Gradient Flows and the Curvature of Theory Space

TL;DR

This paper explores the geometric structure of theory space in multiscalar models, showing it is curved and linking the potential to a monotonic function that extends the $a$- and $F$-theorems to gradient flows in $d=4- ext{} obreakdash- ext{} obreakdash ext{dimensions}$.

Contribution

It derives the Ricci scalar for the theory space metric and connects the potential to a monotonic function, extending the $a$- and $F$-theorems to gradient flows in $d=4- ext{} obreakdash- ext{} obreakdash ext{dimensions}.

Findings

The theory space metric is curved, as shown by the Ricci scalar.

The potential $ ilde{F}$ is linked to a monotonic function interpolating between $a$- and $F$-theorems.

The $ ilde{F}$-theorem extends perturbatively to gradient flow in $d=4- ext{} obreakdash- ext{} obreakdash ext{dimensions}$.

Abstract

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 $d=4-\varepsilon$.