Virasoro conformal blocks at rational central charge

Zamolodchikov's recursive representation of Virasoro conformal blocks is a powerful computational tool at generic central charge, but it is singular at rational central charge. In which cases is it possible to write a recursive representation at rational central charge?

Motivations

Virasoro conformal blocks are the basic special functions of w:two-dimensional conformal field theory. They are essential ingredients for computing correlation functions, which are some of the most important observables. In particular, they play a fundamental role in the w:conformal bootstrap approach.

Zamolodchikov's recursive representation is a powerful tool for numerically computing Virasoro conformal blocks. This representation involves a power series $H_{\mathrm{\Delta }}(\{{\mathrm{\Delta }}_i\}|q)$, which obeys the recursive equation

$H_{\mathrm{\Delta }}(\{{\mathrm{\Delta }}_i\}|q)=1+\sum _{m,n=1}^{\mathrm{\infty }}\frac{(16q{)}^{mn}}{\mathrm{\Delta }-{\mathrm{\Delta }}_{(m,n)}}{R}_{m,n}{H}_{{\mathrm{\Delta }}_{(m,-n)}}(\{{\mathrm{\Delta }}_i\}|q).$

Here $\mathrm{\Delta }$ is the channel conformal dimension of the conformal block, while $\{{\mathrm{\Delta }}_i\}=\{{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4\}$ are the conformal dimensions of the fields in the corresponding four-point function, and $q$ is a combination of their positions. The residues $R_{m,n}$ are explicitly known functions of the central charge and of $\{{\mathrm{\Delta }}_i\}$. The poles ${\mathrm{\Delta }}_{m,n}$ are functions of the central charge.

At rational central charge, the recursive representation becomes singular due to coincidences of the type ${\mathrm{\Delta }}_{(m,n)}={\mathrm{\Delta }}_{(m^{\prime },-n^{\prime })}$, and also because the residues $R_{m,n}$ themselves have poles. Conformal blocks themselves are well-defined, and all the singularities must cancel, as can be checked in examples. However, for generic dimensions $\{{\mathrm{\Delta }}_i\}$, these cancellations lead to higher order poles at $\mathrm{\Delta }={\mathrm{\Delta }}_{(m,n)}$, and it is not clear that an explicit recursive relation can be found. (In the limit from generic central charge, subleading terms can contribute.)

For some special values of the conformal dimensions, the situation can be more favourable. In particular, in w:minimal models, the conformal dimensions $\mathrm{\Delta },\{{\mathrm{\Delta }}_i\}$ are of the type ${\mathrm{\Delta }}_{(m,n)}$, and this leads to the vanishing of many residues $R_{m,n}$.

Questions

• Find a recursive recursion representation of the conformal blocks that appear in minimal models, by taking limits of Zamolodchikov's recursive representation when the central charge becomes rational.

• Compare with the AGT-inspired representation of Bershtein and Foda.

• Generalize to other CFTs with discrete spectrums.

Technical challenges

Given the conformal dimension ${\mathrm{\Delta }}_{(m,n)}$ of a minimal model field, there exist several possible continuations to generic central charge, due to identities of the type ${\mathrm{\Delta }}_{(m,n)}={\mathrm{\Delta }}_{(m^{\prime },n^{\prime })}$ that hold at generic central charge.

Recent results on $c=1$ blocks suggest that it is crucial that our continuation respects all fusion rules of the channel field. A natural way to do this is to fix the four external fields' Kac indices relative to the center of the Kac table. Then at least one of the two indices of each field is half-integer. With this continuation, it is plausible that generic $c$ blocks have the desired limit in all cases.

However, in contrast to the $c=1$ case, the terms in the recursion will not be individually finite, and there will be cancellations of divergences, leading to poles of arbitrarily high orders in the channel dimension. Is this compatible with the claims of Bershtein and Foda?

Compared to 4-point blocks on the sphere, 1-point blocks on the torus lead to similar difficulties, but are technically simpler, because they have fewer parameters. In the case of 1-point blocks on the torus, the rational $c$ limit has been investigated in more detail.