General three-point function in Toda CFT

Determine the general expression for the three-point structure constants in Toda conformal field theory for all parameter ranges, extending beyond the partial results currently known.

Background

Recent probabilistic work has constructed Toda CFT and, in special cases, established the Fateev–Litvinov formula for three-point functions. However, a complete closed-form expression valid across all parameters has not been obtained.

The authors explicitly note the absence of a general formula, marking this as a key missing ingredient for a full bootstrap of Toda CFT.

References

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 on Path integrals for other CFTs (Further developments, related results and open problems)