Impact of topology in causal dynamical triangulations quantum gravity

We investigate the impact of spatial topology in 3 + 1-dimensional causal dynamical triangulations (CDT) by performing numerical simulations with toroidal spatial topology instead of the previously used spherical topology. In the case of spherical spatial topology, we observed in the so-called phase C an average spatial volume distribution n(t) which after a suitable time redefinition could be identified as the spatial volume distribution of the four-sphere. Imposing toroidal spatial topology, we find that the average spatial volume distribution n(t) is constant. Bymeasuring the covariance matrix of spatial volume fluctuations, we determine the form of the effective action. The difference compared to the spherical case is that the effective potential has changed such that it allows a constant average n(t). This is what we observe and this is what one would expect from a minisuperspace GR action, where only the scale factor is kept as dynamical variable. Although no background geometry is put in by hand, the full quantum theory of CDT is also with toroidal spatial toplogy able to identify a classical background geometry around which there are well-defined quantum fluctuations.