Conjectures on counting associative 3-folds in $G_2$-manifolds

There is a strong analogy between compact, torsion-free $G_2$-manifolds $(X,\phi ,\ast \phi )$ and Calabi-Yau 3-folds $(Y,J,g,\omega )$. We can also generalize $(X,\phi ,\ast \phi )$ to ‘tamed almost $G_2$-manifolds’ $(X,\phi ,\psi )$, where we compare $\phi$ with $\omega$ and $\psi$ with $J$. Associative 3-folds in $X$, a special kind of minimal submanifold, are analogous to $J$-holomorphic curves in $Y$. Several areas of Symplectic Geometry – Gromov-Witten theory, Quantum Cohomology, Lagrangian Floer cohomology, Fukaya categories – are built using ‘counts’ of moduli spaces of $J$-holomorphic curves in $Y$, but give an answer depending only on the symplectic manifold $(Y,\omega )$, not on the (almost) complex structure $J$. We investigate whether it may be possible to define interesting invariants of tamed almost $G_2$-manifolds $(X,\phi ,\psi )$ by ‘counting’ compact associative 3-folds $N\subset X$, such that the invariants depend only on $\phi$, and are independent of the 4-form $\psi$ used to define associative 3-folds. We conjecture that one can define a superpotential $\Phi_\psi:{\mathcal U}\to\Lambda_{>0}${}^{\mathrm{\prime }}countin{g}^{\mathrm{\prime }}associative$\mathbb Q$-homology3-spheres$N\subset X$whichisdeformation-invariantin$\psi$for$\varphi$ fixed, up to certain reparametrizations $\mathrm{{\rm Y}}:\mathcal{U}\to \mathcal{U}$ of the base $\mathcal{U}=$Hom$(H_3(X;\mathbb{Z}),1+{\mathrm{\Lambda }}_{>0})$, where ${\mathrm{\Lambda }}_{>0}$ is a Novikov ring. Using this we define a notion of ‘$G_2$ quantum cohomology’. These ideas may be relevant to String Theory or M-Theory on $G_2$-manifolds. We also discuss Donaldson and Segal’s proposal in arXiv:0902.3239, section 6.2, to define invariants ‘counting’ $G_2$-instantons on tamed almost $G_2$-manifolds $(X,\phi ,\psi )$, with ‘compensation terms’ counting weighted pairs of a $G_2$-instanton and an associative 3-fold, and suggest some modifications to it.