1D composite fermions: Bogoliubov-like mode in the Tonks-Girardeau gas

We reformulate 1D boson-fermion duality in path-integral terms. The result is a 1D counterpart of the boson-fermion duality in the 2D Chern-Simons gauge theory. The theory is consistent and enables, using standard resummation techniques, to obtain the long-wavelength asymptotics of the collective mode in 1D boson systems at the Tonks-Girardeau regime. The collective mode has the dispersion of Bogoliubov phonons: ω(q) = q(U(q)/m)1/2, where is the bosons density and U(q) is a Fourier component of the two-body potential.