Elliptic genera of Berglund–Hübsch Landau–Ginzburg orbifolds

Mirror symmetry was originally formulated as a correspondence between the N = (2, 2) superconformal field theories constructed for a Calabi-Yau n-fold X and for its mirror partner X∨. On the level of cohomology groups, there is a 90-degree rotation of the Hodge diamond, i.e. hp,q(X,C) = hn−p,q(X∨,C). Batyrev’s construction of Calabi-Yau hypersurfaces in Gorenstein Fano toric varieties associated to a pair of reflexive polytopes ([B]) is a prolific source of examples of mirror Calabi-Yau varieties. This construction was later generalized by Borisov to Calabi-Yau complete intersections in Gorenstein Fano toric varieties ([B1]), and further by Batyrev and Borisov to the mirror duality of reflexive Gorenstein cones ([BB1]). They proved that the stringtheoretic Hodge numbers of (singular) Calabi-Yau varieties arising from their constructions satisfy the expected mirror duality ([BB2]). Around the same time, physicists Berglund and Hübsch proposed a way to construct mirror pairs of (2, 2)-superconformal field theories in the formalism of orbifold Landau-Ginzburg theories ([BH]). They considered a nondegenerate invertible polynomial potential W whose transpose W∨ is again a non-degenerate invertible potential. They claimed that there exists a suitable group H such that the Landau-Ginzburg orbifolds W and W∨/H form a mirror pair. Recently, Krawitz found a general construction of the dual group G∨ for any subgroup G of diagonal symmetries of W , and proved an “LG-to-LG” mirror symmetry theorem for the pair (W/G,W∨/G∨) at the level of double-graded state spaces ([K]). Under a certain CY condition, the polynomials W , W∨ define CalabiYau hypersurfacesXW ,XW∨ in (usually different) weighted projective spaces.