Description
Conformal blocks of the Liouville CFT are known to have very
simple analytic structure with respect to the positions of degenerate
fields. The corresponding monodromy is « quantum » (operator-valued) as
it involves shifts of internal momenta. In the quasiclassical limit, the
BPZ equation satisfied by the simplest nontrivial example of such
conformal block reduces to Heun equation. I will explain how careful
analysis of the limit allows to solve the Heun connection problem in
terms of quasiclassical Virasoro conformal blocks, generalizing a
conjectural relation between quasiclassical Liouville CFT and Heun
accessory parameter function found by Zamolodchikov in 1986. I will then
discuss how this conjecture can be checked with the help of the
classical Darboux theorem relating the Heun connection matrix to the
large-order asymptotics of the coefficients of the corresponding
Frobenius solutions.
