D-manifolds, a new theory of derived differential geometry
Seminar Room 1, Newton Institute
I describe a new class of geometric objects I call "d-manifolds". D-manifolds are a kind of "derived" smooth manifold, where "derived" is in the sense of the derived algebraic geometry of Jacob Lurie, Bertrand Toen, etc. The definition draws on ideas of Jacob Lurie, David Spivak. The original aim of the project, which I believe I have achieved, is to find the "right" definition of the Kuranishi spaces of Fukaya, Oh, Ohta and Ono, which is the geometric structure on moduli spaces of J-holomorphic curves in a symplectic manifold.
The definition of d-manifolds involves doing algebraic geometry over smooth functions (C-infinity rings); roughly speaking, a d-manifold is a differential-geometric analogue of a scheme with a perfect obstruction theory. D-manifolds form a strict 2-category dMan. It is a 2-subcategory of the larger 2-category of "d-spaces" dSpa. The definition does not involve localization of categories, so we have very good control of what 1-morphisms and 2-morphisms are.
The 2-categories dMan and dSpa have some very nice properties. All fibre products exist in dSpa, and a fibre product of d-manifolds is a d-manifold under weak transversality condition. For example, any fibre product of two d-manifolds over a manifold is a d-manifold. You can glue d-manifolds by equivalences of open d-submanifolds (a kind of pushout in dMan) provided the glued topological space is Hausdorff. There is a notion of "virtual cotangent bundle" of a d-manifold, which lives in a 2-category of virtual vector bundles, and a 1-morphism of d-manifolds is etale (a local equivalence) iff it induces an equivalence of virtual cotangent bundles. And so on.
There are also good notions of d-manifolds with boundary and d-manifolds with corners, and orbifold versions of all this, d-orbifolds.
D-manifolds and d-orbifolds have applications to moduli spaces and enumerative invariants in both differential and algebraic geometry. Almost any moduli space which is used to define some kind of counting invariant should have a d-manifold or d-orbifold structure. Any moduli space of solutions of a smooth nonlinear elliptic p.d.e. on a compact manifold has a d-manifold structure. Any C-scheme with a perfect obstruction theory has a d-manifold structure. In symplectic geometry, Kuranishi spaces and polyfold structures on moduli spaces of J-holomorphic curves induce d-orbifold structures. So much of Gromov-Witten theory, Donaldson-Thomas theory, Lagrangian Floer cohomology, Symplectic Field Theory,... can be rewritten in this language.