q-Deformed de Sitter/Conformal Field Theory Correspondence

TL;DR

This work addresses formulating a holographic description of de Sitter space that preserves unitarity and yields a finite-dimensional Hilbert space via a $q$-deformation of the isometry group. It generalizes the 2D construction to dS$_3$/CFT$_2$ by building left-right tensor products of cyclic $U_q(su(1,1)_R)$ representations at a root of unity, which contract to the $SL(2,\mathbb{C})$ principal series as $N\to\infty$. The paper computes an entanglement entropy across the cosmological horizon using a brick-wall–like regularization in the quantum-deformed setting, finding a finite $N$-dependent entropy tied to the representation data and horizon thermodynamics. These results point to a finite-$N$ holographic description of de Sitter space with prospects for a full $q$-deformed Virasoro symmetry and deeper links between quantum groups and cosmological entropy.

Abstract

Unitary principal series representations of the conformal group appear in the dS/CFT correspondence. These are infinite dimensional irreducible representations, without highest weights. In earlier work of Guijosa and the author it was shown for the case of two-dimensional de Sitter, there was a natural q-deformation of the conformal group, with q a root of unity, where the unitary principal series representations become finite-dimensional cyclic unitary representations. Formulating a version of the dS/CFT correspondence using these representations can lead to a description with a finite-dimensional Hilbert space and unitary evolution. In the present work, we generalize to the case of quantum-deformed three-dimensional de Sitter spacetime and compute the entanglement entropy of a quantum field across the cosmological horizon.

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