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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.

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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