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

TL;DR

This work demonstrates that the partition function of 4d ${\rm N}=1$ theories on ${T^2\times S^2}$ localizes to the elliptic genus of a 2d ${\rm N}=(0,2)$ theory, formalized through a JK-residue expression for the 4d one-loop determinant. The authors derive a precise main formula, show how nontrivial 4d data (gauge, matter, and fluxes) reduce to a 2d spectrum on ${T^2}$, and connect 4d Seiberg duality to 2d ${\rm (0,2)}$ triality. They also explore a range of examples—K3, E-strings, and M-strings—where 4d indices reproduce known 2d elliptic genera, revealing potential new 4d dualities. The results provide a bridge between higher-dimensional indices and 2d CFT data, suggesting deeper hidden structures and relations among dualities across dimensions.

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.