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

TL;DR

This work establishes a concrete bridge between protected operator algebras in 4d $\mathcal{N}=2$ SCFTs and 3d $\mathcal{N}=4$ TQMs arising upon $S^1$ reduction, by analyzing the Cardy (high-temperature) limit of the 4d theory on $S^1\times S^3$ and correlating it with 3d data. It provides explicit dictionaries between selected 4d OPE data (including chiral algebras, W-algebras, and minimal nilpotent Higgs-branch structures) and 3d 1d star-products in the TQM, verified through localization results and Schur index analyses in both Lagrangian and non-Lagrangian theories. A central outcome is the identification of 3d mirrors for various Argyres-Douglas theories and a consistent picture where the 3d limit often reduces to free twisted hypermultiplets and, in some cases, a T[SU(2)] sector, with precise FI/mass deformations encoding R-symmetry mixing. These results reinforce chiral algebra proposals and open avenues for understanding extended operator sectors, line/surface defects, and large-N holographic extensions in the protected sector. Overall, the paper demonstrates that high-temperature reductions faithfully reproduce a rich 3d protected sector, tying together VOA data, 3d mirrors, and exact 3d partition functions across a broad landscape of 4d SCFTs.

Abstract

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.

Figures (2)

• Figure 1: Schematic RG flow of Coulomb branch chiral operators in $(A_1, A_{2n-2})$ and $(A_1, D_{2n+1})$ theories. The plot for $(A_1, D_{2n+1})$ contains $(A_1, A_{2n-2})$ as a subplot, and has one more special operator corresponding to the $T[SU(2)]$ in 3d.
• Figure 2: Constructions of 3d mirrors for (a) $(A_1, A_{2n-2})$ and (b) $(A_1, D_{2n+1})$ from the 5d MSYM compactified on a cigar with one or two punctures, and the corresponding 4d MSYM on the interval.