The VMC survey -- XLVI. Stellar proper motions in the centre of the Large Magellanic Cloud

We present proper motion (PM) measurements within the central region of the Large Magellanic Cloud (LMC) using near-infrared data from the VISTA survey of the Magellanic Cloud system (VMC). This work encompasses 18 VMC tiles covering a total sky area of $\sim$28~deg$^2$. We computed absolute stellar PMs from multi-epoch observations in the $K_s$ filter over time baselines between $\sim$12 and 47 months. Our final catalogue contains $\sim$6,322,000 likely LMC member stars with derived PMs. We employed a simple flat-rotating disc model to analyse and interpret the PM data. We found a stellar centre of rotation ($\alpha_0$ = 79.95 deg +0.22 -0.23, $\delta_0$ = -69.31 deg +0.12 -0.11) that is in agreement with that resulting from Hubble Space Telescope data. The inferred viewing angles of the LMC disc (i = 33.5 deg +1.2 -1.3, $\Theta$ = 129.8 deg +1.9 -1.9) are in good agreement with values from the literature but suggest a higher inclination of the central parts of the LMC. Our data confirm a higher rotation amplitude for the young ($\lesssim$0.5~Gyr) stars compared to the intermediate-age/old ($\gtrsim$1~Gyr) population, which can be explained by asymmetric drift. We constructed spatially resolved velocity maps of the intermediate-age/old and young populations. Intermediate-age/old stars follow elongated orbits parallel to the bar's major axis, providing first observational evidence for $x_1$ orbits within the LMC bar. In the innermost regions, the motions show more chaotic structures. Young stars show motions along a central filamentary bar structure.