A singular integration by parts formula for the exponential Euclidean QFT on the plane

TL;DR

This paper introduces a new characterization of Euclidean quantum field theory with exponential interaction on the plane using a renormalized integration by parts formula, linking it to the Gaussian free field and ensuring well-posedness.

Contribution

It provides a novel renormalized integration by parts formula for exponential Euclidean QFT on , connecting it to the Gaussian free field and addressing singularity issues.

Findings

Established a well-posedness framework for the singular IbP formula.

Linked the measure to the Gaussian free field via Wasserstein distance.

Ensured control of the renormalized IbP formula through coupling with GFF.

Abstract

We give a novel characterization of the Euclidean quantum field theory with exponential interaction $\nu$ on $\mathbb{R}^2$ through a renormalized integration by parts (IbP) formula, or otherwise said via an Euclidean Dyson-Schwinger equation for expected values of observables. In order to obtain the well-posedness of the singular IbP problem, we import some ideas used to analyse singular SPDEs and we require the measure to "look like" the Gaussian free field (GFF) in the sense that a suitable Wasserstein distance from the GFF is finite. This guarantees the existence of a nice coupling with the GFF which allows to control the renormalized IbP formula.