Coupling to matter fields

4.9 Coupling to matter fields

Ambjørn et al. [7 ] considered the influence of both Ising spins and Gaussian scalar fields on bulk geometric quantities. The phase structure remained essentially unchanged, and no improvement in the scaling behaviour of $⟨R⟩$ was found. Taken together with their results on higher-derivative gravity [16

], one finds a universal linear dependence of the cosmological constant $κc (κ ) 4 2$ , with slope $∼ 2.5$ .

Coupling to $mathbbZ2$ -spin variables $s (l)$ located on edges $l$ was considered by Ambjørn et al. [15 ], who added a Wilson loop term $∑ ∏ SW [s;T ] = − β b∈T o(b) l∈bs(l)$ to the action. The matter sector behaves largely as expected when $κ2$ is varied between the crumpled and the elongated gravity phase. However, in the common critical region of both sectors, where a priori one might have expected interesting effects, the critical behaviour seems to agree with that of the pure gravity system.

More recently, Bilke et al. [52 ] have reported a non-trivial back-reaction of matter on geometry, when considering coupling to several non-compact $U (1)$ -gauge fields. Their study was in part motivated by a continuum analysis of the dynamics of the conformal factor of Antoniadis et al. [21 ]. Including three fields $U (1)$ -fields seems to lead to a total suppression of the branched polymer phase, which is replaced by a new weak-coupling phase with negative susceptibility exponent $γ$ and a fractal dimension $≈ 4$ . These are clearly interesting results, but should be treated with some caution because of the small lattice sizes involved ( $N4 ≤ 16k$ ).