Convergence of classical conformal blocks

We give a recursive method to compute the classical conformal blocks in Liouville field theory. The values of the expansion coefficients are given by an algebraic scheme which works to all orders. The algebraic expression of the intervening matrices are explicitly given. With regard to the problem of the convergence of the series we rigorously prove that it has a finite (non zero) convergence radius. We then comment on the relation of the conformal block problem with the Riemann-Hilbert problem.

💡 Research Summary

The paper presents a rigorous recursive scheme for computing classical conformal blocks (CCBs) in Liouville field theory and proves that the resulting power series in the modulus x has a finite, non‑zero radius of convergence. Starting from the classical limit of the quantum four‑point function, the authors relate the CCB to the accessory parameter C(x) that appears in the Fuchsian differential equation governing the Liouville solution. By expanding the potential Q(z) in powers of x and performing a non‑linear change of variable z(v)=v−B₀−B₁/v−B₂/v²+…, they map the original equation to a new one whose Schwarzian derivative encodes the deformation. The coefficients B_k are determined order by order from a linear system A(N)·b=V(N), where A(N) is an upper‑triangular, nested matrix whose N‑th column is given explicitly (see equations (44)–(46)). Because A(N) is triangular, its inverse can be obtained without computing determinants, using a simple column‑exchange formula (eqs. (49)–(50)). This yields the expansion coefficients C^{(N)} for the accessory parameter, and consequently for the CCB, to any desired order. The authors verify that the first few coefficients agree with known classical limits.

The convergence analysis proceeds in two steps. First, using a Green‑function technique they show that the trace of the monodromy matrix remains −2 cos πλ_ν for |x|<1, establishing convergence of the trace series. Second, employing tools from analytic geometry they prove that C(x) itself is analytic in a disk of finite radius around the origin; the lower bound on this radius depends explicitly on the Liouville parameters (δ₀, δ₁, δ_∞, etc.). This distinguishes C(x), which is a genuine holomorphic function of x, from the accessory parameter C_L(x, \bar{x}) that appears in the uniformization problem and is only real‑analytic.

Finally, the work connects the CCB problem to the classical Riemann‑Hilbert problem: determining a Fuchsian differential equation with prescribed monodromy data. The recursive construction provides an explicit solution to this inverse monodromy problem in the four‑point case. The paper concludes by noting that the method extends to all orders, that the convergence is rigorously established, and that open directions include generalizations to higher‑point configurations, torus topology, and sharper numerical estimates of the convergence radius.