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-wave-length asymptotics of the collective mode in 1D boson systems at the Tonks-Girardeau regime. The collective mode has the dispersion of Bogoliubov phonons: $\omega(q)=q \sqrt{\bar\rho U(q)/m}$, where $\bar\rho$ is the bosons density and $U(q)$ is a Fourier component of the two-body potential.