2024/09/10 by Cho, Cheol-Hyun, Choa, Dongwook, Jeong, Wonbo · 1 citation
#13C14 #32S50 #53D37 #FOS: Mathematics #Rings and Algebras (math.RA) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2410.14678
We find a Floer theoretic approach to obtain the transpose polynomial WT of an invertible curve singularity W. This gives an intrinsic construction of the mirror transpose polynomial and enables us to define a canonical A_∞-functor that takes Lagrangians in the Milnor fiber of W and converts them into matrix factorizations of WT. We find Lagrangians in the Milnor fiber of W that are mirror to the indecomposable matrix factorizations of WT when WT is ADE singularity and discover that Auslander-Reiten exact sequences can be realized as surgery exact triangles of Lagrangians in the mirror. There are two primary steps in the Floer theoretic method for obtaining a transposition polynomial: To get a Lagrangian L and corresponding disc potential function WL, we first determine the quotient X by the maximal symmetry group for the Milnor fiber. Second, we define a class Γ of symplectic cohomology of X based on the monodromy of the singularity W. Another disc counting function, g, is defined by the closed-open image of Γ on L. We demonstrate that restricting to the hypersurface g = 0 transforms the disc potential function WL into the transpose polynomial W T. This second step is the mirror of taking the cone of quantum cap action by the monodromy class Γ.