The rank-$n$ $E_8$ compact duality conjecture

Let $n\in\mathbb{Z}_+$, let $X_{E_8^n,sing}$ be the TCS $G_2$ manifold with the tuned building block producing an $E_8-I_n$ intersection, and let $S_i\subset X_{E_8^n,sing}$ for $1\leq i\leq n$ be the surfaces contracted in the singular limit. Let $Y_{E_8}$ be the elliptic Calabi–Yau fourfold and let the $E_8$ singular locus lie on the base.

Rank-$n$ $E_8$ compact duality conjecture. The following theories are equivalent:

• M-theory on $X_{E_8^n,sing}$ in the limit that the $n$ surfaces $S_i\subset X_{E_8^n,sing}$ are contracted.
• F-theory on $Y_{E_8}$ with $G_4=0$ and $n$ D3-branes on the $E_8$ singular locus on the base.

This generalizes the one-D3-brane compact duality conjecture and is motivated by the rank-$n$ $E_8$ theory; only the rank-one and rank-two cases are described as well studied, while the asserted statement for arbitrary positive $n$ remains open.