1

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“…Moreover, let L(M ) be the frame-bundle over M and P → M a principal fiber bundle over M with structural group SO (1,3).…”

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confidence: 99%

“…Referring to [1] for the proof, it is easy to see that the only morphisms satisfying condition (10) are necessarily of the form:…”

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confidence: 99%

A first-order purely frame-formulation of general relativity

Cianci

Vignolo

Bruno

2005

Class. Quantum Grav.

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“…J -bundles have been recently used to provide new geometric formulations of gauge theories and GR [1][2][3][29][30][31][32].…”

Section: The Geometric Frameworkmentioning

confidence: 99%

“…In some previous papers [1][2][3], a new geometric approach for Gauge Theories and General Relativity (GR), in the tetrad-affine formulation, has been proposed. It is called the J -bundles framework (from now on the J -bundles).…”

Section: Introductionmentioning

confidence: 99%

“…In view of this fact, the basic idea developed in [1][2][3] consists of defining a suitable quotient space of the first jet-bundle, making equivalent two sections which have a first order contact with respect to the exterior differentiation (or, equivalently, with respect to the exterior covariant differentiation), instead of the whole set of derivatives. The resulting fiber coordinates of the so defined new spaces are exactly the antisymmetric combinations appearing in the Lagrangian densities.…”

Section: Introductionmentioning

confidence: 99%

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f(R) GRAVITY WITH TORSION: A GEOMETRIC APPROACH WITHIN THE $\mathcal{J}$-BUNDLES FRAMEWORK

Capozziello

Cianci

Stornaiolo

et al. 2008

Int. J. Geom. Methods Mod. Phys.

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We discuss the f (R)-theories of gravity with torsion in the framework of J -bundles. Such an approach is particularly useful since the components of the torsion and curvature tensors can be chosen as fiber J -coordinates on the bundles and then the symmetries and the conservation laws of the theory can be easily achieved. Field equations of f (R)-gravity are studied in empty space and in presence of various forms of matter as Dirac fields, Yang-Mills fields and spin perfect fluid. Such fields enlarge the jet bundles framework and characterize the dynamics. Finally we give some cosmological applications and discuss the relations between f (R)-gravity and scalar-tensor theories.

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A square‐torsion modification of Einstein‐Cartan theory

2012

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In the present paper we consider a theory of gravity in which not only curvature but also torsion is explicitly present in the Lagrangian, both with their own coupling constant. In particular, we discuss the couplings to Dirac fields and spin fluids: in the case of Dirac fields, we discuss how in our approach, the Dirac self-interactions depend on the coupling constant as a parameter that may even make these non-linearities manifest at subatomic scales, showing different applications according to the value of the parameter we have assigned; in the case of spin fluids, we discuss FLRW cosmological models arising from the proposed theory

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The geometrical framework for Yang–Mills theories

Cited by 12 publications

References 4 publications

A first-order purely frame-formulation of general relativity

A first-order purely frame-formulation of general relativity

f(R) GRAVITY WITH TORSION: A GEOMETRIC APPROACH WITHIN THE $\mathcal{J}$-BUNDLES FRAMEWORK

A square‐torsion modification of Einstein‐Cartan theory

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