Lagrangian treatment. Introduction

2.1 Lagrangian treatment. Introduction

This area of research was inspired by the success of non-perturbative lattice methods in treating non-abelian gauge theories [173 ]. To apply some of their techniques, gravity has to be brought into a gauge-theoretic, first-order form, with the pure-gravity Lagrangian

where the $SO (3,1)$ -valued spin-connection $AIJ μ$ (with curvature $R$ ) and the vierbein $I eμ$ are considered as independent variables. The important feature of (1

) is that $IJ A μ$ is a gauge potential, and that the action – in addition to its diffeomorphism-invariance – is invariant under local frame rotations. Variation with respect to $eIμ$ leads to the metricity condition,

which can be solved to yield the unique torsion-free spin connection $A = A[e]$ compatible with the $eμI$ . There are some obvious differences with usual gauge theories: i) the action (1

) is linear instead of quadratic in the curvature two-form $RIJμν$ of $A$ , and ii) it contains additional fields $eIμ$ . Substituting the solution to (2

) into the action, one obtains $∫ S[e] = d4x(det e)R M$ , where $R$ denotes the four-dimensional curvature scalar. This expression coincides with the usual Einstein action $∫ d4x√det--gR$ only for $det e > 0$ .

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 $G$ is $SO (3,1)$ or its “Euclideanized” form $SO (4)$ , or a larger group containing it as a subgroup or via a contraction limit. Local curvature terms are represented by (the traces of) $G$ -valued Wilson holonomies $U ⊓⊔$ 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 $G$ , and does not contain any translation generators that appear when $G$ is the Poincaré group.

When discretizing conformal gravity (where $G = SO (5,1)$ ) 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.