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Conformal bootstrap for Toda CFT with W-algebra symmetry

Develop a complete conformal bootstrap for Toda conformal field theory, whose symmetry algebra is the W-algebra, including the determination of three-point structure constants and conformal blocks consistent with the bootstrap constraints.

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Background

Toda CFT is expected to be governed by an extended W-algebra symmetry, which significantly complicates the bootstrap analysis relative to Virasoro-based Liouville theory.

A full bootstrap solution would parallel the program completed for Liouville CFT and clarify the role of extended symmetries in non-rational CFTs.

References

In physics, it is argued that the symmetry algebra of the Toda CFT is the W-algebra, which is strictly bigger than the Virasoro algebra. Understanding the conformal bootstrap in this context is a challenging open problem.

— 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