2.2 Smolin’s lattice model

The first gauge-theoretic model for lattice gravity is due to Smolin [186], based on the continuum formulation of MacDowell and Mansouri [157], with de Sitter gauge group O (3,2) or O(4,1), and a Lagrangian of 2 R-type,
∫ S = d4x 𝜖μνρσ𝜖 R&tidle;IJ R&tidle;KL , (3 ) IJKL μν ρσ
where the components of &tidle; R μν are related to those of the usual curvature tensor by
&tidle;RIμJν = RIJμν ± 1-(eIμeJν − eJμeIν). (4 ) l2
Although the underlying gauge potentials Aμ are O (3,2)- or O (4,1)-valued, the action is only invariant under the 6-dimensional subgroup of Lorentz transformations. The theory contains a dimensionful parameter l. The gauge potentials associated with the internal 5-direction are identified with the frame fields I eμ, and the action can be decomposed into the usual Einstein-term (1View Equation) plus a cosmological constant term with λ ∼ 1l2 and a topological R ∧ R-term. Smolin analyzed its lattice discretization, and found both a weak- and a strong-coupling phase, with respect to the dimensionless coupling constant √G- g ∼ l. He performed a weak-coupling expansion about flat space and rederived the usual propagator. In the strong-coupling regime he found massive excitations and a confining property for spinors.
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