An initially puzzling feature of is that it comes out as a total -derivative. Restoring the interpretation of as a field on the 2D base space, however, it has a natural interpretation: An immediate consequence of Equations (3.83, 3.85) is that contour integrals of the form
Another intriguing property of Equation (3.85) can be seen from the second line in Equation (3.84). The first two terms correspond to the beta function of a sigma-model without coupling to 2D gravity (i.e. with nondynamical and ). The last term is crucial for all the subsequent properties of the flow (3.83). Comparing with Equation (3.75) one sees that it describes a backreaction of the scale dependent area radius on the coupling flow, which is mediated by the quantum dynamics of the other fields. We shall return to this point below.
As usual fixed points of the flow correspond to zeros of the generalized beta function. The flow has two fixed points, a degenerate Gaussian one corresponding to formally, and a non-Gaussian fixed point which is of main interest here. We postpone the discussion of the Gaussian fixed point and focus on the non-Gaussian one here. The defining relation for such a nontrivial fixed point amounts by Equation (3.85) to the differential equation
The renormalized action (at or away from the fixed point) has no direct significance in the quantum theory. It is however instructive to note that in Equation (3.56) can for any not be written in the form for any metric with two commuting Killing vectors. In accordance with the general renormalization group picture one can of course write . Interestingly, in the 2-Killing vector reduction one can write the renormalized Lagrangian as a sum of two terms which are reductions of . Since the parameter in can be changed at will by redefining , we may assume that , for some . Then
The origin of this feature is the seeming violation of scale invariance on the level of the renormalized action. Recall from after Equation (3.62) that , so that for the action is no longer scale invariant. However this is precisely the property which allows one to cancel the (otherwise) anomalous term in the trace of the would-be energy momentum tensor, as discussed before, rendering the system conformally invariant at the fixed point. Due to the lack of naive scale invariance on the level of the renormalized action the dynamics of quantum gravidynamics is different from that of quantum general relativity, in the sector considered, even at the fixed point. The moral presumably generalizes: The form of the (bare and/or renormalized) action may have to differ from the Einstein–Hilbert action in order to incorporate the physics properties aimed at.
In the present context the most important feature of the fixed point is its ultraviolet stability: For all linearized perturbations around the fixed point function (3.89) are driven back to the fixed point. Since the fixed point (3.89) has the form of a power series in the loop counting parameter , the proper concept of a “linearized perturbation” has the form for the explicit expressions). The lowest order equation (3.93) is homogeneous and its solution is given by  . Since always, Equation (3.94) shows that the lowest order perturbation will always die out for , for arbitrary smooth initial data prescribed at . It can be shown that this continues to hold for all higher order irrespective of the signs of the coefficients , .
Result (UV stability):
Given smooth initial with , , the solution of the linearized flow equations (3.93) is unique and satisfies
where the convergence is uniform in .
The situation is illustrated in Figure 1. The proof of this result is somewhat technical and can be found in .
Often the stability properties of fixed points are not discussed by solving the linear flow equations directly, but by studying the spectral properties of the linearized perturbation operator (the “stability matrix” in Equation (A.10)). Since the generalized couplings here are functions, the linearized perturbation operator is a formal integral operator,all (nontrivial) eigenfunctions of are normalizable; the spectrum is “purely discrete” and consists of the entire halfplane . Indeed, the general solution to is , with . The first term merely corresponds to a change of normalization of and we may set , without loss of generality. The second term corresponds to Equation (3.94) with . This clearly confirms the above result from a different perspective. For the parameters are not spectral values for . Moreover, since the kernels are distributions it is not quite clear which precise functional analytic setting one should choose for the full spectral problem . This is why above we adopted the direct strategy and determined the solutions of the linearized flow equations. Their asymptotic decay shows the ultraviolet stability of the fixed point unambiguously and independent of functional analytical subtleties.
We can put this result into the context the general discussion in Section 2 and arrive at the following conclusion:
With respect to the non-Gaussian fixed point all couplings in the generating functional are asymptotically safe. All symmetry reduced gravity systems satisfy the Criteria (PTC1) and (PTC2) to all loop orders of sigma-model perturbation theory. As explained in Section 3.2 from the viewpoint of the graviton loop expansion the distinction between a perturbative and a non-perturbative treatment is blurred here.
It is instructive to compare these properties to that of the Gaussian fixed point. The Gaussian fixed point of the flow (3.83) is best understood in analogy to the Gaussian fixed-point of a conventional nonlinear sigma-model. For a nonlinear sigma-model with Lagrangian (with satisfying Equation (3.12)) the beta function has only the trivial zero . As the renormalized Lagrangian blows up, but in an expansion around one can see that for the interaction terms vanish. In this sense the fixed point is Gaussian. This holds irrespective of the sign of , which however determines the stability properties of the flow. The stability ‘matrix’ vanishes so that the linearized stability analysis is empty. By direct inspection of the differential equation one sees that the unstable manifold of is one-dimensional for (typical for compact) and empty for (typical for noncompact). Indeed, , and if one insists on for positivity-of-energy reasons, the flow will be attracted to for iff . In particular for these models are, based on the Criterion (PTC2) of Section 2, not expected to have a genuine continuum limit.
The Gaussian fixed point of the symmetry reduced gravity theories can be analyzed similarly. In terms ofnot all couplings contained in the generating functional are asymptotically safe. That is, there exists initial data (with , for ) for which the asymptotics does not vanish identically in . To see this, it suffices to note that the right-hand-side of Equation (3.99) to quadratic order reads , with . Since always, initial data for which is a strictly increasing function of will give rise to solutions having the tendency to be driven towards larger values (pointwise in ) as increases. Conversely, only initial data for which is strictly decreasing in can be expected to give rise to a solution which vanishes identically in as . A rigorous theorem describing the stable and the unstable manifold is presumably hard to come by, but for our purposes it is enough to know that there exist initial data which do not give rise to solutions decaying to for . For example or , , have this property. The upshot is that the Gaussian fixed point of the symmetry truncated gravity theories is not UV stable, the Condition (PTC2) is not satisfied, and one can presumably not use it for the construction of a genuine continuum limit.
