Tales of the trace anomaly from six dimensions

We investigate the trace anomaly of a quantum field theory on a curved background with (background) spacetime-dependent couplings in six dimensions. The spacetime dependence of the couplings allows one to study the trace anomaly away from conformal fixed points. In particular the possibility of a strong a-theorem in six dimensions in multi-flavor φ^3 theory is considered. Contrary to the case in two and four dimensions, we find that in perturbation theory the relevant quantity \tilde{a}, the natural extension of the infamous a-quantity away from fixed points, increases monotonically along flows away from the trivial fixed point. We also present results from a related theory in six dimensions where \tilde{a} decreases away from UV fixed points. These results suggest that some new intuition about the a-theorem is in order.