Connecting Lyman-$\alpha$ and ionizing photon escape in the Sunburst Arc

We investigate the Lyman-$\alpha$ (Ly$\alpha$) and Lyman continuum (LyC) properties of the Sunburst Arc, a $z=2.37$ gravitationally lensed galaxy with a multiply-imaged, compact region leaking LyC and a triple-peaked Ly$\alpha$ profile indicating direct Ly$\alpha$ escape. Non-LyC-leaking regions show a redshifted Ly$\alpha$ peak, a redshifted and central Ly$\alpha$ peak, or a triple-peaked Ly$\alpha$ profile. We measure the properties of the Ly$\alpha$ profile from different regions of the galaxy using $R\sim5000$ Magellan/MagE spectra. We compare the Ly$\alpha$ spectral properties to LyC and narrowband Ly$\alpha$ maps from Hubble Space Telescope (HST) imaging to explore the subgalactic Ly$\alpha-$LyC connection. We find strong correlations (Pearson correlation coefficient $r>0.6$) between the LyC escape fraction ($f_{\rm esc}^{\rm LyC}$) and Ly$\alpha$ (1) peak separation $v_{\rm{sep}}$, (2) ratio of the minimum flux density between the redshifted and blueshifted Ly$\alpha$ peaks to continuum flux density $f_{\rm{min}}/f_{\rm{cont}}$, and (3) equivalent width. We favor a complex \ion{H}{1} geometry to explain the Ly$\alpha$ profiles from non-LyC-leaking regions and suggest two \ion{H}{1} geometries that could diffuse and/or rescatter the central Ly$\alpha$ peak from the LyC-leaking region into our sightline across transverse distances of several hundred parsecs. Our results emphasize the complexity of Ly$\alpha$ radiative transfer and its sensitivity to the anisotropies of \ion{H}{1} gas on subgalactic scales. Large differences in the physical scales on which we observe spatially variable direct escape Ly$\alpha$, blueshifted Ly$\alpha$, and escaping LyC photons in the Sunburst Arc underscore the importance of resolving the physical scales that govern Ly$\alpha$ and LyC escape.