Irregular Liouville correlators and connection formulae for Heun functions

TL;DR

The paper establishes a precise bridge between irregular Liouville CFT correlators and Heun-type differential equations by analyzing five-point blocks with degenerate insertions and their confluences. Using the AGT correspondence and the Nekrasov-Shatashvili limit, it derives explicit combinatorial expressions for conformal blocks and their semiclassical limits, which correspond to solutions and connection matrices of Heun, confluent Heun, and their reduced forms. It provides detailed dictionaries between CFT data (momenta, DOZZ/OPE coefficients, irregular states) and the parameters of these linear ODEs, along with complete connection formulae across all confluence families. The results offer concrete tools to study isomonodromic deformations, gauge-theory dualities, and classical uniformization, with potential extensions to higher-point blocks, higher-rank singularities, and q-difference deformations. Overall, the work turns the Heun equation’s connection problem into a controlled CFT computation, illuminating deep links between conformal blocks, gauge theory, and special functions.

Abstract

We perform a detailed study of a class of irregular correlators in Liouville Conformal Field Theory, of the related Virasoro conformal blocks with irregular singularities and of their connection formulae. Upon considering their semi-classical limit, we provide explicit expressions of the connection matrices for the Heun function and a class of its confluences. Their calculation is reduced to concrete combinatorial formulae from conformal block expansions.

Figures (2)

• Figure 1: Confluence diagram of conformal blocks.
• Figure 2: Arm length $A_{\tilde{Y}} (s)=4$ (white circles) and leg length $L_Y(s)=2$ (black dots) of a box at the site $s = (2,2)$ for the pair of superimposed diagrams $Y$ (solid lines) and $\tilde{Y}$ (dotted lines).