Virtual technique for orbifold Fredholm systems

In this paper, we review the the theory of virtual manifold/orbifolds developed by the first named author and Tian and develop the virtual technique for any orbfiold Fredholm system with compact moduli space M. This provides a description of M in terms of a virtual orbifold system {(VI , EI , σI )} Here {VI } is a virtual orbifold, and {EI } is a finite rank virtual orbifold bundle with a virtual section {σI } such that the zero sets {σ −1 I (0)} form a cover of the underlying moduli space M. A virtual orbifold system can be thought as a special class of Kuranishi structures on a moduli problem developed by Fukaya and Ono. Under some assumptions which guarantee the existence of a partition of unity and a virtual Euler form, we show that the virtual integration is well-defined for the resulting virtual orbifold system.