Resurgence in Liouville Theory

TL;DR

This paper explores the asymptotic nature of the perturbative expansion in Liouville theory, revealing factorial growth, Borel plane singularities linked to complex instantons, and the interplay of multiple saddle points in a trans-series framework.

Contribution

It demonstrates that Liouville theory's perturbative series is asymptotic with instanton contributions, providing a concrete example of summing perturbative and non-perturbative effects.

Findings

Perturbative coefficients grow factorially, indicating an asymptotic series.

Singularities in the Borel plane are associated with complex instantons.

Perturbative expansions around different saddle points combine in a trans-series structure.

Abstract

Liouville conformal field theory is a prototypical example of an exactly solvable quantum field theory, in the sense that the correlation functions in an arbitrary background can be determined exactly using only the constraints of unitarity and crossing symmetry. For example, the three point correlation functions are given by the famous formula of Dorn-Otto-Zamolodchikov-Zamolodchikov (DOZZ). Unlike many other exactly solvable theories, Liouville theory has a continuously tunable parameter -- essentially $\hbar$ -- which is related to the central charge of the theory. Here we investigate the nature of the perturbative expansion in powers of $\hbar$, which is the loop expansion around a semi-classical solution. We show that the perturbative coefficients grow factorially, as expected of a Feynman diagram expansion, and take the form of an asymptotic series. We identify the singularities in the Borel plane, and show that they are associated with complex instanton solutions of Liouville theory; they correspond precisely to the complex solutions described by Harlow, Maltz, and Witten. Both single- and multi-valued solutions of Liouville appear. We show that the perturbative loop expansions around these different saddle points mix in the way expected for a trans-series expansion. Thus Liouville theory provides a calculable example of a quantum field theory where perturbative and instanton contributions can be summed up and assembled into a finite answer.