Liouville field theory with heavy charges. I. The pseudosphere

TL;DR

This paper develops a perturbative approach to quantum Liouville theory on the pseudosphere with heavy charges, providing exact Green functions and one-loop corrections, and validating results against bootstrap and perturbative methods.

Contribution

It introduces a method to compute exact Green functions and quantum corrections in Liouville theory with heavy charges on the pseudosphere, connecting semiclassical and quantum results.

Findings

Exact Green function on the pseudosphere with one finite charge

One-loop quantum corrections to semiclassical limits

Agreement with bootstrap and double perturbative expansions

Abstract

We work out the perturbative expansion of quantum Liouville theory on the pseudosphere starting from the semiclassical limit of a background generated by heavy charges. By solving perturbatively the Riemann-Hilbert problem for the Poincare' accessory parameters, we give in closed form the exact Green function on the background generated by one finite charge. Such a Green function is used to compute the quantum determinants i.e. the one loop corrections to known semiclassical limits thus providing the resummation of infinite classes of standard perturbative graphs. The results obtained for the one point function are compared with the bootstrap formula while those for the two point function are compared with the existing double perturbative expansion and with a degenerate case, finding complete agreement.