Braided $$\varvec{L_{\infty }}$$-algebras, braided field theory and noncommutative gravity

We define a new homotopy algebraic structure, that we call a braided $$L_\infty $$ L ∞ -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, which we compare to the more conventional theories with star-gauge symmetries. We apply our formalism to introduce a braided version of general relativity without matter fields in the Einstein–Cartan–Palatini formalism. In the limit of vanishing deformation parameter, the braided theory of noncommutative gravity reduces to classical gravity without any extensions.

Introducing matter fields in model of noncommutative gravity

Theory founded on SO(2,3)* gauge symmetry. One significant feature of this approach is that gravitational field, given by the vierbein, becomes manifest only after a suitable gauge fixing and it is formally united with other gauge fields. Starting from a model of pure noncommutative gravity, we extend it by introducing fermions and Yang-Mills gauge field. Using the enveloping algebra approach and the Seiberg-Witten map we construct corresponding actions and expand them perturbatively in powers of the canonical noncommutativity parameter ???. Unlike in the case of pure noncommutative gravity, first non-vanishing noncommutative corrections are linear in the noncommutativity parameter and they describe the coupling of matter and gauge fields with gravity due to spacetime noncommutativity. This is augmented by the fact that some of these corrections pertain even in flat spacetime where they induce potentially observable noncommutative deformations. We discuss the effects of noncommutativity on electron?s dispersion relation in the presence of constant background magnetic field - Landau levels. Our results could be useful for further investigation of phenothe menological consequences of spacetime noncommutativity.

Yang–Mills theory in the SO(2,3)⋆ model of noncommutative gravity

According to the standard cosmological model, thermodynamic conditions of the early Universe were such that nuclear matter existed in the state of quark–gluon plasma, rather than hadrons. On the other hand, it is generally believed that quantum gravity effects become ever more stronger as we approach the Big Bang, in particular, we expect that the phenomenon of space–time noncommutativity will be significant. Thus we are led to consider the properties of quarks and gluons in noncommutative space–time. For this, we employ the [Formula: see text] model of noncommutative gravity. As a first step towards the full theoretical treatment of the effects of noncommutativity on quark–gluon plasma, our main goal in this paper is to consistently incorporate Yang–Mills gauge fields in the [Formula: see text] framework and investigate their coupling to gravity that arises due to space–time noncommutativity. We construct an action that is invariant under deformed [Formula: see text] gauge transformations and expand it perturbatively in orders of the canonical deformation parameter [Formula: see text] via Seiberg–Witten map. In particular, we analyze the flat-space–time limit and demonstrate that residual noncommutativity induces various new couplings of quarks and gluons.

Relating Noncommutative SO(2,3)☆ Gravity to the Lorentz-Violating Standard-Model Extension

We consider a model of noncommutative gravity that is based on a spacetime with broken local SO(2,3) ☆ symmetry. We show that the torsion-free version of this model is contained within the framework of the Lorentz-violating Standard-Model Extension (SME). We analyze in detail the relation between the torsion-free, quadratic limits of the broken SO(2,3) ☆ model and the Standard-Model Extension. As part of the analysis, we construct the relevant geometric quantities to quadratic order in the metric perturbation around a flat background.