Non-smooth strings and noncommutative geometry

The phase manifold of $d$-dimensional theory of smooth closed
strings may be identified with the space $\Omega(R_d)$ of smooth
loops taking values in the $d$-dimensional Minkowski space $R_d$.
However, the symplectic form $\omega$ of this theory can be extended
to the Sobolev completion of $\Omega(R_d)$ given by the space
$V_d=H_0^{1/2}(S^1,R_d)$ of half-differentiable loops with values in
$R_d$. The group of reparametrizations of such strings coincides
with the group $\text{QS}(S^1)$ of quasisymmetric homeomorphisms of
the circle and its action on the Sobolev space $V_d$ preserves the
symplectic form $\omega$. Taking this into account it is natural to
choose for the phase manifold of the theory of non-smooth strings
the space $V_d$ provided with the action of the group
$\text{QS}(S^1)$. If this action would be smooth we could associate
with this theory a classical system consisting of the phase manifold
$V_d$ and the Lie algebra of the group $\text{QS}(S^1)$. However,
this action is not smooth and we cannot associate any classical Lie
algebra with the group $\text{QS}(S^1)$. Nevertheless, we can
construct a quantum Lie algebra associated with $V_d$. We use for
that an approach based on the Connes noncommutative geometry.

Session commune avec le séminaire d'Analyse complexe et équations différentielles