Ricci curvature and quantum geometry

TL;DR

This paper explores how Ricci curvature emerges from quantum field theory approaches to Riemannian geometry, highlighting its connections to diffusion, optimal transport, and renormalization.

Contribution

It introduces a quantum field theory perspective on Riemannian geometry, emphasizing the role of Ricci curvature in this framework.

Findings

Ricci curvature linked to quantum fluctuations and spectral data.

Connections between Ricci curvature, diffusion, and optimal transport.

Insights into the renormalization group in geometric contexts.

Abstract

We describe a few elementary aspects of the circle of ideas associated with a quantum field theory (QFT) approach to Riemannian Geometry, a theme related to how Riemannian structures are generated out of the spectrum of (random or quantum) fluctuations around a background fiducial geometry. In such a scenario, Ricci curvature with its subtle connections to diffusion, optimal transport, Wasserestein geometry, and renormalization group, features prominently.