Convergence of Virasoro conformal blocks

Virasoro conformal blocks have several series representations, but it is not proved that these series converge. This does not prevent them from being routinely used in numerical calculations. Proofs of convergence would put these calculations on firmer mathematical ground.

Motivations
Virasoro four-point conformal blocks on the sphere are basic special functions of two-dimensional conformal field theory. Understanding their analytic properties, and computing them efficiently, are of fundamental interest.

Type of project
Tools: Known expressions for conformal blocks from the physics literature, plus the mathematical tools that are needed to prove convergence. (Hopefully, elementary tools.)

Chances of success: The question of convergence is fundamental in principle, but not crucial in practice for physicists. The lack of proofs so far does not imply that the problem is particularly difficult. Chances of success seem quite good, at least with Zamolodchikov's recursive representation, which is reasonably simple and explicit.

Length and difficulty: As series, conformal blocks depend on a number of parameters: the central charge and conformal dimensions. They have poles as functions of these parameters, and the convergence properties therefore presumably depend on the parameters' values. Studying convergence for simple and/or generic cases may be relatively quick and easy, but dealing with all possible cases may take more time.

Known results
Conformal blocks are usually expanded as series either in the cross-ratio $$z$$ or the nome $$q=q(z)$$. The radius of convergence of both expansions is expected to be 1, but this has not been proved. Examples such as hypergeometric conformal blocks show that the radius is at most 1 for the expansion in $$z$$: the question is whether it is always 1.

Expansion in the cross-ratio
From their definition as sums over descendant states, four-point conformal blocks have a representation as series in the cross-ratio $$z\in\mathbb{C}$$ of the four fields' positions. The radius of convergence is expected to be 1: intuitively, convergence occurs when the two fields whose OPE defines the block are closer to one another than to the other remaining fields.

The formula for the series is explicit enough, except for the Shapovalov matrix $$Q$$. Deriving the needed properties of that matrix may require some algebra.

The expansion in the cross-ratio can alternatively be computed using the AGT relation with Nekrasov partition functions. However, it is not clear how to deduce that the series converges.

Expansion in the nome
This expansion follows from Zamolodchikov's recursive representation of conformal blocks. The radius of convergence is again expected to be one, but the region $$\{|q|<1\}$$ corresponds to a dense subset of the complex $$z$$-plane.

This representation involves infinite sums whose coefficients are known explicitly. Further formal manipulations are possible, which may help prove convergence. For certain rational values of the central charge, the recursive representation apparently diverges, whereas the blocks are finite: the divergences can be eliminated by resumming the expansion, but there is no known way to do this systematically.

Work to be done
It seems reasonable to start with Zamolodchikov's recursive representation, which is known more explicitly and is more useful in practice. The main question will be for which values of the parameters convergence occurs. The coefficients in the series expansion have poles as functions of the central charge and conformal dimensions: this may mean that convergence depends on the number-theoretic properties of the central charge. Such a phenomenon has been found to occur in the case of sums over conformal blocks in certain conformal field theories.