Virasoro blocks and quasimodular forms

Peer review ratings

This is a commentary of the preprint by Das, Datta and Raman.

Context and general comments

Virasoro conformal blocks are functions that play an important role in two-dimensional conformal field theory. They do not have simple expressions in terms of well-known special, functions, although they have many remarkable properties. Computing the blocks efficiently is important for numerical calculations of correlation functions.

This preprint focuses on expanding Virasoro conformal blocks around $h_{p}\to \infty$, where $h_{p}$ is the intermediate conformal dimension. Using insight from the 2d/4d correspondence (also known as the AGT correspondence), the authors predict that the coefficients of this expansion should have simple properties under modular transformations. And indeed they succeed in writing the first few coefficients in terms of Eisenstein series, which are quasimodular forms. This leads to an improved analytic understanding and/or better numerical algorithms for Virasoro conformal blocks in some regimes.

Validity

The proposed expressions for the first few coefficients of the sphere four-point blocks and torus one-point blocks in the large $h_{p}$ expansion look plausible. They can be checked by comparing with expressions from Zamolodchikov's recursion. The authors claim they have done these checks, using publicly available code for computing Virasoro conformal blocks, but they do not provide their own code. They also perform a few simple analytic checks of the formulas.

Overall, there is a convincing case that the coefficients can be written in terms of Eisenstein series, but it is not easy to check the specific expressions from the publicly available material.

Interest

As is well explained in the Introduction, the expression of the leading order coefficient in terms of the weight-2 Eisenstein series was already known.

The present article pushes the computation to the next few orders, which gives a much better idea of the expansion to all orders. In the case of the sphere four-point block, the results are however restricted to the case where the four fields all have the same dimension. There is also the interesting (although not unexpected) remark that the expansion simplifies significantly if the expansion parameter is the momentum rather than the conformal dimension (in the article's terminology, this is called the Liouville parametrization), and if we consider the logarithm of the block rather than the block itself.

This article's results are substantial, but they are neither the first nor probably the last word on the subject. As explained in the Conclusions, it remains to generalize the results in various directions, and to "pin down the CFT origin of the modular anomaly in the blocks".

Clarity

The article is clear and well-written, with a level of technical detail that is enough for precisely understanding the statements, while still allowing the article to remain concise. The choice of the material that is relegated to the Appendices is judicious: standard facts on Eisenstein series, and higher orders in the expansions, should certainly be kept out of the main text, but can be helpful to some readers.