Exact expressions for the 5-Point Liouville conformal block with a level-two degenerate field insertion

Abstract

IIn this paper we investigate the 5-point Liouville conformal block with a level two degenerate field insertion. Our main tool is the BPZ differential equation, which, upon placing three of the insertions at the standard positions $\infty$, $1$, and $0$, reduces to a linear differential equation which is of order two in the degenerate insertion point $z$, and order one in the remaining point $x$. In a previous paper, it was conjectured that the solution could be expressed in terms of a single hypergeometric function and its derivative, with coefficients computable via recursive relations up to the desired order $x^k$. In this paper, we simplify these recursion relations and provide a rigorous inductive proof of the conjecture. A key advantage of our representation of the 5-point conformal block is that the hypergeometric connection formulae readily relate its different regions of analyticity. In the quasi-classical limit, the 5-point BPZ equation reduces to the Heun equation. Consequently, we recover a recently proposed representation of the Heun equation in terms of a single hypergeometric function, which has proven to be highly effective in the analysis of gravitational perturbations of black holes.