Conjectures on counting associative 3-folds in $G_2$-manifolds
  There is a strong analogy between compact, torsion-free $G_2$-manifolds $(X,\varphi,*\varphi)$ and Calabi-Yau 3-folds $(Y,J,g,\omega)$. We can also generalize $(X,\varphi,*\varphi)$ to 'tamed almost $G_2$-manifolds' $(X,\varphi,\psi)$, where we compare $\varphi$ 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 $(X,\varphi,\psi)$ by 'counting' compact associative 3-folds $N\subset X$, such that the invariants depend only on $\varphi$, and are independent of the 4-form $\psi$ used to define associative 3-folds.   We conjecture that one can define a superpotential $\Phi_\psi:{\mathcal U}\to\Lambda_{>0}$ 'counting' associative $\mathbb Q$-homology 3-spheres $N\subset X$ which is deformation-invariant in $\psi$ for $\varphi$ fixed, up to certain reparametrizations $\Upsilon:{\mathcal U}\to{\mathcal U}$ of the base ${\mathcal U}=$Hom$(H_3(X;{\mathbb Z}),1+\Lambda_{>0})$, where $\Lambda_{>0}$ 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 $(X,\varphi,\psi)$, with 'compensation terms' counting weighted pairs of a $G_2$-instanton and an associative 3-fold, and suggest some modifications to it. 
Dominic Joyce
math.DG hep-th