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

TL;DR

This paper analyzes the topologically twisted index of 4D $ ext{SU}(N)$ super-Yang-Mills on $T^2 imes S^2$, revealing its decomposition into orbifold sectors and its relation to the elliptic genus of a 2D $ ext{(0,2)}$ theory, linking high-temperature behavior to central charge.

Contribution

It shows that the index decomposes into orbifold sectors and can be expressed as the elliptic genus of a reduced 2D theory, providing new insights into its high-temperature limit and central charge connection.

Findings

Index receives contributions from orbifold sectors $T^2/\mathbb Z_m \times \mathbb Z_n$

The index equals the elliptic genus of a 2D $\mathcal N=(0,2)$ theory

Confirms the link between the index's high-temperature limit and the right-moving central charge

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.