N=2 Landau-Ginzburg模型とトポロジカル・シグマ模型におけるミラー対称性
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We first review mirror symmetry in the N=2 Landau-Ginzburg model. This mirror symmetry exists only in the superconformal field theory. The correspondence between the Landau-Ginzburg model and the string theory is proved, and if the Hodge number of the deformation of the complex structure on a Calabi-Yau manifold was equal to the Hodge number of the deformation of the Kahler structure on other Calabi-Yau manifold and vice versa, then there necessarily exists a pair of Calabi-Yau manifolds with different structures. The manifolds with such a relation are called mirror pairs. In the case of a Calabi-Yau threefold, the Hodge numbers of the deformation of a comelex structure and a Kahler structure correspond to the Hodge numbers h^<2,1> and h^<1,1> respectively. If either value could be decided, the other value can be decided through the Euler number of the manifold. Physically, by introducing the Witten index Tr(-1)^F, the numbers of the states of classification is obtained by means of the U(1) charges in the chiral ring of the Landau-Ginzburg model. Each Hodge number correspond to the numbers of the states in the chiral ring and the numbers of the states of the mirror pair satisfy the relation h^<1,1>(M)=h^<2,1>(W). The mirror manifolds can be constructed explicitly by orbifoldizing the coordinates of the two kinds of target spaces, i.e, complex projective space and the weighted complex projective space. Moreover, by introducing the complete intersection Calabi-Yau manifolds we could define more generic forms of the mirror manifolds. Next, the topological A and B model are constructed by twisting the N=(2, 2) nonlinear sigma model, and it is shown that the physical states of these models are classified by the BRST cohomology groups. The physical states of these two topological sigma models are related through the mirror symmetry. Finally, we will briefly mention the twistor string approach to the B-model, which is recently extensively investigated in the Maximally Helicity Violating amplitude in the QCD and graviton scattering amplitude.
- 東海大学の論文
- 2006-03-30
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- N=2 Landau-Ginzburg模型とトポロジカル・シグマ模型におけるミラー対称性