AN ${\rm SL}(2, {\mathbb C})$ ACTION ON CHIRAL RINGS AND THE MIRROR MAP

Abstract

Each transversal degree-d hypersurface ℳ in a weighted projective space defines a Landau-Ginzburg orbifold, the superpotential of which equals the defining polynomial of ℳ. For a generic such ℳ with trivial canonical class, the degree-0 (mod d) subring of the Jacobian ring (that is, the (c, c)-ring of the Landau-Ginzburg orbifold) is shown to admit an [Formula: see text] action and the corresponding Lefschetz-type decomposition. This leads to a general definition of a “large complex structure” limit, the mirror of the “large volume” limit, and the mirror images on ⊕qH3−q,q of the Hodge *-operator, duality and inner product on ⊕qHq,q.