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 ${\Bbb P}(w_0, \ldots, w_d)$ 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 $h^{p,q}_{\rm str}(X^{\vee}) = h^{d-1-p,q}_{\rm orb}(X)$ for all $p, q$ and $d$ as it is expected in mirror symmetry.