Mirror symmetry in 3d in 3d mirror symmetry

Given a compact CY3 $Y$, its A-side (resp. B-side) universal intermediate Jacobian $X$ (resp. $X^{!}$) admits a natural hyperkahler structure. Both $X$ and $X^{!}$ determines 3d Rozansky-Witten theories, in both A-model and B-model. We describe some surprising 3d mirror symmetry phenomena between $X$ and $X^{!}$. This includes (i) a 3d SYZ construction of $X^{!}$ from $X$ via moduli of certain 3d A-branes in $X$ constructed from $D^{b}\left( Y,\Omega \right)$ with varying stability conditions given by $\varpi$; (ii) The 3d B-brane on $X^{!}$, constructed from varying $D^{b}\left( Y,\Omega \right)$ with fixed $\varpi$, is realized as the 3d SYZ transformation of a cotangent fiber 3d A-brane in $X$. Under mirror symmetry between $Y$ and $Y^{\vee }$, the roles for $X$ and $$ got interchanged. We also construct 3d A- and B-branes in $X$ and $X^{!}$ via DT theory and discrete symmetries on $Y$ and $Y^{\vee }$.