Higher Quantum Teichmuller theory
Alexander Goncharov
Given topological surface S with punctures and a split reductive group G with trivial center, e.g. PGL(n), we quantize the moduli space of flat G-connections on S.
Namely, we define a non-commutative q-deformation A(G,S;q) of the algebra of regular functions on this moduli space -- it is a *-algebra. The mapping class group M(S) of S acts by its automorphisms.
We define the following data:
1) A unitary representation of the group M(S) in an infinite dimensional Hilbert space H
2) A *-representation of the tensor product of A(G,S;q) and A(G*,S;q*) in a Hilbert space H, equivariant under the group M(S).
Here G* is the Langlands dual group, and q=exp(2pi i h), q*=exp(2pi i /h) in the simply-laced case.
Finally, the Hilbert space H is conjectured to be the space of conformal blocks for the higher Liouville theory related to G.