Schwarzian functional integrals calculus

TL;DR

This paper develops a rigorous mathematical framework for functional integration in Schwarzian theories, simplifying complex integrals to explicit ordinary integrals and analyzing correlation functions on different diffeomorphism groups.

Contribution

It provides a complete, mathematically rigorous set of rules for Schwarzian functional integrals, reducing them to explicit calculations without relying on conjectures.

Findings

Explicit evaluation of two-point correlation functions

Explicit evaluation of four-point correlation functions

Comparison of results between real line and circle cases

Abstract

We derive the general rules of functional integration in the theories of Schwarzian type, thus completing the elaboration of Schwarzian functional integrals calculus initiated in \cite{(BShExact)}, \cite{(BShCorrel)}. Our approach is mathematically rigorous and does not contain any unproved conjectures. It is based on the analysis of the properties of the measures on the groups of diffeomorphisms, and does not appeal for the experience from other physical models. Its great merit consists in reducing a problem of functional integration to that of the only functional integral (\ref{E}) that is calculated explicitly with the result written in the form of the ordinary integral. We evaluate two-point and four-point correlation functions defined as functional integrals over the groups $Diff^{1}_{+}(\textbf{R}) $ and $Diff^{1}_{+}(S^{1})\,,$ and discuss the difference between the results in the two cases.