Gradient flows and the curvature of theory space

A bstract The metric and potential associated with the gradient property of renormalisation group flow in multiscalar models in d = 4 − ε dimensions are studied. The metric is identified with the Zamolodchikov metric of nearly marginal operators on the sphere. An explicit form for the associated Ricci scalar in d = 4 − ε is derived, which shows that the space of multiscalar field theories is curved. The potential is identified with a quantity F F ~ that was previously proposed as a weakly monotonic function interpolating between the a -theorem in four dimensions and the F -theorem in three dimensions. This implies that the F F ~ -theorem can be extended perturbatively to a theorem about gradient flow in d = 4 − ε.