Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 7

4d/2d → 3d/1d: A song of protected operator algebras

Pages: 2011 – 2075



Mykola Dedushenko (Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, New York, U.S.A.; and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, Calif., U.S.A.)

Yifan Wang (Joseph Henry Laboratories, Princeton University, Princeton, New Jersey, U.S.A.; Center of Mathematical Sciences & Applications and Jefferson Physical Laboratory, Harvard University, Cambridge, Massachusetts, U.S.A.)


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 $\href{}{\textrm{arXiv:1911.05741}}$.

Published 30 August 2023