The study of the moduli space plays an important role in classical enumerative geometry and its interaction with string theory in physics. Given X=[P1/Zr] and let x＇ = （[0]a , [∞]b） the 2-tuple of twisted sectors on X , we construct in this paper two different compactifications of the moduli space M0,2（X, d[P1/Zr], x＇）： Nonlinear Sigma Model Mx＇d and Linear Sigma Model Nx＇d . Relations between Mx＇d and Nx＇d are studied and a new gluing recursive relation on Nx＇d is derived from Mx＇d due to virtual localization formula.
In this paper, we generalize the construction of the inverse transgression map done by Adem, A., Ruan, Y. and Zhang, B. in [A stringy product on twisted orbifold K-theory. Morfismos, 11, 33 64 （2007）] and give a different proof to the statement that the image of the inverse transgression map for a gerbe with connection over an orbifold is an inner local system on its inertia orbifold.
In this paper,we study the global singular symplectic flops related to the following affine hypersurface with cyclic quotient singularities,Vr,b={(x,y,z,t)∈C4|xy-z2r+t2=0}/μr(a,-a,b,0),r 2,where b=1 appears in Mori’s minimal model program and b=1 is a new class of singularities in symplectic birational geometry.We prove that two symplectic 3-orbifolds which are singular flops to each other have isomorphic Ruan cohomology rings.The proof is based on the symplectic cutting argument and virtual localization technique.
