Supersymmetric index on T^2 x S^2 and elliptic genus

TL;DR

This paper explores the partition function of 4d supersymmetric theories on T^2 x S^2, showing it equals the elliptic genus of 2d theories, revealing new dualities and connections to string theory objects.

Contribution

It establishes a link between 4d supersymmetric partition functions and 2d elliptic genera, and identifies new dualities and string theory correspondences.

Findings

Partition function on T^2 x S^2 equals elliptic genus of 2d (0,2) theories.

Identification of 4d theories corresponding to elliptic genera of K3, M-strings, and E-strings.

Discovery of two distinct 4d theories with the elliptic genus of K3, suggesting new dualities.

Abstract

We study partition function of four-dimensional $\mathcal{N}=1$ supersymmetric field theory on $T^2 \times S^2$. By applying supersymmetry localization, we show that the $T^2 \times S^2$ partition function is given by elliptic genus of certain two-dimensional $\mathcal{N}=(0,2)$ theory. As an application, we discuss a relation between 4d Seiberg duality duality and 2d $(0,2)$ triality, proposed by Gadde, Gukov and Putrov. In other examples, we identify 4d theories giving elliptic genera of K3, M-strings and E-strings. In the example of K3, we find that there are two 4d theories giving the elliptic genus of K3. This would imply new four-dimensional duality.