General three-point function in Toda conformal field theory

Determine the general expression for the three-point structure constants in Toda conformal field theory beyond the parameter ranges where the Fateev–Litvinov formula has been proved.

Background

Toda CFT generalizes Liouville theory to fields valued in the Cartan algebra of a semisimple Lie algebra and is closely connected to the AGT correspondence. Recent works have constructed Toda path integrals and certain boundary theories.

While specific cases of the three-point function are known (via Fateev–Litvinov) in restricted parameter regimes, a general closed-form expression remains unknown.

References

Cercl e has been able to prove the formula of Fateev-Litvinov for the $3$-point function for certain range of parameters. The general expression of the $3$-point function is not known.

— Two Decades of Probabilistic Approach to Liouville Conformal Field Theory  (2509.21053 - Rhodes et al., 25 Sep 2025) in Section 7 (Further developments, related results and open problems) — Path integrals for other CFTs