$2+2=4$

Abstract

Motivated by the observation that $2+2=4$, we consider four-dimensional $\mathcal{N}=2$ superconformal field theories on $S^2\times \hat{I}\pounds$, turning on a suitable rigid supergravity background. On the one hand, reduction of a four-dimensional theory $T$ on a Riemann surface $\hat{I}\pounds$ leads to a family $\mathcal{F}[T,\hat{I}\pounds ]$ of two-dimensional $(2,2)$ unitary SCFTs, a two-dimensional analog of the four-dimensional theories of class $\mathcal{S}$. On the other hand, reduction on $S^2$ yields a non-unitary two-dimensional CFT $\mathcal{C}[T]$ whose chiral algebra is the same as the one associated to $T$ by the standard SCFT/VOA correspondence. This construction upgrades the vertex operator algebra to a full-fledged two-dimensional CFT. What's more, it leads to a novel 2d/2d correspondence, a "$2+2=4$" analog of the "$4+2=6$" AGT correspondence: the $S^2$ partition function of $\mathcal{F}[T;\hat{I}\pounds ]$ is computed by correlation functions of $\mathcal{C}[T]$ on $\hat{I}\pounds$. The elliptic genus of $\mathcal{F}[T;\hat{I}\pounds ]$ is instead computed by a topological QFT $\mathcal{E}[T]$ on $\hat{I}\pounds$. A central question is whether one can give a purely two-dimensional presentation of the family $\mathcal{F}[T;\hat{I}\pounds ]$ of $(2,2)$ theories. We propose an algorithm to realize the $(2,2)$ theories as gauged linear sigma models when $T$ is an Argyres-Douglas theory of type $(A_1,A_{2k})$ and $\hat{I}\pounds$ an $n$-punctured sphere. We perform stringent checks of our conjecture for $k=1$ and $k=2$.

62+22 pages