Kinematic analysis of $\mathbf{z = 4.3}$ galaxies in the SPT2349$-$56 protocluster core

SPT2349$-$56 is a protocluster discovered in the 2500 deg$^2$ South Pole Telescope (SPT) survey. In this paper, we study the kinematics of the galaxies found in the core of SPT2349$-$56 using high-resolution (1.55 kpc spatial resolution at $z = 4.303$) redshifted [CII] 158-$μ$m data. Using the publicly available code 3D-Barolo, we analyze the seven far-infrared (FIR) brightest galaxies within the protocluster core. Based on conventional definitions for the detection of rotating discs, we classify six sources to be rotating discs in an actively star-forming protocluster environment, with weighted mean $V_{\mathrm{rot}}/σ_{\mathrm{disp}} = 4.5 \pm 1.3$. The weighted mean rotation velocity ($V_{\mathrm{rot}}$) and velocity dispersion ($σ_{\mathrm{disp}}$) for the sample are $ 357.1 \pm 114.7$ km s$^{-1}$ and $43.5 \pm 23.5$ km s$^{-1}$, respectively. We also assess the disc stability of the galaxies and find a mean Toomre parameter of $Q_\mathrm{T} = 0.9 \pm 0.3$. The galaxies show a mild positive correlation between disc stability and dynamical support. Using the position-velocity maps, we find that five sources further classify as disturbed discs, and one classifies as a strictly rotating disc. Our sample joins several observations at similar redshift with high $V/σ$ values, with the exception that they are morphologically disturbed, kinematically rotating and interacting galaxies in an extreme protocluster environment.