Computational techniques in FJRW theory with applications to Landau–Ginzburg mirror symmetry

Abstract

New Computational Techniques in FJRW Theory with Applications to Landau Ginzburg Mirror Symmetry Amanda Francis Department of Mathematics, BYU Doctor of Philosophy Mirror symmetry is a phenomenon from physics that has inspired a lot of interesting mathematics. In the Landau-Ginzburg setting, we have two constructions, the A and B models, which are created based on a choice of an affine singularity with a group of symmetries. Both models are vector spaces equipped with multiplication and a pairing (making them Frobenius algebras), and they are also Frobenius manifolds. We give a result relating stabilization of singularities in classical singularity to its counterpart in the Landau-Ginzburg setting. The A model comes from so-called FJRW theory and can be defined up to a full cohomological field theory. The structure of this model is determined by a generating function which requires the calculation of certain numbers, which we call correlators. In some cases the their values can be computed using known techniques. Often, there is no known method for finding their values. We give new computational methods for computing concave correlators, including a formula for concave genus-zero, four-point correlators and show how to extend these results to find other correlator values. In many cases these new methods give enough information to compute the A model structure up to the level of Frobenius manifold. We give the FJRW Frobenius manifold structure for various choices of singularities and groups.