Elliptic genera of Berglund-Hübsch Landau-Ginzburg orbifolds (Q2630389)

Mirror symmetry is a remarkable duality between a pair of Calabi-Yau \(n\)-folds, exhibiting as the ``mirror'' flip of the respective Hodge diamonds, or physically, as the equivalence of a pair of \(\mathcal{N}=(2,2)\) super-conformal field theories, especially coming from orbifold Landau-Ginzburg theories. Of the many interesting properties of the mirror pair, the equality between the elliptic genus is an important consequence. In general, the elliptic genus \(Ell\) for a complex manifold \(M\) is given in terms of the Euler characteristic of a certain doubly-graded bundle and when \(M\) is Calabi-Yau, this genus become a weak Jacobi form of weight 0 and index half of the dimension of \(M\). Moreover, as shown by \textit{L. A. Borisov} and \textit{A. Libgober} [Invent. Math. 140, No. 2, 453--485 (2000; Zbl 0958.14033)], \(Ell\) is, up to a factor of \(y^{-\dim(M)/2}\), the supertrace of the operator \(y^{J_0} q^{L_0}\) on the chiral de Rham complex (a sheaf of vertex superalgebras). For mirror pairs, these agree up to a sign. Using physical arguments, Berglund-Henningson computed \(Ell\) for arbitrary LG orbifolds and showed that they are equal (up to sign) for mirror pairs in a particular limit. In this paper, using the vertex algebra technique, the author nicely proves this rigorously.