Matching branches of a nonperturbative conformal block at its singularity divisor

A conformal block is a function of many variables, usually represented as a formal series with coefficients that are certain matrix elements in the chiral {(}i.e., Virasoro{\rm)} algebra. A nonperturbative conformal block is a multivalued function defined globally over the space of dimensions and has many branches and, perhaps, additional free parameters not seen at the perturbative level. We discuss additional complications of the nonperturbative description that arise because all the best-studied examples of conformal blocks are at the singularity locus in the moduli space {\rm(}at divisors of the coefficients or, simply, at zeros of the Kac determinant{\rm).} A typical example is the Ashkin–Teller point, where at least two naive nonperturbative expressions are provided by the elliptic Dotsenko–Fateev integral and by the celebrated Zamolodchikov formula in terms of theta constants, and they differ. The situation is somewhat similar at the Ising and other minimal model points.