The topologically twisted index of $\mathcal N=4$ super-Yang-Mills on $T^2\times S^2$ and the elliptic genus

Abstract

We examine the topologically twisted index of $\mathcal N=4$ super-Yang-Mills with gauge group $SU(N)$ on $T^2\times S^2$, and demonstrate that it receives contributions from multiple sectors corresponding to the freely acting orbifolds $T^2/\mathbb Z_m\times\mathbb Z_n$ where $N=mn$. After summing over these sectors, the index can be expressed as the elliptic genus of a two-dimensional $\mathcal N=(0,2)$ theory resulting from Kaluza-Klein reduction on $S^2$. This provides an alternate path to the 'high-temperature' limit of the index, and confirms the connection to the right-moving central charge of the $\mathcal N=(0,2)$ theory.

Figures (1)

• Figure 1: Regions of vanishing determinant for $n'=2,\ldots,7$. The black regions correspond to $1+\tilde{\mathbb B}^0=0$, while the gray regions correspond to non-trivial $1+\tilde{\mathbb B}^0$, but still with vanishing determinant. The determinant evaluates to $n'^2$ in the unshaded regions. The yellow triange corresponds to the region $0<d_1\le d_2\le d_3<1$.