At this point it may be instructive to contrast the quantum properties of the dimensionally reduced gravity theories with those of the same noncompact sigma-model without coupling to gravity (which effectively amounts to setting constant in Equation (3.16)). The qualitative differences are summarized in Table 2.
The comparison highlights why the above conclusion is surprising and significant. While the noncompact sigma-models are renormalizable with just one relevant coupling (denoted by in the table), at least in the known constructions they do not have a fixed point at which they are conformally invariant. Their gravitational counterparts require infinitely many relevant couplings for their UV renormalization. This infinite coupling flow has a nontrivial UV fixed point at which the theory is conformally invariant. Most importantly the stability properties of the renormalization flow are reversed (compared to the flow of ) for all of the infinitely many relevant couplings. As there appears to be no structural reason for this surprising reversal in the reduced theory itself, we regard it as strong evidence for the existence of an UV stable fixed point for the full renormalization group dynamics.
In the table we anticipated that at the fixed point the trace anomaly of the would-be energy momentum tensor vanishes for the symmetry truncated gravity theories. This allows one to construct quantum counterparts of the constraints and as well-defined composite operators. In detail this comes about as follows. Taking the trace in Equation (3.81) gives , again modulo the equations of motion operator. The first term has a nonzero trace anomaly given by). The improvement term is determined by the function in Equation (3.81), which in turn is largely determined by and thus cannot be freely chosen as a function of .
It is therefore a very nontrivial match that (i) upon insertion of the fixed point the function becomes stationary (-independent), (ii) the equation turns out to determine completely, and (iii) the so-determined function has the property that precisely cancels the second term in Equation (3.100) evaluated for . Thus the trace anomaly vanishes precisely at the fixed point of the coupling flowproperty of the system, rather than one which is used to define the renormalized target space metric . The latter is already determined by the warped product structure (3.59, 3.60) and the renormalization result (3.57). At the fixed point the first term in Equation (3.100) vanishes, but the second then is completely determined. On the other hand by Equation (3.81) one is not free to choose the improvement potential as a function of -independent of , so the cancellation is not built in. One can also verify that the only solution for such that is with the above , such that Equation (3.101) holds. That is, scale invariance implies here conformal invariance. Since the target space metric (3.60) has indefinite signature this does not follow from general grounds [176, 194].
Due to Equation (3.101) we can now define the quantum constraints by[127, 128] for an anomaly-free implementation of the Virasoro constraints in essentially noninteracting systems). In the case at hand the proper positivity requirement will be determined by the quantum observables commuting with the constraints. Their construction and the exploration of the physical state space is a major desideratum. In summary, the systems should at the fixed point be described by one whose physical states can be set into correspondence to the above quantum observables. The infrared problem has not been investigated so far, but based on results in the polarized subsector  one might expect it to be benign.
Despite the fact that the system is conformally invariant at the fixed point there are still scale dependent running parameters. At first sight this seems paradoxical. However, already the example of the massless continuum limit in a 3D scalar field theory exhibits this behavior. The remaining scale parameter is related to the direction of instability within the critical manifold, pointing from the Gaussian to the Fisher–Wilson fixed point, in the direction of coarse graining. The systems considered here provide an intriguing other example of this phenomenon. The critical manifold can be identified with the subset of parameter values where the system is scale (and here conformally) invariant. This fixes , but the inessential parameters contained in the renormalized fields are left unconstrained. This allows one to introduce a running parameter as follows. One evaluates the running coupling function at the ‘comoving’ field and setsvalue of – and hence on . Combining Equation (3.83) with Equation (3.75) one finds the following flow equation: fixed set of field coordinates. Moreover in the technical sense is an “inessential” parameter. The fact that the scale dependence of is governed by the beta function means that for increasing it will be driven away from the fixed point . The condition can be traded for the specification of the Gaussian fixed point . Thus the parameter flow may be viewed as a coupling flow emanating (in the direction of increasing ) from the Gaussian fixed point. At the non-Gaussian fixed point, on the other hand, governs the scale dependence of the ‘rod field’ via
For orientation we summarize here our results on the renormalization of the symmetry reduced Quantum Einstein Gravity theories (3.1):
So far we considered the renormalization of the symmetry reduced theories in its own right, leaving the embedding into the full Quantum Gravidynamics open. The proposed relation to qualitative aspects of the Quantum Gravidynamics in the extreme UV has already been mentioned. Here we offer some tentative remarks on the embedding otherwise. The constructions presented in this section can be extended to dimensions in the spirit of an -expansion. At the same time this mimics quantum aspects of the 1-Killing vector reduction. One finds that the qualitative features of the renormalization flow – non-Gaussian fixed point and asymptotic safety – are still present . A cosmological ‘constant’ term can likewise be included and displays a similar pattern as outlined at the end of Section 3.2. The advantage of this setting is that the UV cutoff can strictly be removed, which is hard to achieve with a nonperturbative technique. The extension of these results from a quasi-perturbative analysis to a nonperturbative one, ideally via controlled approximations, is an important open problem. The same holds for the analysis of the 1-Killing vector reduction, which holds the potential for cosmological applications. These truncations can be viewed as complementary to the ‘hierarchical’ truncations used in Section 4: A manifest truncation is initially imposed on the functional integral, but the infinite coupling renormalization flow can then be studied in great detail, often without further approximations.
© Max Planck Society and the author(s)