Tameness of Irregular Conformal Blocks in Liouville Conformal Field Theory

Abstract

Building on the principle that physical observables should be definable within o-minimal structures, this study investigates tameness in the context of two-dimensional Liouville conformal field theory. The computation of conformal blocks, fundamental building blocks of correlation functions in CFT, is significantly simplified using the Coulomb gas formalism. Particular attention is given to irregular conformal blocks, which arise from the insertion of an irregular operator at infinity. This operator gives rise to the Stokes phenomenon, necessitating a detailed analysis of their asymptotic behaviour to understand tameness. We address this challenge through two complementary approaches: an exploration of Borel summability techniques and the geometric interpretation provided by Lefschetz thimbles, which illustrate how integration paths change across Stokes lines. Together, these methodologies offer insights into the definability of irregular conformal blocks within the o-minimal structure \(\mathbb{R}_{\mathrm{an}, H}\).