Pohlmeyer reduction of Newton-Cartan strings

Core claim

On the winding branch of the non-relativistic string, static gauge fixes the longitudinal projections of the light-cone tangents to $\tau^0(\partial_\pm X)=\kappa/2$ and $\tau^1(\partial_\pm X)=\pm\kappa/2$, and the only remaining data are the transverse projections. The paper's central claim is that the invariant oriented area $\chi=(\tau^1\wedge E^2)(\partial_+X,\partial_-X)/(2\tau^0(\partial_+X)\tau^0(\partial_-X))=\sin\vartheta\, \partial_\tau\varphi/\kappa$ is the Pohlmeyer field of the sNC string: it obeys the tachyonic Klein-Gordon equation $\partial_+\partial_-\chi=\kappa^2\chi/4$, and finite energy forces $\chi$ to vanish at the poles, converting the unstable long-wavelength tachyon into Dirichlet segment modes of real frequencies $\omega_n=\kappa\sqrt{n^2-1}$ on the $2\kappa$ segments between poles. The reduced dynamics is integrable in the zero-curvature sense: the connection $A_+=(\lambda\kappa/2)J+\partial_+\chi\, P_2$, $A_-=-(\kappa/2\lambda)(J+\chi P_1)$, valued in the Euclidean algebra $e(2)$ (one rotation plus two commuting translations, the contraction of $su(2)$), is flat precisely when the reduced equation holds, and at $\lambda=1$ it is the Maurer-Cartan form of the intrinsic moving frame $F=e^{\Upsilon}e^{\vartheta J}$; the spectral parameter is the rapidity of a worldsheet boost. The monodromy yields conserved charges only at the resonant parameters $\lambda^\pm_n=n\pm\sqrt{n^2-1}$, which select the $n$-th mode and its two chiralities and reproduce their quadratic oscillator invariants, with the static rigid-rotation mode $n=1$ invisible. In the $1/c^2$ expansion the same field is the next-to-leading-order object $\chi=(c/2)(\pi-\psi)+O(c^{-2})$, and the intrinsic Lax pair is the sum of the leading-order and next-to-leading-order sine-Gordon Lax pairs under the Inönü-Wigner contraction, so the tachyonic mass of the reduced equation is the coupling of the NLO transverse field to the frozen LO longitudinal motion.