Correlation functions in the Schwarzian theory

TL;DR

This work introduces a mathematically rigorous framework for Schwarzian theory based on a quasi-invariant measure on the diffeomorphism group, enabling exact evaluation of correlation functions that were previously approached via heuristic Wiener-measure or Liouville methods. By constructing a measure $\mu_{\sigma}$ on $Diff^{1}_{+}([0,1])$ and exploiting its quasi-invariance under $Diff^{3}_{+}$, the authors derive explicit integral representations for mean values and multi-point correlators, culminating in a finite, normalized two-point function $G_{2}(0,t)$ and a general N-point structure. A key technical device is the function $\mathcal{E}_{\sigma}(u,v)$ and its properties, together with an SL(2,\mathbb{R}) regularization and normalization, which render the Schwarzian path integrals well-defined. The approach has potential applications to SYK, 2D dilaton gravity, and higher-dimensional or supersymmetric generalizations of theories with Schwarzian-like diffeomorphism symmetry. The results provide concrete, computable expressions and highlight singular behavior at the endpoints, offering a solid foundation for further analytical and numerical exploration in diffeomorphism-invariant quantum theories.

Abstract

A mathematically correct approach to study theories with infinite-dimensional groups of symmetries is presented. It is based on quasi-invariant measures on the groups. In this paper, the properties of the measure on the group of diffeomorphisms are used to evaluate the functional integrals in the Schwarzian theory. As an important example of the application of the new technique, we explicitly evaluate the correlation functions in the Schwarzian theory.

Figures (3)

• Figure 1: The form of the correlation function $G_{2}\left(0,\,t \right)$ for $\sigma=2\pi\,.$
• Figure 2: The form of the symmetric function $\mathcal{E}_{1}(x,\,x)\,.$
• Figure 3: The form of the function $\mathcal{E}_{\sigma}(x,\,y)\,.$