Gradient Flows and the Curvature of Theory Space

William H. Pannell; Andreas Stergiou

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arXiv · 2502.06940

Gradient Flows and the Curvature of Theory Space

William H. Pannell·Andreas Stergiou

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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 $d=4-\varepsilon$.

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William H. Pannell, Andreas Stergiou. 2025-02-10. Gradient Flows and the Curvature of Theory Space. https://arxiv.org/abs/2502.06940

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