Homotopy L-infinity spaces and Kuranishi manifolds, I: categorical structures

الأصل · EN

Motivated by the definition of homotopy L∞ spaces, we develop a new theory of Kuranishi manifolds, closely related to Joyce's recent theory. We prove that Kuranishi manifolds form a 2-category with invertible 2-morphisms, and that certain fiber product property holds in this 2-category. In a subsequent paper, we construct the virtual fundamental cycle of a compact oriented Kuranishi manifold, and prove some of its basic properties. Manifest from this new formulation is the fact that [0,1]-type homotopy L∞ spaces are naturally Kuranishi manifolds. The former structured spaces naturally appear as derived enhancements of Maurer-Cartan moduli spaces from Chern-Simons type gauge theory. In this way, Kuranishi manifolds theory can be applied to study path integrals in such type of gauge theories.