4d/2d $\rightarrow $ 3d/1d: A song of protected operator algebras

Understand

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.