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 [16Jump To The Next Citation Point], 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).


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