Gravity and Yang-Mills in d=2+ε

This thesis explores the ultraviolet behavior of gravity coupled to Yang-Mills fields within the Asymptotic Safety scenario. We employ a perturbative approach based on the dimensional expansion in d=2-ε, which allows for a controlled analytical continuation and circumvents ambiguities associated with heat kernel methods for quantizing the metric. We calculate the one-loop beta functions for the Einstein-Yang-Mills system and analyze the renormalization group flow. To interpret the physical implications, we evaluate the flow on-shell using two different schemes to handle the equations of motion. In both schemes, we identify a non-Gaussian fixed point, suggesting the theory could be asymptotically safe. However, we discover a discrepancy between the schemes in the limit d->4, where in one case the non-Gaussian fixed point merges with the Gaus- sian fixed point. This result highlights a potential scheme dependence in the on-shell analysis. A parallel investigation in d = 4 - ε dimensions confirms the perturbative non-renormalizability of the theory, as no interacting fixed point is found. Our findings support the utility of the d = 2 - ε expansion as a tool to investigate quantum gravity while also underscoring the challenges in extrapolating results to four dimensions.