Kavli Affiliate: R. Morris
| First 5 Authors: Alex Mitchell, Tim R. Morris, Dalius Stulga, ,
| Summary:
We study an $f(R)$ approximation to asymptotic safety, using a family of
non-adaptive cutoffs, kept general to test for universality. Matching solutions
on the four-dimensional sphere and hyperboloid, we prove properties of any such
global fixed point solution and its eigenoperators. For this family of cutoffs,
the scaling dimension at large $n$ of the $n^text{th}$ eigenoperator, is
$lambda_npropto b, nln n$. The coefficient $b$ is non-universal, a
consequence of the single-metric approximation. The large $R$ limit is
universal on the hyperboloid, but not on the sphere where cutoff dependence
results from certain zero modes. For right-sign conformal mode cutoff, the
fixed points form at most a discrete set. The eigenoperator spectrum is
quantised. They are square integrable under the Sturm-Liouville weight. For
wrong sign cutoff, the fixed points form a continuum, and so do the
eigenoperators unless we impose square-integrability. If we do this, we get a
discrete tower of operators, infinitely many of which are relevant. These are
$f(R)$ analogues of novel operators in the conformal sector which were used
recently to furnish an alternative quantisation of gravity.
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