Mirror symmetry for quasi-smooth Calabi-Yau hypersurfaces in weighted projective spaces

Abstract

We consider a $d$-dimensional well-formed weighted projective space $\mathbb{P}(\overline{w})$ as a toric variety associated with a fan $\Sigma (\overline{w})$ in $N_{\overline{w}}\otimes \mathbb{N}$ whose $1$-dimensional cones are spanned by primitive vectors $v_0,v_1,\dots ,v_d\in N_{\overline{w}}$ generating a lattice $N_{\overline{w}}$ and satisfying the linear relation ${\sum }_i{w}_i{v}_i=0$. For any fixed dimension $d$, there exist only finitely many weight vectors $\overline{w}=(w_0,\dots ,w_d)$ such that $\mathbb{P}(\overline{w})$ contains a quasi-smooth Calabi-Yau hypersurface $X_w$ defined by a transverse weighted homogeneous polynomial $W$ of degree $w={\sum }_{i=0}^d{w}_i$. Using a formula of Vafa for the orbifold Euler number ${\chi }_{\mathrm{orb}}(X_w)$, we show that for any quasi-smooth Calabi-Yau hypersurface $X_w$ the number $(-1{)}^{d-1}{\chi }_{\mathrm{orb}}(X_w)$ equals the stringy Euler number ${\chi }_{\mathrm{str}}(X_{\overline{w}}^{\ast })$ of Calabi-Yau compactifications $X_{\overline{w}}^{\ast }$ of affine toric hypersurfaces $Z_{\overline{w}}$ defined by non-degenerate Laurent polynomials $f_{\overline{w}}\in \mathbb{C}[N_{\overline{w}}]$ with Newton polytope $\text{conv}(\{v_0,\dots ,v_d\})$. In the moduli space of Laurent polynomials $f_{\overline{w}}$ there always exists a special point $f_{\overline{w}}^0$ defining a mirror $X_{\overline{w}}^{\ast }$ with a $\mathbb{Z}/w\mathbb{Z}$-symmetry group such that $X_{\overline{w}}^{\ast }$ is birational to a quotient of a Fermat hypersurface via a Shioda map.

20 pages, 2 figures

References in corpus (3)

• FJRW rings and Landau-Ginzburg Mirror Symmetry
• On the Fine Interior of Three-dimensional Canonical Fano Polytopes
• Mirror Symmetry for Calabi-Yau complete intersections in Fano toric varieties

Cited by in corpus (2)

• Calabi-Yau Four/Five/Six-folds as ${\mathbb{P}}_{\text{w}}^n$ Hypersurfaces: Machine Learning, Approximation, and Generation
• On the stringy Hodge numbers of mirrors of quasi-smooth Calabi-Yau hypersurfaces