The compact TCS $G_2$ duality conjecture with one D3-brane

Let $X_{E_n,sing}$ be the singular TCS $G_2$-manifold, let $S_i\subset X_{E_n,sing}$ be a surface that can be contracted, and let $Y_{E_n}$ be the corresponding elliptic Calabi–Yau fourfold. The F-theory compactification has vanishing flux $G_4=0$ and a D3-brane supported on the $E_n$ singular locus of the base.

Compact TCS $G_2$ duality conjecture. The following theories are equivalent:

• M-theory on $X_{E_n,sing}$ in the limit that a surface $S_i\subset X_{E_n,sing}$ is contracted.
• F-theory on $Y_{E_n}$ with $G_4=0$ and a single D3-brane on the $E_n$ singular locus on the base.

This is the paper’s main finite-Kovalev-parameter conjecture, extending the local duality to compact TCS $G_2$ manifolds; the source gives no resolution status.