XXXVI Workshop on Geometric Methods in Physics 2-8.07.2017 VI School on Geometry and Physics 26-30.06.2017
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# Armen Sergeev

## 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.

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