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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.

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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)