Most of the lattice gauge formulations I will discuss below share some common features. The lattice geometry is hypercubic, defining a natural global coordinate system for labelling the lattice sites and edges. The gauge group is or its “Euclideanized” form , or a larger group containing it as a subgroup or via a contraction limit. Local curvature terms are represented by (the traces of) -valued Wilson holonomies around lattice plaquettes. The vierbeins are either considered as additional fields or identified with part of the connection variables. The symmetry group of the lattice Lagrangian is a subgroup of the gauge group , and does not contain any translation generators that appear when is the Poincaré group.
When discretizing conformal gravity (where ) or higher-derivative gravity in first-order form, the metricity condition on the connection has to be imposed by hand. This leads to technical complications in the evaluation of the functional integral.
The diffeomorphism invariance of the continuum theory is broken on the lattice; only the local gauge invariances can be preserved exactly. The reparametrization invariance re-emerges only at the linearized level, i.e., when considering small perturbations about flat space.
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