On the stringy Hodge numbers of mirrors of quasi-smooth Calabi-Yau hypersurfaces

Abstract

Mirrors $X^{\vee }$ of quasi-smooth Calabi-Yau hypersurfaces $X$ in weighted projective spaces $\mathbb{P}(w_0,\dots ,w_d)$ can be obtained as Calabi-Yau compactifications of non-degenerate affine toric hypersurfaces defined by Laurent polynomials whose Newton polytope is the lattice simplex spanned by $d+1$ lattice vectors $v_i$ satisfying the relation ${\sum }_i{w}_i{v}_i=0$. In this paper, we compute the stringy $E$-function of mirrors $X^{\vee }$ and compare it with the Vafa's orbifold $E$-function of quasi-smooth Calabi-Yau hypersurfaces $X$. As a result, we prove the equalities of Hodge numbers $h_{\mathrm{str}}^{p,q}(X^{\vee })=h_{\mathrm{orb}}^{d-1-p,q}(X)$ for all $p,q$ and $d$ as it is expected in mirror symmetry.

arXiv comment18 pages

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