  Superconformal field theories (SCFT) are known to possess solvable yet
nontrivial sectors in their full operator algebras. Two prime examples are the
chiral algebra sector on a two dimensional plane in four dimensional
$\mathcal{N}=2$ SCFTs, and the topological quantum mechanics (TQM) sector on a
line in three dimensional $\mathcal{N}=4$ SCFTs. Under Weyl transformation,
they respectively map to operator algebras on a great torus in $S^1\times S^3$
and a great circle in $S^3$, and are naturally related by reduction along the
$S^1$ factor, which amounts to taking the Cardy (high-temperature) limit of the
four dimensional theory on $S^1\times S^3$. We elaborate on this relation by
explicit examples that involve both Lagrangian and non-Lagrangian theories in
four dimensions, where the chiral algebra sector is generally described by a
certain W-algebra, while the three dimensional descendant SCFT always has a
(mirror) Lagrangian description. By taking into account a subtle R-symmetry
mixing, we provide explicit dictionaries between selected operator product
expansion (OPE) data in the four and three dimensional SCFTs, which we verify
in the examples using recent localization results in four and three dimensions.
Our methods thus provide nontrivial support for various chiral algebra
proposals in the literature. Along the way, we also identify three dimensional
mirrors for Argyres-Douglas theories of type $(A_1, D_{2n+1})$ reduced on
$S^1$, and find more evidence for earlier proposals in the case of $(A_1,
A_{2n-2})$, which both realize certain superconformal boundary conditions for
the four dimensional $\mathcal{N}=4$ super-Yang-Mills. This is a companion
paper to arXiv:1911.05741.
