$\mathbb{T}^4/\mathbb{Z}_2$, $\mathbb{T}^2/\mathbb{Z}_2$, and $\mathbb{CP}^1$

Question

I was stuck by reading this figure:

It looks that $\mathbb{T}^4/\mathbb{Z}_2$, $\mathbb{T}^2/\mathbb{Z}_2$, and $\mathbb{CP}^1$ are somehow related.

Are there some easier explanations from math directly?

It is a figure 2 in this ref: https://link.springer.com/content/pdf/10.1007%2FJHEP07%282017%29088.pdf