The light asymptotic limit of $W$ algebra conformal blocks

$W$ algebras are natural extensions of the Virasoro algebra, the symmetry algebra of local conformal field theories in two dimensions. Conformal field theories with $W$ algebra symmetry include $W$ minimal models and conformal Toda theories, which are generalizations of Virasoro minimal models and Liouville theory respectively. In particular, $sl_N$ conformal Toda theory is based on the $W_N$ algebra, which has $N-1$ generators with spins $2,3,\dots, N$, and reduces to the Virasoro algebra in the case $N=2$.

The problem of solving conformal Toda theory

 

Solving $sl_{N\geq 3}$ conformal Toda theory is an outstanding problem. One may think that this is due to the complexity of the $W_N$ algebra, with its quadratic commutators. I would argue that this is rather due to the complexity of the fusion ring of $W_{N}$ representations, with its infinite fusion multiplicities. Due to these fusion multiplicities, solving $sl_N$ conformal Toda theory does not boil down to computing three-point function of primary fields: rather, one should also compute three-point functions of infinitely many descendent fields.

The interest of the light asymptotic limit

 

This is where the light asymptotic limit comes to the rescue. In this limit, the infinite-dimensional $W_N$ algebra reduces to the finite-dimensional $sl_N$ algebra, and $sl_N$ conformal Toda theory becomes tractable, while infinite fusion multiplicities are still present. In the case $N=3$ for example, four-point conformal blocks can be computed as three-dimensional integrals in this limit.
In their recent article, Hasmik Poghosyan, Rubik Poghossian and Gor Sarkissian have further shown how formulas for conformal blocks from the AGT correspondence simplify in the light asymptotic limit. More specifically, they have shown that in the light asymptotic limit, certain $W_N$ four-point blocks can be written as sums over $\frac{N(N-1)}{2}$ positive integers. The $W_N$ four-point blocks in question involve two generic fields and two almost fully degenerate fields, so they are not the most general $W_N$ four-point blocks. Rather, they constitute one of two classes of $W_N$ four-point blocks for which explicit formulas are known from the AGT correspondence. (There are two classes because there are two varieties of almost fully degenerate fields.)
The sum over $\frac{N(N-1)}{2}$ integers is a feature of the $W_N$ algebra, not of these particular four-point blocks. It is expected that in the light asymptotic limit, any $W_N$ four-point block can be written as a sum over $\frac{N(N-1)}{2}$ integers, or equivalently as an integral over $\frac{N(N-1)}{2}$ real variables. The results of Poghosyan, Poghossian and Sarkissian provide substantial support for this conjecture.

Suggestive formulas from the AGT correspondence

 

Thanks to work by Vladimir Mitev and Elli Pomoni, there is some hope that the AGT correspondence can be used for computing arbitrary correlation functions in conformal Toda theory, and not just particular correlation functions with no fusion multiplicities. When given a (very complicated) formula for a Toda correlation function, the challenge will be to understand how it can be decomposed into structure constants and conformal blocks. This decomposition depends on how fusion multiplicity is parametrized. So far, a simple parametrization of fusion multiplicity is known only in the light asymptotic limit. So this limit may have a useful role to play in solving conformal Toda theory: either as a toy model and source of inspiration, or as the starting point of a perturbative expansion.