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

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.

28 pages

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• Rainbow tensor model with enhanced symmetry and extreme melonic dominance
• Knot invariants from Virasoro related representation and pretzel knots
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• Decomposing Nekrasov Decomposition
• $q$-Virasoro/W Algebra at Root of Unity and Parafermions
• On determinant representation and integrability of Nekrasov functions
• On modular transformations of non-degenerate toric conformal blocks
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
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• On new exact conformal blocks and Nekrasov functions
• Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models