The Close AGN Reference Survey (CARS): tracing the circumnuclear star formation in the super-Eddington NLS1 Mrk 1044

The host galaxy conditions for rapid supermassive black hole growth are poorly understood. Narrow-line Seyfert 1 (NLS1) galaxies often exhibit high accretion rates and are hypothesized to be prototypes of active galactic nuclei (AGN) at an early stage of their evolution. We present VLT MUSE NFM-AO observations of Mrk 1044, the nearest super-Eddington accreting NLS1. Together with archival MUSE WFM data we aim to understand the host galaxy processes that drive Mrk 1044's black hole accretion. We extract the faint stellar continuum emission from the AGN-deblended host and perform spatially resolved emission line diagnostics with an unprecedented resolution. Combining both MUSE WFM and NFM-AO observations, we use a kinematic model of a thin rotating disk to trace the stellar and ionized gas motion from 10$\,$kpc down to 30$\,$pc around the nucleus. Mrk 1044's stellar kinematics follow circular rotation, whereas the ionized gas shows tenuous spiral features in the center. We resolve a compact star forming circumnuclear ellipse (CNE) that has a semi-minor axis of 306$\,$pc. Within this CNE, the gas is metal rich and its line ratios are entirely consistent with excitation by star formation. With an integrated SFR of $0.19 \pm 0.05 \,{\rm M}_\odot\,{\rm yr}^{-1}$ the CNE contributes 27% of the galaxy-wide star formation. We conclude that Mrk 1044's nuclear activity has not yet affected the circumnuclear star formation. Thus, Mrk 1044 is consistent with the idea that NLS1s are young AGN. A simple mass budget consideration suggests that the circumnuclear star formation and AGN phase are connected and the patterns in the ionized gas velocity field are a signature of the ongoing AGN feeding.