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

Understand

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.