Planar string as an anyon model: cusps, braids and soliton exitations

We investigate the strings in (2 + 1)D space-time, which may be considered as excitations of straight lines on the spatial plane. We also propose the hamiltonian description of such objects that differs from the standard hamiltonian description of the string. The hamiltonian variables of suggested dynamical system are separated into two independent groups: the "internal" and "external" variables. The first ones are invariant under space-time transformations and are connected with the second form of the world-sheet. The "external" variables define the embedding of the world-sheet into space-time. The constructed phase space is nontrivial because the finite number of constraints entangles the variables from these groups. First group of the variables constitute the coefficients for the pair of first-order spectral problems; the solution of these problems is necessary for the reconstruction of the string world-sheet. We consider the excitations, which correspond to "N-soliton" solution of the spectral problem, and demonstrate that the reconstructed string has cuspidal points. World lines of such points form the braids of various topologies. The quantization scheme is suggested. The validity of such approach to avoid the critical dimensions in quantized theory is discussed.