Curved momentum spaces from quantum groups with cosmological constant

TL;DR

This paper constructs the momentum space geometry for quantum deformations of de Sitter symmetries with a cosmological constant, revealing a curved, de Sitter-like structure that generalizes known flat-space results.

Contribution

It explicitly constructs the momentum space for the $"kappa$-deformation of de Sitter algebra in lower dimensions, incorporating the cosmological constant and unifying translation and boost momenta.

Findings

Momentum space has de Sitter geometry including translation and boost momenta.

Reduces to known $ ext{kappa}$-Poincaré momentum space as $\Lambda o 0$.

Method applicable to higher-dimensional and other quantum deformations.

Abstract

We bring the concept that quantum symmetries describe theories with nontrivial momentum space properties one step further, looking at quantum symmetries of spacetime in presence of a nonvanishing cosmological constant $\Lambda$. In particular, the momentum space associated to the $\kappa$-deformation of the de Sitter algebra in (1+1) and (2+1) dimensions is explicitly constructed as a dual Poisson-Lie group manifold parametrized by $\Lambda$. Such momentum space includes both the momenta associated to spacetime translations and the `hyperbolic' momenta associated to boost transformations, and has the geometry of (half of) a de Sitter manifold. Known results for the momentum space of the $\kappa$-Poincar\'e algebra are smoothly recovered in the limit $\Lambda\to 0$, where hyperbolic momenta decouple from translational momenta. The approach here presented is general and can be applied to other quantum deformations of kinematical symmetries, including (3+1)-dimensional ones.