2 + 2 = 4 a 2d/2d unitary/non-unitary correspondence

A

bstract

Motivated by the observation that 2 + 2 = 4, we consider four-dimensional

= 2 superconformal field theories on S ² × Σ, 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

$$ \mathcal{F}\left[T,\Sigma \right] $$ F T Σ

of two-dimensional (2, 2) unitary SCFTs, a two-dimensional analog of the four-dimensional theories of class

$$ \mathcal{S} $$ S

. On the other hand, reduction on S ² yields a non-unitary two-dimensional CFT

$$ \mathcal{C}\left[T\right] $$ 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

$$ \mathcal{F}\left[T;\Sigma \right] $$ F T Σ

is computed by correlation functions of

$$ \mathcal{C}\left[T\right] $$ C T

on Σ. The elliptic genus of

$$ \mathcal{F}\left[T;\Sigma \right] $$ F T Σ

is instead computed by a topological QFT

$$ \mathcal{E}\left[T\right] $$ E T

on Σ. A central question is whether one can give a purely two-dimensional presentation of the family

$$ \mathcal{F}\left[T;\Sigma \right] $$ 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 _{2 k} ) and Σ an n -punctured sphere. We perform stringent checks of our conjecture for k = 1 and k = 2.