$2+2=4$

Abstract

Motivated by the observation that $2+2=4$, we consider four-dimensional $\mathcal{N}=2$ superconformal field theories on $S^2\times\Sigma$, turning on a suitable rigid supergravity background. On the one hand, reduction of a four-dimensional theory ${T}$ on a Riemann surface $\Sigma$ leads to a family $\mathscr{F}[{T}, \Sigma]$ of two-dimensional $(2,2)$ unitary SCFTs, a two-dimensional analog of the four-dimensional theories of class $\mathscr{S}$. On the other hand, reduction on $S^2$ yields a non-unitary two-dimensional CFT $\mathscr{C}[{T}]$ 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 $\mathscr{F}[{T}; \Sigma]$ is computed by correlation functions of $\mathscr{C}[{T}]$ on $\Sigma$. The elliptic genus of $\mathscr{F}[{T}; \Sigma]$ is instead computed by a topological QFT $\mathscr{E}[T]$ on $\Sigma$. A central question is whether one can give a purely two-dimensional presentation of the family $\mathscr{F}[{T}; \Sigma]$ 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 $\Sigma$ an $n$-punctured sphere. We perform stringent checks of our conjecture for $k=1$ and $k=2$.