Matching branches of non-perturbative conformal block at its singularity divisor

TL;DR

This paper explores the complex structure of non-perturbative conformal blocks at singularities, comparing different expressions and analyzing their branch matching issues in the moduli space.

Contribution

It investigates the matching of branches of non-perturbative conformal blocks at singular points, highlighting discrepancies between known formulas at special loci.

Findings

Different non-perturbative expressions differ at singular points.

Multiple branches of conformal blocks are analyzed at divisors in moduli space.

The study clarifies the structure of conformal blocks at special algebraic points.

Abstract

Conformal block is a function of many variables, usually represented as a formal series, with coefficients which are certain matrix elements in the chiral (e.g. Virasoro) algebra. Non-perturbative conformal block is a multi-valued function, defined globally over the space of dimensions, with many branches and, perhaps, additional free parameters, not seen at the perturbative level. We discuss additional complications of non-perturbative description, caused by the fact that all the best studied examples of conformal blocks lie at the singularity locus in the moduli space (at divisors of the coefficients or, simply, at zeroes of the Kac determinant). A typical example is the Ashkin-Teller point, where at least two naive non-perturbative expressions are provided by elliptic Dotsenko-Fateev integral and by the celebrated Zamolodchikov formula in terms of theta-constants, and they are different. The situation is somewhat similar at the Ising and other minimal model points.