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

TL;DR

This work develops a covariant, spin-parity based analysis of quantum torsion fluctuations on maximally symmetric spaces to compute 1-loop radiative corrections to the $R$ and $R^2$ operators. Using a Schwinger-DeWitt heat-kernel approach and a careful treatment of kinematical and dynamical zero-modes, the authors extract the logarithmic divergences and obtain the beta function for the $R^2$ (Starobinsky) term in a torsionful MAG setting with a classical metric. The results highlight how torsion loops contribute to the running of curvature invariants and demonstrate a pathway to incorporate torsion quantum effects in the path integral, albeit with limitations due to neglecting metric fluctuations and restricting to maximally symmetric backgrounds. These findings offer qualitative guidance for embedding torsion-induced quantum corrections into inflationary dynamics and RG analyses of MAGs, and set the stage for more complete treatments that include nontrivial torsion backgrounds and dynamical metrics.

Abstract

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 $β$-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.