M-theory–F-theory duality for a class of TCS $G_2$-compactifications

Let $X$ be the TCS $G_2$-manifold constructed from the building blocks $Z_+$ and $Z_-$, and let $Y$ be the elliptic Calabi–Yau fourfold described as the fiber product of $Z_+$ with $dP_9$ over the common $P^1$. The flux on $Y$ is $G_4=0$.

M-theory–F-theory duality conjecture. The following physical theories are equivalent:

• M-theory on $X$.
• F-theory on $Y$ with $G_4=0$ and $12$ D3-branes.

This is the main proposed duality for the compactification class considered in the paper; its status is not established in the source.