Conformal dynamics of quantum gravity with torsion

The trace anomaly induced dynamics of the conformal factor is investigated in four-dimensional quantum gravity with torsion. The constraints for the coupling constants of torsion matter interaction are obtained in the infrared stable fixed point of the effective scalar theory. ⋆ On leave from Pedagogical Institute, 634041 Tomsk, Russia In a recent paper [1] the trace anomaly induced dynamics of the conformal factor in four-dimensional quantum gravity has been investigated. The conformal sector was described by spacetime metrics of the form: gμν(x) = e ημν , where e is the conformal factor and ημν is the flat metric. The generalization to curved fiducial metric was considered in refs.[2,3]. The effective action for σ can be obtained by integrating the general form of trace anomaly in curved spacetime [4,3]. It turns out that the resulting action is local in σ and possesses a non-trivial, infrared stable critical point which can be computed to all orders in perturbation theory [1]. In particular, the anomalous scaling dimension of the conformal field e was found using the same line of reasoning as in two-dimensional gravity [5]. It was argued that the effective σ theory describes four-dimensional quantum gravity at large distances and provides a framework for a dynamical solution of the cosmological constant problem [1]. The purpose of this note is to generalize the results of ref.[1] in the case of quantum gravity with torsion. Torsion interactions start becoming non-trivial in four dimensions. In fact in two dimensions there is no minimal coupling of matter with torsion. Furthermore, in the non-minimal case [6] torsion appears quadratically and, after functional integration, 2d induced gravity with torsion is equivalent to 2d gravity without torsion up to coupling constants redefinitions. In four dimensions minimal interactions of spinors with torsion are non-trivial. Moreover, the torsion dependence of trace anomaly is much more complicated than in two dimensions [7], so that one may expect to find some new properties of the conformal factor dynamics in this case. Let us start with the free actions for scalar and spinor fields in curved spacetime in the presence of torsion T γβ. Recall that in a theory with torsion the connection Γβγ is not symmetric in the lower indices: Γβγ − Γγβ = T γβ 6= 0. (1)