“Gradient flow and the curvature of theory space”
The study of monotonicity within renormalization group flow has attracted considerable attention over the years, and gradient flow is the strongest possible version of monotonicity one could demand. After a brief review of monotonicity theorems in general and gradient flow in particular, I will detail the advances in the study of gradient flow in multiscalar systems. Perturbatively, gradient flow can be reduced to a set of constraint equations on the coefficients appearing in the beta function, and using recent results for the generic multiscalar beta function I will show that these constraints are satisfied through all known loop orders. The monotonic quantity appearing in this solution is connected to a weak monotonicity conjecture by Fei, Giombi, Klebanov and Tarnopolsky in d=4-ϵ, suggesting that this conjecture can be strengthened. Interestingly, gradient flow produces a natural metric on the space of couplings, and I will investigate the geometric properties of this metric by writing down its Ricci scalar at next-to-leading order.
Host: David Poland