Equivalence of Liouville conformal blocks with Teichmüller space quantization

Prove the conjectured equivalence between the Liouville conformal blocks quantization and the Chekhov–Fock–Kashaev quantization of Teichmüller space, establishing a precise correspondence between these two quantization frameworks.

Background

Teichmüller space admits a quantization via the Chekhov–Fock–Kashaev approach. Independently, Liouville conformal blocks define a quantization structure through the conformal bootstrap. A long-standing conjecture asserts these quantizations coincide.

The authors recall Verlinde’s conjecture and subsequent supporting evidence by Teschner, noting that a complete proof remains open.