Gradient Flows and the Curvature of Theory Space

The metric and potential associated with the gradient property of renormalisation group flow in multiscalar models in $d=4-\varepsilon$ 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 $d=4-\varepsilon$ is derived, which shows that the space of multiscalar field theories is curved. The potential is identified with a quantity $\widetilde{F}$ 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 $\widetilde{F}$-theorem can be extended perturbatively to a theorem about gradient flow in $d=4-\varepsilon$.