\(W\) 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

\(N=2\).

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.