WKB-asymptotics for multipoint Virasoro conformal blocks and applications

Abstract

We study multipoint Virasoro conformal blocks on the sphere in the comb channel. We arrive at the asymptotic expression for these blocks at large intermediate dimensions, applying WKB method for "classical BPZ equation", which is used to study (classical) Virasoro blocks via monodromy method. Several applications of this asymptotic are discussed, such as the possibility to generalize Zamolodchikov's elliptic recursion and numerical evaluation of amplitudes in minimal string theory. Our expressions pass nontrivial checks, such as agreement with known exact expressions for 5-point blocks in special cases and the usual series expansion of Virasoro blocks computed using AGT correspondence.

Figures (9)

• Figure 1: Monodromy in the comb channel of the $5$-point block.
• Figure 2: Comb channel for the $n$-point conformal block.
• Figure 3: $A$ and $B$ periods on the hyperelliptic curve.
• Figure 4: Relative difference between elliptic recursion at 8th order and other methods with fixed intermediate momenta $P_{i1}=1, P_{i2}=1.5$, for real $x$ (left) and $\text{Arg}\,x = \pi/3$ (right).
• Figure 5: Relative difference between elliptic recursion for 8th order and other methods for fixed $x = 0.3$ and $P_{i2} = 1.5$ as a function of $P_{i1}$.