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

Book section

There is a strong analogy between compact, torsion-free G2-manifolds (X, ϕ, ∗ϕ) and Calabi–Yau 3-folds (Y, J, g, ω). We can also generalize (X, ϕ, ∗ϕ) to ‘tamed almost G2-manifolds’ (X, ϕ, ψ), where we compare ϕ with ω and ψ 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,ω), not on the (almost) complex structure J. We investigate whether it may be possible to define interesting invariants of tamed almost G2-manifolds (X, ϕ, ψ) by ‘counting’ compact associative 3- folds N ⊂ X, such that the invariants depend only on ϕ, and are independent of the 4-form ψ used to define associative 3-folds. We conjecture that one can define a superpotential Φψ : U → Λ>0 ‘counting’ associative Q-homology 3-spheres N ⊂ X which is deformation-invariant in ψ for ϕ fixed, up to certain reparametrizations Υ : U→U of the base U = Hom(H3(X; Z), 1+Λ>0), where Λ>0 is a Novikov ring. Using this we define a notion of ‘G2 quantum cohomology’. We also discuss Donaldson and Segal’s proposal from their 2011 work to define invariants ‘counting’ G2-instantons on tamed almost G2-manifolds, with ‘compensation terms’ counting weighted pairs of a G2-instanton and an associative 3-fold, and suggest some modifications to it.

Authors: Joyce, D

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: American Mathematical Society
• Host title: Proceedings of Symposia in Pure Mathematics
• Volume: 99
• Pages: 97-160
• Series: Proceedings of Symposia in Pure Mathematics
• Publication date: 2018-06-20
• ISBN: 9781470440947
• Keywords: math.DG; hep-th