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- W2542202295 abstract "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 $Nsubset X$, 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 $Phi_psi:{mathcal U}toLambda_{>0}$ 'counting' associative $mathbb Q$-homology 3-spheres $Nsubset X$ 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." @default.
- W2542202295 created "2016-11-04" @default.
- W2542202295 creator A5023239521 @default.
- W2542202295 date "2016-10-31" @default.
- W2542202295 modified "2023-09-27" @default.
- W2542202295 title "Conjectures on counting associative 3-folds in $G_2$-manifolds" @default.
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