Conformal Toda theory

Conformal Toda theory is a generalization of Liouville theory with a larger chiral symmetry algebra, namely a W-algebra. While Liouville theory is solved, conformal Toda theory is more challenging.

Difficulties
In CFT with Virasoro symmetry, three-point functions of primary fields determine three-point functions of descendant fields via conformal Ward identities. In other words, fusion multiplicity is always one. This is no longer true with W-symmetry, where fusion multiplicity is in general infinite. And there is no canonical way to parametrize this multiplicity. For example, three-point functions of all $$W_3$$-descendants of the primary three-point functions $$\left\langle V_1 V_2 V_3\right\rangle$$ are determined if we know the infinite family of descendant three-point functions $$\left\langle W_{-1}^k V_1 V_2 V_3\right\rangle$$ with $$k\in\mathbb{N}$$, but this family is not natural or symmetric.

Known results
Fateev and Litvinov have applied the known methods of two-dimensional CFT to conformal Toda theory, in particular Coulomb gas integrals and analytic bootstrap equations. These methods work for correlation functions that have sufficiently many vanishing null vectors. In particular, fusion multiplicities are finite. (Actually always one?) The resulting formulas for three-point structure constants generalize the DOZZ formula of Liouville theory, in particular they are built on Barnes' double Gamma function.

In the light asymptotic limit, when the central charge becomes infinite while conformal dimensions remain fixed, conformal $$\mathfrak{sl}_N$$ Toda theory reduces to harmonic analysis on the group $$SL_N$$. This leads to a canonical treatment of the infinite fusion multiplicities, and correlation functions may be written as integrals over $$SL_N(\mathbb{C})/SU_N$$.

The problem becomes somewhat more tractable in q-deformed conformal field theory. While the q-deformation's meaning in field theory is not clear, it makes it possible to compute three-point structure constants of generic primary fields, without the null vectors of Fateev and Litvinov. However, knowing primary three-point functions is not enough to solve the theory. Moreover, it is not known how to take the limit $$q\to 1$$, where we should recover the Toda CFT.

In the boundary Toda CFT, some observables such as bulk one-point functions do not involve fusion multiplicities, and are tractable.

Considerations of diagonality
By a nice basis for fusion multiplicities, we may mean a basis where correlation functions are diagonal in fusion multiplicities. What does this correspond to in the chiral CFT, in terms of conformal blocks? Use formal crossing symmetry, and the relation between the fusing matrix and the three-point structure constant.

Direct bootstrap approach
In order to use the analytic bootstrap approach that works for Liouville theory, one should study a four-point function of one fully degenerate field and three generic primary fields, in order to deduce shift equations for three-point structure constants. While in Liouville's case such a four-point function is hypergeometric, in the Toda case it does not obey any nontrivial differential equation. This can be interpreted as a manifestation of the infinite fusion multiplicities.

However, we do not really need to determine this four-point function, but only the corresponding fusing matrix elements. Presumably, these might be derived using the Pentagon identity. In order for these matrix elements to be simple enough, it is probably needed to have a good parametrization of fusion multiplicities.

Detour via minimal models
The solution of Liouville theory could in principle be derived from A-series Virasoro minimal models. The central charges of these models are indeed dense in the half-line $$c\leq 1$$, and taking a limit where the central charge and conformal dimensions become irrational, we recover Liouville theory with $$c\leq 1$$. Liouville theory with more generic complex central charges cannot in principle be deduced from $$c\leq 1$$, but it is very similar, and could potentially be guessed.

It may be possible to follow a similar approach for conformal Toda theory. In particular, fusion multiplicities are finite in W-minimal models. It is possible to study them in the simplest minimal models, before increasing the models' size, and hopefully find a simple way to describe them in general. It is then necessary to understand arbitrary W-minimal models, so that one can take a limit where the central charge becomes irrational.

This approach looks tedious, but at least it decomposes the problem into a sequence of presumably tractable steps. To some extent, this approach was already used in the boundary theory.