FJRW Rings and Landau-Ginzburg Mirror Symmetry.

Abstract

In this thesis, we study applications of the Berglund–Huebsch transpose construction to Landau-Ginzburg (LG) mirror symmetry. Given an invertible quasihomogeneous potential W, a dual potential W^T is obtained by transposition of the exponent matrix of W.

By the work of Fan–Jarvis–Ruan, one can associate a LG A-model to each pair consisting of a potential W and an admissible group G of symmetries of W. On the other hand, Intriligator-Vafa have produced the LG B-model state space associated to such a pair.

The first step in this work is to define, given an invertible potential W and group of symmetries G, a dual group G^T of symmetries of W^T. We then prove that, at the level of (bi-graded) state spaces, the LG A-model of the pair (W, G) is isomorphic to the LG Bmodel of (W^T, G^T).

In the case where G = G^max is the maximal diagonal symmetry group of W, the dual group G^T is trivial, and the LG B-model is just the local algebra of W^T. In particular, both the A-model and the B-model are Frobenius algebras in this case, and we prove that the mirror map preserves this structure.

Building on work of Kaufmann, we produce a product structure on the LG B-model orbifolded by a general diagonal symmetry group, and present examples which suggest the mirror map respects this product in non-trivial cases.

As an additional application, we interpret Arnol’d strange duality of exceptional singularities in the context of LG mirror symmetry.