I had better quote AJL about this, because it is a key point, and I will repost the basic CDT reading list in case anyone wants to refer to the original papers
This is from the introductory paragraph of "Emergence of a 4D World", reference 2 below:
"... a particular case of the more general truth, not always appreciated, that
in any nonperturbative theory of quantum gravity dimension will become a dynamical quantity, along with other aspects of geometry. (By dimension we mean an effective dimension observed at macroscopic scales.)..."
Dimension is just another aspect of geometry. AJL are working with a topological continuum which has no prior geometry and they put the geometry on in a quantum way, and dimensionality emerges, and it is different at different scales, and down near Planck scale it tends to be around D = 2, or even less. But happily enough at big scale, where we like to build our houses and go snow-boarding etc, it is right around D = 4.
Here is the short list of CDT references
1.
http://arxiv.org/hep-th/0105267
Dynamically Triangulating Lorentzian Quantum Gravity
J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (AEI, Golm)
41 pages, 14 figures
Nucl.Phys. B610 (2001) 347-382
"Fruitful ideas on how to quantize gravity are few and far between. In this paper, we give a complete description of a recently introduced non-perturbative gravitational path integral whose continuum limit has already been investigated extensively in d less than 4, with promising results. It is based on a simplicial regularization of Lorentzian space-times and, most importantly, possesses a well-defined, non-perturbative Wick rotation. We present a detailed analysis of the geometric and mathematical properties of the discretized model in d=3,4..."
2.
http://arxiv.org/abs/hep-th/0404156
Emergence of a 4D World from Causal Quantum Gravity
J. Ambjorn (1 and 3), J. Jurkiewicz (2), R. Loll (3) ((1) Niels Bohr Institute, Copenhagen, (2) Jagellonian University, Krakow, (3) Spinoza Institute, Utrecht)
11 pages, 3 figures; final version to appear in Phys. Rev. Lett
Phys.Rev.Lett. 93 (2004) 131301
"Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over geometries in nonperturbative quantum gravity, with a positive cosmological constant. We present evidence that a macroscopic four-dimensional world emerges from this theory dynamically."
3.
http://arxiv.org/abs/hep-th/0411152
Semiclassical Universe from First Principles
J. Ambjorn, J. Jurkiewicz, R. Loll
15 pages, 4 figures
Phys.Lett. B607 (2005) 205-213
"Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over space-time geometries in nonperturbative quantum gravity. We show that the macroscopic four-dimensional world which emerges in the Euclidean sector of this theory is a bounce which satisfies a semiclassical equation. After integrating out all degrees of freedom except for a global scale factor, we obtain the ground state wave function of the universe as a function of this scale factor."
4.
http://arxiv.org/abs/hep-th/0505113
Spectral Dimension of the Universe
J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)
10 pages, 1 figure
SPIN-05/05, ITP-UU-05/07
"We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be "self-renormalizing" at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction."