نظریه گراموف-ویتن از نسبت‌های انواع فرمت کالیبی-یاو

"We construct a global B-model for any quasi-homogeneous polynomial f that has properties similar to the properties of the physic's B-model on a Calabi-Yau manifold. The main ingredients in our construction are K. Saito's theory of primitive forms and Givental's higher genus reconstruction. More precisely, we consider the moduli space M[unfilled bullet]mar of the so-called marginal deformations of f. For each point [sigma] [is an element of] M[unfilled bullet]mar we introduce the notion of an opposite subspace in the twisted de Rham cohomology of the corresponding singularity f[sigma] and prove that opposite subspaces are in one-to-one correspondence with the splittings of the Hodge structure in the vanishing cohomology of f[sigma]. Therefore, according to M. Saito, an opposite subspace gives rise to a semi-simple Frobenius structure on the space of miniversal deformations of f[sigma]. Using Givental's higher genus reconstruction we define a total ancestor potential A[sigma](h,q) whose properties can be described quite elegantly in terms of the properties of the corresponding opposite subspace. For example, if the opposite subspace corresponds to the splitting of the Hodge structure given by complex conjugation, then the total ancestor potential is monodromy invariant and it satisfies the BCOV holomorphic anomaly equations. The coefficients of the total ancestor potential could be viewed as quasi-modular forms on M[unfilled bullet]mar in a certain generalized sense. As an application of our construction, we consider the case of a Fermat polynomial W that defines a Calabi-Yau hypersurface XW in a weighted-projective space. We have constructed two opposite subspaces and proved that the corresponding total ancestor potentials can be identified with respectively the total ancestor potential of the orbifold quotient XW/G̃W and the total ancestor potential of FJRW invariants corresponding to (W,GW). Here GW is the maximal group of diagonal symmetries of W and GW is a quotient of GW by the subgroup of those elements that act trivially on XW . In particular, our result establishes the so-called Landau-Ginzburg/Calabi-Yau correspondence for the pair (W,GW)"-- Provided by publisher Cover 1 Title page 2 Chapter 1. Introduction 8 Chapter 2. Global CY-B-model and quasi-modular forms 12 Chapter 3. Global Landau-Ginzburg B-model at genus zero 18 3.1. A family of polynomials 18 3.2. The twisted de Rham cohomology 19 3.3. The Gauss-Manin connection and the higher residue pairing 20 Chapter 4. Opposite subspaces 22 4.1. Symplectic vector space and semi-infinite VHS 22 4.2. Definition and first properties 23 4.3. Homogeneous opposite subspaces over the marginal moduli 25 4.4. The complex conjugate opposite subspace 30 4.5. Opposite subspaces for Fermat polynomials 31 4.6. The Cecotti-Vafa structure 32 Chapter 5. Quantization and Fock bundle 36 5.1. Givental’s quantization formalism 36 5.2. Frobenius structures 37 5.3. From an opposite subspace to a Frobenius structure 38 5.4. The total ancestor potential 40 5.5. The abstract Fock bundle 44 5.6. Abstract modular forms 51 5.7. The holomorphic anomaly equations 52 5.8. Fermat simple elliptic singularity of type E6^{(1,1)} 54 Chapter 6. Mirror symmetry for orbifold Fermat CY hypersurfaces 58 6.1. Orbifold Gromov–Witten theory 58 6.2. I-function 61 6.3. Mirror symmetry for D-modules 63 Chapter 7. Mirror symmetry for Fermat CY singularities 76 Appendix A. A proof of Proposition 7.0.1 86 Bibliography 96 Back Cover 104