Abstract

The Orbifold Landau-Ginzburg Mirror Symmetry Conjecture states that for a quasihomogeneous singularity W and a group G of symmetries of W, there is a dual singularity WT and dual group GT such that the orbifold A-model of W/G is isomorphic to the orbifold B-model of WT/GT. The Landau-Ginzburg A-model is the Frobenius algebra HW,G constructed by Fan, Jarvis, and Ruan, and the B-model is the Orbifold Milnor ring of WT . The unorbifolded conjecture has been verified for Arnol'd's list of simple, unimodal and bimodal quasi-homogeneous singularities with G the maximal diagonal symmetry group by Priddis, Krawitz, Bergin, Acosta, et al. [9], and by Fan-Shen [4] and Acosta [1] for all two dimensional invertible singularities and by Krawitz for all invertible singularities of 3 dimensions and greater in [8]. Based on this Krawitz posed the Orbifold Landau-Ginzburg Mirror Symmetry Conjecture, where the A-model is still the Frobenius algebra HW,G constructed by Fan, Jarvis, and Ruan but constructed with respect to a proper subgroup G of the maximal group of symmetries GW and the B-model is the orbifold Milnor ring of WT orbifolded with respect to a non-trivial group K in SLn of order [GW :J]. I verify this Orbifold Landau-Ginzburg Mirror Symmetry Conjecture for all unimodal and bimodal quasi-homogeneous singularities in Arnol'd's list with G = J < GW, being the minimal admissible diagonal symmetry group. I also discuss some axioms and properties of these singularities.

Degree

College and Department

Physical and Mathematical Sciences; Mathematics

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