Three Dimensional Quantum Geometry and Deformed Poincare Symmetry

TL;DR

This work constructs a non-commutative 3D geometry from the quantum double $\mathcal{D}(SU(2))$, viewed as a deformation of the Euclidean isometry group $ISU(2)$, enabling a covariant QFT on a curved momentum space with bounded momenta. The authors develop the convolution algebra $C(SU(2))^*$ and its star-product $\star$ to realize the non-commutative space $C_{\ell_P}(\mathbb{E}^3)$, and show how translations induce a deformed addition of momenta while rotations act via the adjoint representation. They further establish a fuzzy-space formulation $\text{Mat}(\mathbb{C})$ through a Fourier transform on $C(SU(2))^*$, revealing a hierarchy of concentric fuzzy spheres and a concrete map between matrix data and function space. The coordinate operators satisfy an $\mathfrak{su}(2)$ algebra, yielding a discrete spectrum of spatial coordinates and a well-defined, UV-regular propagator. Overall, the paper provides a coherent framework linking deformed isometries, non-commutative geometry, and QFT in three dimensions, with clear routes to extensions to Lorentzian signatures and nonzero cosmological constants.

Abstract

We study a three dimensional non-commutative space emerging in the context of three dimensional Euclidean quantum gravity. Our starting point is the assumption that the isometry group is deformed to the Drinfeld double D(SU(2)). We generalize to the deformed case the construction of the flat Euclidean space as the quotient of its isometry group ISU(2) by SU(2). We show that the algebra of functions becomes the non-commutative algebra of SU(2) distributions endowed with the convolution product. This construction gives the action of ISU(2) on the algebra and allows the determination of plane waves and coordinate functions. In particular, we show that: (i) plane waves have bounded momenta; (ii) to a given momentum are associated several SU(2) elements leading to an effective description of an element in the algebra in terms of several physical scalar fields; (iii) their product leads to a deformed addition rule of momenta consistent with the bound on the spectrum. We generalize to the non-commutative setting the local action for a scalar field. Finally, we obtain, using harmonic analysis, another useful description of the algebra as the direct sum of the algebra of matrices. The algebra of matrices inherits the action of ISU(2): rotations leave the order of the matrices invariant whereas translations change the order in a way we explicitly determine.