We present a noncommutative extension of Einstein-Hilbert gravity in the context of twist-deformed space-time, with a $\star$-product associated to a quite general triangular Drinfeld twist. In particular the $\star$-product can be chosen to be the usual Groenewald-Moyal product. The action is geometric, invariant under diffeomorphisms and centrally extended Lorentz $\star$-gauge transformations. In the commutative limit it reduces to ordinary gravity, with local Lorentz invariance and usual real vielbein. This we achieve by imposing a charge conjugation condition on the noncommutative vielbein. The theory is coupled to fermions, by adding the analog of the Dirac action in curved space. A noncommutative Majorana condition can be imposed, consistent with the $\star$-gauge transformations. Finally, we discuss the noncommutative version of the Mac-Dowell Mansouri action, quadratic in curvatures.
We investigate a relation of the contravariant geometry to the emergent gravity from noncommutative gauge theories. We give a refined formulation of the contravariant gravity and provide solutions to the contravariant Einstein equation. We linearize the equation around background solutions, including curved ones. A noncommutative gauge theory on the Moyal plane can be rewritten as an ordinary gauge theory on a curved background via the Seiberg-Witten map, which is known as the emergent gravity. We show that this phenomenon also occurs for a gauge theory on a noncommutative homogeneous Kähler b…
We summarise recent perspectives on symmetries of noncommutative field theories based on homotopy algebras. We show how these viewpoints naturally lead to a new class of noncommutative field theories which possess braided gauge symmetries, and explain in detail their uses in gravity. We review how these considerations lead to a new theory of noncommutative gravity in four dimensions within the Einstein-Cartan-Palatini formalism.
We unveil the geometric nature of the multiplet of fundamental fermions in the Standard Model of fundamental particles as a noncommutative analogue of de Rham forms on the internal finite quantum space.
A solution of the so-called fermion doubling problem in Connes' Noncommutative Standard Model has been given by Barrett in 2006 in the form of Majorana-Weyl conditions on the fermionic field. These conditions define a ${\cal U}_{J,χ}$-invariant subspace of the correct physical dimension, where ${\cal U}_{J,χ}$ is the group of Krein unitaries commuting with the chirality and real structure. They require the KO-dimension of the total triple to be $0$. In this paper we show that this solution is, up to some trivial modifications, and under some mild assumptions on the finite triple, the only one …
We define a new homotopy algebraic structure, that we call a braided $L_\infty$-algebra, and use it to systematically construct a new class of noncommutative field theories, that we call braided field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which do not act on the solutions of the field equations. We use Drinfel'd twist deformation quantization techniques to generate new noncommutative deformations of classical field theories with braided gauge symmetries, …
We propose a way to understand the 3 fermion generations by the algebraic structures of noncommutative geometry, which is a promising framework to unify the standard model and general relativity. We make the tensor product extension and the quaternion extension on the framework. Each of the two extensions alone keeps the action invariant, and we consider them as the almost trivial structures of the geometry. We combine the two extensions, and show the corresponding physical effects, i.e., the emergence of 3 fermion generations and the mass relationships among those generations. We define the c…
Based on the construction of the 4-dim noncommutative gravity model described in our previous work, first, a more extended description of the covariant noncommutative space (fuzzy 4-dim de Sitter space), which accommodates the gravity model, is presented and then the corresponding field equations, which are obtained after variation of the previously proposed action, are extracted. Also, a spontaneous breaking of the initial symmetry is performed, this time induced by the introduction of an auxiliary scalar field, and its implications in the reduced theory, which is produced after considering t…
In our previous work we have constructed a model of noncommutative (NC) gravity based on $SO(2,3)_\star$ gauge symmetry. In this paper we extend the model by adding matter fields: fermions and a $U(1)$ gauge field. Using the enveloping algebra approach and the Seiberg-Witten map we construct actions for these matter fields and expand the actions up to first order in the noncommutativity (deformation) parameter. Unlike in the case of pure NC gravity, first non-vanishing NC corrections are linear in the noncommutativity parameter. In the flat space-time limit we obtain a non-standard NC Electrod…
Massive particles coupled with 2+1 dimensional gravity and noncommutative field theory
最近，在李代数非对易空间[x^i，x^j]= 2 i kappa <$^{ijk}x_k（i，j，k= 1，2，3）（其中k =4 pi G）中，有质量粒子与无自旋粒子耦合的Ponzano-Regge模型的有效场论由三维欧氏非对易标量场给出.本文研究了与2+1维爱因斯坦引力耦合的无自旋大质量粒子与洛伦兹非对易场论之间是否存在联系。指出2+1维Einstein引力中无自旋质量粒子的动量空间一般不同于SL（2，R）/Z_2群空间所给出的非对易场论的动量空间。
