Toric Virasoro conformal blocks

This is a commentary of a recent article by Nikita Nemkov, based on the report I wrote for the journal JHEP. I am making this text public because it might be useful to the community, but is kept confidential by the journal (as is unfortunately common practice). This blog post omits the parts of the report that deal with technical details and suggested improvements. Only the general commentary is reproduced, in a slightly modified form. Making it public implies renouncing anonymity. But I had already renounced anonymity by engaging in private correspondence with the author while studying his article. This made the process easier and more efficient, and I am grateful to Nikita Nemkov for his prompt and detailed answers. As a result, the article underwent very important improvements between Arxiv's second version and the JHEP version. I had never seen an author make such extensive improvements following a referee's suggestions.

Toric conformal blocks

Virasoro conformal blocks are fundamental objects in two-dimensional conformal field theory, that can be considered as generalizations of the characters of representations of the Virasoro algebra. Correlation functions can be written as combinations of conformal blocks and structure constants, where conformal blocks are universal and structure constants are model-dependent. The structure constants must obey conformal bootstrap equations, whose coefficients can be deduced from the behaviour of conformal blocks under certain transformations. Therefore, the transformation properties of conformal blocks are essential in the bootstrap approach to two-dimensional conformal field theory.

The article by Nikita Nemkov deals with toric conformal blocks that are relevant for computing one-point functions on the torus. These are the simplest non-trivial Virasoro conformal blocks. Together with the more complicated sphere four-point conformal blocks, they are enough for writing a generating set of the conformal bootstrap equations. More specifically, the behaviour of these blocks under modular transformations of the torus is described by a toric modular kernel, which appears in the bootstrap equations.

This toric modular kernel is already known in closed form. However, as far as I know, a clear and elementary derivation of this kernel has not appeared in the literature before Nemkov's article. Moreover, this article gives alternative formulas for this kernel.

Fusion algebra

In order to compute the toric modular kernel, the author derives and solves consistency relations of the fusion algebra. These relations are analogous to the Pentagon equation for the fusing matrix that describes transformation properties of sphere four-point blocks. As usual in theories with Virasoro symmetry, these consistency relations are vastly overdetermined. So the author only considers special cases where one field is degenerate. The special cases that correspond to the two simplest degenerate fields yield tractable equations for the toric modular kernel, that have a unique solution. Since the existence of the toric modular kernel is guaranteed on general grounds, this unique solution is expected to also be a solution of the general consistency relations.

The toric modular kernel is a function \(\mathcal{M}_{\alpha\alpha'}(\mu)\) of three parameters $\alpha$, $\alpha'$ and $\mu$, that are equivalent to the conformal dimensions of three Virasoro primary fields. It also depends on the central charge of the Virasoro algebra, via a parameter $b$ such that $b$ and $\frac{1}{b}$ correspond to the same central charge. The special equations essentially determine how the kernel behaves under shifts of $\alpha$, $\alpha'$ and $\mu$ by $\frac{b}{2}$ and $\frac{1}{2b}$. In other words, they amount to a system of difference equations.

Difference equations

The toric modular kernel should be a solution of a second-order difference equation on $\alpha$, that involves three values $\alpha, \alpha -\frac{b}{2}, \alpha + \frac{b}{2}$. We now need the statement that the space of solutions has dimension two, as a vector space on the field of functions of $\alpha$ with the period $\frac{b}{2}$. That statement is probably well-known and anyway quite easy to prove, as the author has shown to me in private communication. The author finds two independent solutions by using a series expansion in powers of $e^{4\pi i \alpha}$ as an ansatz. The relevant combination of the two solutions is determined by the expected symmetries of the kernel, more specifically reflection symmetry in $\alpha$.

So the dependences of the kernel on $\alpha$, $\alpha'$ and $\mu$ are determined by the difference equations, up to periodic functions with period $\frac{b}{2}$. This remaining freedom is then fixed by the dual difference equations, that are obtained by the replacement $b\to \frac{1}{b}$. The resulting kernel is found to agree with the previously known formula. However, solving the difference equations produces two alternative series formulas for the kernel, whereas the previously known formula was an integral expression. One of the series formulas can easily be shown to agree with the integral expression, but the equivalence of the two series formulas is not so easy to see.

Conclusion

This article presents a non-trivial technical result on a fundamental object of two-dimensional CFT. The proof uses well-known techniques in an efficient and rigorous way, there is no doubt that it is correct.

It would be interesting to compare the present formulas for the modular kernel with the formulas that follows from its relations with the fusing matrix.
Curiously, there are two independent such relations: a not very intuitive relation that involves two different values of the central charge, and a more intuitive but also more complicated relation that follows from the Moore-Seiberg equations. The second relation appeared in Appendix D of a long paper by Teschner and Vartanov, which also provides the precise normalization of the modular kernel, and a proof that it solves all the consistency relations of the fusion algebra, not just the degenerate relations. (I am grateful to Jörg Teschner for pointing this out.) So by now there is no shortage of derivations of the modular kernel, although Nemkov's derivation is probably the simplest.