Quantum Anti-de Sitter space and sphere at roots of unity

TL;DR

This paper constructs a q-deformed Anti-de Sitter space and sphere at roots of unity, exploring their algebraic structures, symmetries, and implications for scalar fields with high-energy cutoffs, relevant for quantum gravity models.

Contribution

It introduces a covariant algebra of functions on q-deformed AdS and spheres at roots of unity, analyzing star-structures and scalar fields with intrinsic cutoffs, extending quantum geometry frameworks.

Findings

Scalar fields have a natural high-energy cutoff.

Fields are realized on orbifolds involving finite groups.

Analogous structures are established for q-deformed spheres.

Abstract

An algebra of functions on q-deformed Anti-de Sitter space AdS_q^D is defined which is covariant under U_q(so(2,D-1)), for q a root of unity. The star-structure is studied in detail. The scalar fields have an intrinsic high-energy cutoff, and arise most naturally as fields on orbifolds AdS_q^D \times S^D/G if D is odd, and AdS_q^D \times S_{\chi}^{2D-1}/G if D is even. Here G is a finite abelian group, and S_{\chi} is a certain ``chiral sector'' of the classical sphere. Hilbert spaces of square integrable functions are discussed. Analogous results are found for the q-deformed sphere S_q^D.