In this paper, we explore the virtual technique that is very useful in studying the moduli problem from a differential geometric point of view. We introduce a class of new objects ＂virtual manifolds/orbifolds＇, on w...In this paper, we explore the virtual technique that is very useful in studying the moduli problem from a differential geometric point of view. We introduce a class of new objects ＂virtual manifolds/orbifolds＇, on which we develop the integration theory. In particular, the virtual localization formula is obtained.展开更多
In this paper, by using the de Rham model of Chen-Ruan cohomology, we define the relative Chen-Ruan cohomology ring for a pair of almost complex orbifold （G, H） with H being an almost sub-orbifold of G. Then we use ...In this paper, by using the de Rham model of Chen-Ruan cohomology, we define the relative Chen-Ruan cohomology ring for a pair of almost complex orbifold （G, H） with H being an almost sub-orbifold of G. Then we use the Gromov Witten invariants of G, the blow-up of G along H,to give a quantum modification of the relative Chen-Ruan cohomology ring H^R（G, H） when H is a compact symplectic sub-orbifold of the compact symplectic orbifold G.展开更多
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 ...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.展开更多
The bubble tree compactified instanton moduli space -Mκ （X） is introduced. Its singularity set Singκ（X） is described. By the standard gluing theory, one can show that- Mκ（X） - Singκ（X） is a topological orb...The bubble tree compactified instanton moduli space -Mκ （X） is introduced. Its singularity set Singκ（X） is described. By the standard gluing theory, one can show that- Mκ（X） - Singκ（X） is a topological orbifold. In this paper, we give an argument to construct smooth structures on it.展开更多
Consider a Hamiltonian action of S1 on （Cn^n＋1,ωstd）, we shown that the Hamiltonian Gromov-Witten invariants of it are well-defined. After computing the Hamiltonian Gromov-Witten invariants of it, we construct a r...Consider a Hamiltonian action of S1 on （Cn^n＋1,ωstd）, we shown that the Hamiltonian Gromov-Witten invariants of it are well-defined. After computing the Hamiltonian Gromov-Witten invariants of it, we construct a ring homomorphism from HS1,CR（X, R） to the small orbifold quantum cohomology of X//rS^1 and obtain a simpler formula of the Gromov-Witten invariants for weighted projective space.展开更多
基金supported by an NSF key projecta 973 project 2006CB805900
文摘In this paper, we explore the virtual technique that is very useful in studying the moduli problem from a differential geometric point of view. We introduce a class of new objects ＂virtual manifolds/orbifolds＇, on which we develop the integration theory. In particular, the virtual localization formula is obtained.
基金supported by National Natural Science Foundation of China(Grant Nos.11071173 and 11221101)
文摘In this paper, by using the de Rham model of Chen-Ruan cohomology, we define the relative Chen-Ruan cohomology ring for a pair of almost complex orbifold （G, H） with H being an almost sub-orbifold of G. Then we use the Gromov Witten invariants of G, the blow-up of G along H,to give a quantum modification of the relative Chen-Ruan cohomology ring H^R（G, H） when H is a compact symplectic sub-orbifold of the compact symplectic orbifold G.
基金supported by Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20100181110071)National Natural Science Foundation of China (Grant No. 11071176),supported by National Natural Science Foundation of China (Grant Nos. 11071173 and 11221101)Hundred Talents Program for Young Teachers (Grant No. SWJTU12BR028)
文摘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.
基金Supported by NSF Key Project 10631050A0109a 973 project 2006CB805900
文摘The bubble tree compactified instanton moduli space -Mκ （X） is introduced. Its singularity set Singκ（X） is described. By the standard gluing theory, one can show that- Mκ（X） - Singκ（X） is a topological orbifold. In this paper, we give an argument to construct smooth structures on it.
基金partially supported by NSFC(Grant Nos.11021101 and 11426233)
文摘Consider a Hamiltonian action of S1 on （Cn^n＋1,ωstd）, we shown that the Hamiltonian Gromov-Witten invariants of it are well-defined. After computing the Hamiltonian Gromov-Witten invariants of it, we construct a ring homomorphism from HS1,CR（X, R） to the small orbifold quantum cohomology of X//rS^1 and obtain a simpler formula of the Gromov-Witten invariants for weighted projective space.
