Extended Lyman-$\alpha$ emission towards the SPT2349-56 protocluster at $z=4.3$

International audienceContext. Deep spectroscopic surveys with the Atacama Large Millimeter/submillimeter Array (ALMA) have revealed that some of the brightest infrared sources in the sky correspond to concentrations of dusty star-forming galaxies (DSFG) at high redshift. Among these, the SPT2349-56 protocluster system at z = 4.304 is amongst the most extreme examples due to its high source density and integrated star formation rate. Aims. We conducted a deep Lyman-$\alpha$ line emission survey around SPT2349-56 using the Multi-Unit Spectroscopic Explorer (MUSE) at Very Large Telescope (VLT) in order to characterize this uniquely dense environment. Methods. Taking advantage of the deep three-dimensional nature of this survey, we performed a sensitive search for Lyman-$\alpha$ emitters (LAEs) toward the core and northern extension of the protocluster, which correspond to the brightest infrared regions in this field. Using a smoothed narrowband image extracted from the MUSE datacube around the protocluster redshift, we searched for possible extended structures. Results. We identify only three LAEs at z = 4.3 in this field, in concordance with expectations for blank-fields, and an extended Lyman-$\alpha$ structure spatially associated with core of the protocluster. All the previously-identified DSFGs in this field are undetected in Lyman-$\alpha$ emission, consistent with the conspicuous dust obscuration in these systems. We find an extended Lyman-$\alpha$ structure, about $60 \times 60$ kpc$^{2}$ in size, and located 56 kpc west of the protocluster core. Three DSFGs coincide spatially with the location of this structure. We conclude that either the three co-spatial DSFGs or the protocluster core itself are feeding ionizing photons to the Lyman-$\alpha$ structure