Singularities with Symmetries, orbifold Frobenius algebras and Mirror Symmetry

Details Singularities with Symmetries, orbifold Frobenius algebras and Mirror Symmetry Singularities with Symmetries, orbifold Frobenius algebras and Mirror Symmetry

2003-12-22

arxiv.org/abs/math/0312417v2

Mathematics

Algebraic Geometry

Scientific paper

Previously, we introduced a duality transformation for Euler $G$--Frobenius algebras. Using this transformation, we prove that the simple $A,D,E$ singularities and Pham singularities of coprime powers are mirror self--dual where the mirror duality is implemented by orbifolding with respect to the symmetry group generated by the grading operator and dualizing. We furthermore calculate orbifolds and duals to other $G$--Frobenius algebras which relate different $G$--Frobenius algebras for singularities. In particular, using orbifolding and the duality transformation we provide a mirror pairs for the simple boundary singularities $B_n$ and $F_4$. Lastly, we relate our constructions to $r$ spin--curves, classical singularity theory and foldings of Dynkin diagrams.

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