Noncommutative self-dual gravity

García‐Compeán, Hugo; Obregón, O.; Ramı́rez, C.; Sabido, M.

doi:10.1103/physrevd.68.044015

Cited by 83 publications

(117 citation statements)

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“…The application of these ideas leads us to a consistent noncommutative deformation of the full BRST fermionic symmetry of this topological theory given by Eqs. (4), (5) and (6). The results of noncommutative deformations of a standard BRST quantization of Ref.…”

Section: Discussionmentioning

confidence: 99%

“…Formulating a noncommutative topological theory is another way of finding noncommutative topological invariants. Noncommutative classical invariants (as the Euler number or signature) are known in the literature from some years ago in the context of the standard formulation of spectral triples from noncommutative geometry [5] and recently has been also pursued by using the method of Moyal product and the Seiberg-Witten map [6].…”

Section: Introductionmentioning

confidence: 99%

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A noncommutative deformation of topological field theory

García‐Compeán

Paniagua

2005

Gen Relativ Gravit

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Cohomological Yang-Mills theory is formulated on a noncommutative differentiable four manifold through the θ-deformation of its corresponding BRST algebra. The resulting noncommutative field theory is a natural setting to define the θ-deformation of Donaldson invariants and they are interpreted as a mapping between the Chevalley-Eilenberg homology of noncommutative spacetime and the Chevalley-Eilenberg cohomology of noncommutative moduli of instantons. In the process we find that in the weak coupling limit the quantum theory is localized at the moduli space of noncommutative instantons. * This work is dedicated to Professor Alberto García on the occasion of his 60th birthday.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A noncommutative deformation of topological field theory

García‐Compeán

Paniagua

2005

Gen Relativ Gravit

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“…For P s being the principal bundle with fiber Spin c (n), we introduce the complex linear bundle L(λ c ) = P S × Spin c (n) C defined as the factor space of P S × C on equivalence relation 22) where t ∈ Spin c (n). This linear bundle is associated to complex spinor structure on λ c .…”

Section: Almost Complex Spinor Structuresmentioning

confidence: 99%

“…The work of S. Vacaru is supported by a NATO/Portugal fellowship at CENTRA, Instituto Superior Tecnico, Lisbon. The author is grateful to R. Ablamowicz, John Ryan, and B. Fauser for collaboration and support of his participation at "The 6th International Conference on Clifford Algebras," Cookeville, Tennessee, USA (May, [20][21][22][23][24][25] 2002). He would like to thank J. P. S. Lemos, R. Miron, M. Anastasiei, and P. Stavrinos for hospitality and support.…”

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confidence: 99%

“…Introduction. Nowadays, interest has been established in non-Riemannian geometries derived in the low-energy string theory [18,64,65], noncommutative geometry [1,3,8,12,15,22,32,34,53,55,67,109,111,112], and quantum groups [33,35,36,37]. Various types of Finsler-like structures can be parametrized by generic off-diagonal metrics, which cannot be diagonalized by coordinate transforms but only by anholonomic maps with associated nonlinear connection (in brief, N-connection).…”

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confidence: 99%

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Nonlinear connections and spinor geometry

Vacaru¹,

Vicol

2004

International Journal of Mathematics and Mathematical Sciences

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We present an introduction to the geometry of higher-order vector and covector bundles (including higher-order generalizations of the Finsler geometry and Kaluza-Klein gravity) and review the basic results on Clifford and spinor structures on spaces with generic local anisotropy modeled by anholonomic frames with associated nonlinear connection structures. We emphasize strong arguments for application of Finsler-like geometries in modern string and gravity theory, noncommutative geometry and noncommutative field theory, and gravity.2000 Mathematics Subject Classification: 15A66, 58B20, 53C60, 83C60, 83E15.1. Introduction. Nowadays, interest has been established in non-Riemannian geometries derived in the low-energy string theory [18,64,65], noncommutative geometry [1,3,8,12,15,22,32,34,53,55,67,109,111,112], and quantum groups [33,35,36,37]. Various types of Finsler-like structures can be parametrized by generic off-diagonal metrics, which cannot be diagonalized by coordinate transforms but only by anholonomic maps with associated nonlinear connection (in brief, N-connection). Such structures may be defined as exact solutions of gravitational field equations in the Einstein gravity and its generalizations [75,79,80,94,95,96,97,98,99,100,102,103,104,105,109,110,111], for instance, in the metric-affine [19,23,56] Riemann-Cartan gravity [24,25]. Finsler-like configurations are considered in locally anisotropic thermodynamics, kinetics, related stochastic processes [85,96,107,108], and (super-) string theory [84,87,90,91,92].The following natural step in these lines is to elucidate the theory of spinors in effectively derived Finsler geometries and to relate this formalism of Clifford structures to noncommutative Finsler geometry. It should be noted that the rigorous definition of spinors for Finsler spaces and generalizations was not a trivial task because (on such spaces) there are no defined even local groups of automorphisms. The problem was solved in [82,83,88,89,93] by adapting the geometric constructions with respect to anholonomic frames with associated N-connection structure. The aim of this work is to outline the geometry of generalized Finsler spinors in a form more oriented to applications in modern mathematical physics.We start with some historical remarks: the spinors studied by mathematicians and physicists are connected with the general theory of Clifford spaces introduced in 1876 [14]. The theory of spinors and Clifford algebras play a major role in contemporary physics and mathematics. The spinors were discovered by Èlie Cartan in 1913 in 1190 S. I. VACARU AND N. A. VICOL mathematical form in his researches on representation group theory [10,11]; he showed that spinors furnish a linear representation of the groups of rotations of a space of arbitrary dimensions. The physicists Pauli [60] and Dirac [20] (in 1927, resp., for the three-dimensional and four-dimensional space-times) introduced spinors for the representation of the wave functions. In general relativity theory spinors and the Dirac equat...

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Celestial chiral algebras, colour-kinematics duality and integrability

Monteiro

2023

J. High Energ. Phys.

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We study celestial chiral algebras appearing in celestial holography, using the light-cone gauge formulation of self-dual Yang-Mills theory and self-dual gravity, and explore also a deformation of the latter. The recently discussed w1+∞ algebra in self-dual gravity arises from the soft expansion of an area-preserving diffeomorphism algebra, which plays the role of the kinematic algebra in the colour-kinematics duality and the double copy relation between the self-dual theories. The W1+∞ deformation of w1+∞ arises from a Moyal deformation of self-dual gravity. This theory is interpreted as a constrained chiral higher-spin gravity, where the field is a tower of higher-spin components fully constrained by the graviton component. In all these theories, the chiral structure of the operator-product expansion exhibits the colour-kinematics duality: the implicit ‘left algebra’ is the self-dual kinematic algebra, while the ‘right algebra’ provides the structure constants of the operator-product expansion, ensuring its associativity at tree level. In a scattering amplitudes version of the Ward conjecture, the left algebra ensures the classical integrability of this type of theories. In particular, it enforces the vanishing of the tree-level amplitudes via the double copy.

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Noncommutative self-dual gravity

Cited by 83 publications

References 40 publications

A noncommutative deformation of topological field theory

A noncommutative deformation of topological field theory

Nonlinear connections and spinor geometry

Celestial chiral algebras, colour-kinematics duality and integrability

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