Abstract:
The unversal Teichmüller space $\mathcal T$ is the quotient of the group $\mathrm{QS}(S^1)$ of quasisymmetric homeomorphisms of the unit circle $S^1$ modulo Möbius transforms. (Recall that a homoemorphism of the unit circle is called quasisymmetric if it extends to a quasiconformal homeomorphism of the unit disk.) The space $\mathcal T$ contains all classical Teichml̈ler spaces and the quotient $\mathcal S$ of the group $\mathrm{Diff}_+(S^1)$ of
smooth diffeomorphisms of $S^1$ modulo Möbius transforms. Both groups act by change of variable on the Sobolev space $H:=H^{1/2}_0(S^1,\mathbb R)$ of half-differentiable functions on $S^1$.
The quantization problem for the introduced spaces $\mathcal T$ and $\mathcal S$ arises in string theory where both spaces are considered as phase manifolds of this theory. Recall that the solution of the quantization problem for a given phase manifold consists of the choice of a Lie algebra of functions (observables) on this manifold and construction of its irreducible representation in some Hilbert space, called the quantization space.
In the case of the space of smooth diffeomorphisms $\mathcal S$ we take for the algebra of observables the Lie algebra $\mathrm{Vect}(S^1)$ of the Lie group $\mathrm{Diff}_+(S^1)$, consisting of smooth vector fields on the circle. While the role of the quantization space in this case is played by the Fock space $F(H)$, associated with the Sodolev space $H$. The infinitesimal version of the action of the group $\mathrm{Diff}_+(S^1)$ on $H$ generates an
irreducible representation of the Lie algebra $\mathrm{Vect}(S^1)$ in the space $F(H)$, determining the quantization of $\mathcal S$.
In the case of the universal Teichmüller space $\mathcal T$ the situation becomes more complicated since the action of the group $\mathrm{QS}(S^1)$ on $\mathcal T$ is no longer smooth. Thus, there exists no classical Lie algebra, associated with the group $\mathrm{QS}(S^1)$. In this case we use considerations from the noncommutative geometry. With their help we can construct a quantum Lie algebra of observables $\mathrm{Der}^q(\mathrm{QS})$, generated by the quantum differentials, acting in the space $F(H)$. These differentials are generated by the integral operators $d^qh$ on the Sobolev space $H$ with kernels, given essentially by the finite-difference derivatives of homeomorphisms $h\in\mathrm{QS}(S^1)$.