Modular functor conjecture from quantum Teichmüller theory

H Verlinde suggested in 1980’s to use quantization of the Teichmüller spaces of surfaces to study the spaces of conformal blocks for the Liouville conformal field theory. This suggestion initiated and stimulated the development of quantum Teichmüller theory, and the first major steps were taken by Kashaev and by Chekhov and Fock in 1990’s, where the Chekhov-Fock quantization is generalized later by Fock and Goncharov to quantization of cluster varieties. The modular functor conjecture asserts that these quantum theories of Teichmüller spaces indeed yield a 2-dimensional modular functor, which can be viewed as one axiomatization of conformal field theory. The core part of the conjecture says that, for each punctured surface S and an essential simple loop in S, the Hilbert space associated to S by quantum Teichmüller theory should decompose into the direct integral of the Hilbert spaces associated to the surface obtained by cutting S along the loop and shrinking the holes to punctures. I will give an introduction to this story and present some recent developments, including 2405.14727.