$2+2=4$

TL;DR

This work extends the SCFT/VOA correspondence by promoting the associated vertex operator algebra to a full 2d CFT, denoted $\mathscr{C}[T]$, in the setting of 4d $\mathcal{N}=2$ SCFTs on $S^2\times\Sigma$. A 2d/2d map is proposed: the $S^2$ partition function of the $\mathscr{F}[T;\Sigma]$ theory, obtained by reducing on $\Sigma$, is captured by correlators of $\mathscr{C}[T]$ on $\Sigma$, while the elliptic genus is computed by a topological QFT $\mathscr{E}[T]$ on $\Sigma$. The paper develops a rigorous 4d off-shell supergravity background to realize the construction (B-type background), analyzes counterterm ambiguities, and provides concrete 2d constructions via gauged linear sigma models for $(A_1,A_{2k})$ Argyres–Douglas theories on an $n$-punctured sphere, with stringent checks for $k=1,2$ (i.e., $(A_1,A_2)$ and $(A_1,A_4)$). A central result is the explicit 2d/2d dictionary: the $S^2$ partition function of $\mathscr{F}[T;\Sigma]$ equals the correlator of $\mathscr{C}[T]$ on $\Sigma$, while the elliptic genus factorizes through a 2d TQFT $\mathscr{E}[T]$, with the blocks determined by vortex/gluing data linked to puncture types. The minimal $(A_1,A_2)$ example is worked out in depth, establishing a GLSM realization and a consistent TQFT structure for the elliptic genus, and the analysis is extended to $(A_1,A_4)$ with analogous, though more intricate, constructions and checks. Overall, the work provides a concrete, computable 2d framework for studying protected sectors of 4d $\mathcal{N}=2$ theories, opening avenues toward a broader class of AD theories and potential holographic connections.

Abstract

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

Figures (9)

• Figure 1: A sphere as a gluing of two cigars, realizing the combination of a chiral algebra with its anti-chiral counterpart to yield a full two-dimensional CFT.
• Figure 2: Thinking of $S^2_B$ as an $S^1$ fibration of the interval $I$ and reducing along the red $S^1$ fibers produces a 3d $\mathcal{N}=4$ SCFT subjected to an $SU(2)_H$ topological twist. The endpoints of the interval support holomorphic and anti-holomorphic boundary conditions.
• Figure 3: TQFT structure of the elliptic genus of the 2d $(2,2)$ theory $\mathscr{F}[(A_1,A_2),\Sigma_{0,4}]$ obtained compactifying $(A_1,A_2)$ on a sphere with four punctures of type $\phi_{(2,1)}$. We use red and blue colors to distinguish punctures of different sign, i.e. with opposite choice of boundary conditions.
• Figure 4: Check of the associativity property of the TQFT formula for the elliptic genus. We verify that in the case of a sphere with two $\phi_{(2,1)}$ and two $I$ punctures, different decompositions lead to the same answer.
• Figure 5: TQFT structure of the elliptic genus of the 2d $(2,2)$ theory $\mathscr{F}[(A_1,A_2),\Sigma_{0,5}]$ obtained compactifying $(A_1,A_2)$ on a sphere with five punctures of type $\phi_{(2,1)}$.