Radiative corrections to the $R$ and $R^2$ invariants from torsion fluctuations on maximally symmetric spaces

We derive the runnings of the $R$ and $R^2$ operators that stem from integrating out quantum torsion fluctuations on a maximally symmetric Euclidean background, while treating the metric as a classical field. Our analysis is performed in a manifestly covariant way, exploiting both the recently-introduced spin-parity decomposition of torsion perturbations and the heat kernel technique. The Lagrangian we start with is the most general one for 1-loop computations on maximally symmetric backgrounds involving kinetic terms and couplings to the scalar curvature that is compatible with a gauge-like symmetry for the torsion. The latter removes the twice-longitudinal vector mode from the spectrum, and it yields operators of maximum rank four. We also examine the conditions required to avoid ghost instabilities and ensure the validity of our assumption to neglect metric quantum fluctuations, demonstrating the compatibility between these two assumptions. Then, we use our findings in the context of Starobinsky's inflation to calculate the contributions from the torsion tensor to the $\beta$-function of the $R^2$ term. While this result is quantitatively reliable only at the $0$-th order in the slow-roll parameters or during the very early stages of inflation -- due to the background choice -- it qualitatively illustrates how to incorporate quantum effects of torsion in the path integral formalism.