Three-dimensional gravity and deformations of relativistic symmetries

TL;DR

This paper explores the mathematical structures of deformed relativistic symmetries in quantum gravity, focusing on noncommutative spacetime, curved momentum space, and their implications in three-dimensional gravity models.

Contribution

It investigates the properties and relations of $oldsymbol{ ext{kappa}-}$Poincaré algebra, noncommutative geometry, and curved momentum space within quantum gravity frameworks.

Findings

Spectral dimension of $oldsymbol{ ext{kappa}-}$Minkowski space analyzed.

Alternative limits of Chern-Simons theory for 3D gravity presented.

Spaces of momenta for conical defects in higher dimensions discussed.

Abstract

It is possible that relativistic symmetries become deformed in the semiclassical regime of quantum gravity. Mathematically, such deformations lead to the noncommutativity of spacetime geometry and non-vanishing curvature of momentum space. The best studied example is given by the $\kappa$-Poincar\'e Hopf algebra, associated with $\kappa$-Minkowski space. On the other hand, the curved momentum space is a well-known feature of particles coupled to three-dimensional gravity. The purpose of this thesis was to explore some properties and mutual relations of the above two models. In particular, I study extensively the spectral dimension of $\kappa$-Minkowski space. I also present an alternative limit of the Chern-Simons theory describing three-dimensional gravity with particles. Then I discuss the spaces of momenta corresponding to conical defects in higher dimensional spacetimes. Finally, I consider the Fock space construction for the quantum theory of particles in three-dimensional gravity.