28 problems

Let $J$ be a compact $G_2$ manifold, and consider Type IIA and Type IIB string compactifications on $J$. At the level described, both compactifications have the same matter content…

The $\chi$-flow conjecture. The $\chi$-flow on $\Sigma\subset Q$ takes $\Sigma$ diffeomorphically onto the zero section.

Let $M$ be a $G_2$-manifold and consider calibrated fibrations on $M$, analogous to special Lagrangian or holomorphic fibrations on a Calabi–Yau threefold. Conjectural characteriza…

Let a $G_2$ manifold admit a coassociative torus fibration, and let its associated Calabi–Yau three-fold be obtained by brane reduction. Conversely, consider the Kaluza–Klein lift…

Compact lift conjecture. The local construction lifts to the compact TCS $G_2$ manifold as an action on

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

M2–M5 monodromy conjecture. The corresponding monodromy acts by

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

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

Euler characteristic conjecture. The Euler number of $\mathcal{C}$ is

Let $X$ be a twisted connected sum ($TCS$) $G_2$ manifold, and consider its mirror-symmetry description in analogy with special Lagrangian torus fibrations. A $T^4$-fibration is a…

A TCS $G_2$ manifold $X$ is constructed by gluing asymptotically cylindrical Calabi–Yau three-folds, and globally it can be viewed as a $K3$ fibration over $S^3$: … The fiber is a…

Chen's gluing conjecture. Although the standard TCS gluing argument does not apply, it should be possible to glue the modified building blocks to a $G_2$-manifold with conical sing…

Let $M$ be the compact $G_2$ manifold under consideration, and let the associatives $\tilde{C}_\gamma$ be the three-spheres obtained from $S^1_{x_3}$ and sections of the relevant e…

Donaldson–Witten conjecture. The nodal singularities along circles may be replaced by isolated conical singularities with this homogeneous space as the link. The conjecture propose…

For every element $(\gamma,\hat\gamma)$ of $E_8 \oplus E_8$, let $\Sigma_{\gamma}^+$ and $\Sigma_{\hat\gamma}^-$ be three-chains on $Z_\pm$ with boundaries $e_1-e^1$, where…

Let $J$ be a TCS $G_2$-manifold, and let the three-cycles under consideration be the cycles identified by matching dual three-cycles in the TCS construction. Calibrated three-cycle…

Equal-Betti-number mirror conjecture. Any two $G_2$ manifolds with the same value of $b^2+b^3$ should lead to mirror compactifications of Type II strings. The conjecture is invoked…

Singular TCS gauge-theory conjecture. The singular TCS $G_2$-manifolds underlying the M-theory compactifications exist, and they give rise to the same gauge theories as on the F-th…

Coassociative-cycle collapse conjecture. The existence of tensionless strings in the dual F-theory model forces the collapse of a coassociative four-cycle that arises as a componen…

Coassociativity conjecture. The K3 fibres are coassociative four-cycles in $J$; equivalently, they are calibrated by $\star\Phi_3$.

Let $X_L$ and $X_R$ be asymptotically cylindrical Calabi--Yau threefolds used to construct a twisted connected sum $G_2$-manifold by orthogonal gluing, and let an extremal transiti…

Let M-theory denote the proposed eleven-dimensional theory whose compactifications on seven-dimensional manifolds with $G_2$ holonomy yield effective theories with $\mathcal{N}=1$…

Let $X$ be a compact $G_2$-manifold, let $N$ be the singular set, and let $P$ be an $SO(3)$-bundle over $X\backslash N$ induced by restricting a bundle $\tilde{P}$ over $X$. Let…

Effective-associatives conjecture. The appropriate way to study such degenerations is to move to a wall in a “cone of effective associatives.”