Reconstruction of Mirror Symmetry Hypothesis from a Geometrical Point of View

Abstract

In this chapter, we discuss degree d instanton correction of the A-model Yukawa coupling of a quintic hypersurface derived in the previous chapter, from the point of view of geometry of the moduli space of degree d holomorphic maps from \(CP^{1}\) to \(CP^{4}\). First, we naively compactify the moduli space into \(CP^{5(d+1)-1}\) and compute the corresponding instanton correction. Next, we introduce a refined moduli space \(\overline{Mp}_{0,2}(CP^{4},d)\), which is the compactified moduli space of degree d polynomial maps with two marked points, and compute intersection numbers of the moduli space that are related to instanton corrections of the quintic hypersurface [8]. We show that generating functions of these intersection numbers reproduce the period integrals used in the B-model computation. Then we reconstruct the mirror formula of the A-model Yukawa coupling proposed in the previous chapter by using these generating functions. Lastly, we explain the geometrical meaning of the coordinate transformation induced by the mirror map \(u=u(t)\). In some sense, discussions in this chapter correspond to mathematical justification of the mirror symmetry hypothesis in the case of a quintic hypersurface in \(CP^{4}\), which was rigorously done in the celebrated works by Givental [1] and by Lian, Liu and Yau [2]. But we present here our new approach from the point of view of “naive compactification of moduli space”.