Abstract
Being interested in the compatibility of Asymptotic Safety with Hilbert space positivity (unitarity), we consider a local truncation of the functional RG flow which describes quantum gravity in $d>2$ dimensions and construct its limit of exactly two dimensions. We find that in this limit the flow displays a nontrivial fixed point whose effective average action is a non-local functional of the metric. Its pure gravity sector is shown to correspond to a unitary conformal field theory with positive central charge $c=25$. Representing the fixed point CFT by a Liouville theory in the conformal gauge, we investigate its general properties and their implications for the Asymptotic Safety program. In particular, we discuss its field parametrization dependence and argue that there might exist more than one universality class of metric gravity theories in two dimensions. Furthermore, studying the gravitational dressing in 2D asymptotically safe gravity coupled to conformal matter we uncover a mechanism which leads to a complete quenching of the a priori expected Knizhnik-Polyakov-Zamolodchikov (KPZ) scaling. A possible connection of this prediction to Monte Carlo results obtained in the discrete approach to 2D quantum gravity based upon causal dynamical triangulations is mentioned. Similarities of the fixed point theory to, and differences from, non-critical string theory are also described. On the technical side, we provide a detailed analysis of an intriguing connection between the Einstein-Hilbert action in $d>2$ dimensions and Polyakov's induced gravity action in two dimensions.
Highlights
Background independence via background fieldsIn this preparatory section we collect a number of results concerning the implementation of Background Independence in the Effective Average Action (EAA) framework which does employ background fields
We find that in this limit the flow displays a nontrivial fixed point whose effective average action is a non-local functional of the metric
In this paper we started from the Einstein-Hilbert truncation for the effective average action of metric quantum gravity in d > 2 dimensions and constructed its intrinsically 2-dimensional limit
Summary
In this preparatory section we collect a number of results concerning the implementation of Background Independence in the EAA framework which does employ (unspecified!) background fields. These functionals include a purely gravitational piece, Γgkrav, a (for the time being) generic matter action ΓMk , as well as gauge fixing and ghost terms, Γgkf and Γgkh, respectively Concerning the latter, only the following two properties are needed at this point: (i) The hμν-derivative of the gauge fixing functional Γgkf[h; g] ≡ Γgkf[g + h, g] vanishes at hμν = 0. Γgkf drops out of the tadpole equation (2.3), and it follows that ξ = 0 = ξis always a consistent background for the Faddeev-Popov ghosts Adopting this background for the ghosts, (2.3) boils down to the following conditions for self-consistent metric and matter field configurations gksc and Askc, respectively:. And in the following we consider Θk and ΘNk GFP as referring to exactly 2 dimensions, in the sense that the limit has already been taken, and we omit the “O(ε)” symbol
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