Conjectures on counting associative 3-folds in \(G_2\)-manifolds

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 \(N\subset X\), 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 argue that 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, probably will not work.