English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We construct the $\Phi^4_3$ measure on an arbitrary 3-dimensional compactRiemannian manifold without boundary as an invariant probability measure of asingular stochastic partial differential equation. Proving the nontrivialityand the covariance under Riemannian isometries of that measure gives for thefirst time a non-perturbative, non-topological interacting Euclidean quantumfield theory on curved spaces in dimension 3. This answers a longstanding openproblem of constructive quantum field theory on curved 3 dimensionalbackgrounds. To control analytically several Feynman diagrams appearing in theconstruction of a number of random fields, we introduce a novel approach ofrenormalization using microlocal and harmonic analysis. This allows to obtain arenormalized equation which involves some universal constants independent ofthe manifold. We also define a new vectorial Cole-Hopf transform which allowsto deal with the vectorial $\Phi^4_3$ model where $\Phi$ is now a bundle valuedrandom field. In a companion paper, we develop in a self-contained way all thetools from paradifferential and microlocal analysis that we use to build in ourmanifold setting a number of analytic and probabilistic objects.

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