v 2 1 8 D ec 1 99 5 Inequivalence of First and Second Order Formulations in D = 2 Gravity Models

The usual equivalence between the Palatini and metric (or affinity and vielbein) formulations of Einstein theory fails in two spacetime dimensions for its " Kaluza–Klein " reduced (as well as for its standard) version. Among the differences is the necessary vanishing of the cosmological constant in the first order forms. The purely affine Eddington formulation of Einstein theory also fails here. Most general constructions in physical theories are formally valid in all dimensions, even though many properties can be quite different in special D. In this note we present a simple but striking deviation from this rule in the context of some gravity models at D=2, a dimension that is always " more equal " than others. We shall see that the standard Palatini formulations, in which metric and affinity or vielbein and connection are independently varied, no longer coincide with their purely metric or zweibein second order expressions. While this is known [1] for the usual Einstein Lagrangian, it is also true for the more interesting, " Kaluza–Klein reduced " , version involving a Lagrange multiplier [2]. The source of the difference is the Weyl invariance enjoyed (only) by the first order Palatini forms. As a result metricity of the initial, affine, space is no longer recovered from the field equations; the cosmological constant must also vanish. We shall also see that even purely affine expression of Einstein given by Eddington [3] theory is different in 2D. 1 The present results were reported in the Proceedings of the Markov Memorial Quantum Gravity Seminar.