Abstract by Matthew Williams
Classification of \"Elliptic\" Landau-Ginzburg Models
Mirror Symmetry is the study of Calabi-Yau manifolds which come in dual pairs. The physics of string theory produces an A-model and a B-model for each Calabi-Yau manifold. Mirror symmetry says that the A-model for a Calabi-Yau manifold is “the same” as the B-model on its mirror dual—meaning they produce the same physics.
We consider A-models stemming from so called invertible polynomials W with 3 variables together with a group G of symmetry satisfying the Calabi-Yau condition, meaning the sum of their weights is one. There are a total of 12 such polynomials, with each having one or a few possible groups of symmetry. From each of these pairs (W, G), the A-model construction includes a vector space, called the state space. It has been shown that for each of these pairs, the associated state spaces has one of four possible dimensions. It is conjectured that these A-models are uniquely determined by the dimension of the state space, implying all of the A-models with the same dimension are isomorphic to each other. We will first define invertible polynomials and their symmetry groups, and then discuss our progress on this conjecture.