$\Phi^4_3$ measures on compact Riemannian $3$-manifolds

TL;DR

This paper constructs a non-perturbative $ ext{Φ}^4_3$ measure on any compact 3D Riemannian manifold, establishing it as an invariant measure for a singular SPDE and developing new renormalization techniques using microlocal analysis.

Contribution

It introduces a novel renormalization approach for $ ext{Φ}^4_3$ measures on curved spaces, extending quantum field theory to arbitrary compact 3D Riemannian manifolds.

Findings

Constructed the $ ext{Φ}^4_3$ measure on arbitrary compact 3D manifolds.

Proved the measure's invariance and covariance under isometries.

Developed new microlocal analysis methods for renormalization.

Abstract

We construct the $\Phi^4_3$ measure on an arbitrary 3-dimensional compact Riemannian manifold without boundary as an invariant probability measure of a singular stochastic partial differential equation. Proving the nontriviality and the covariance under Riemannian isometries of that measure gives a non-perturbative, non-topological interacting Euclidean quantum field theory on curved spaces in dimension 3. To control analytically several Feynman diagrams appearing in the construction of a number of random fields, we introduce a novel approach of renormalisation using microlocal and harmonic analysis. This allows to obtain a renormalized equation which involves some universal constants independent of the manifold. In a companion paper, we develop in a self-contained way all the tools from paradifferential and microlocal analysis that we use to build in our manifold setting a number of analytic and probabilistic objects.