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

Understand

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.