An Isomorphism Extension Theorem for Landau-Ginzburg B-Models

Landau-Ginzburg mirror symmetry studies isomorphisms between A- and B-models, which are graded Frobenius algebras that are constructed using a weighted homogeneous polynomial $W$ and a related group of symmetries $G$ of $W$. It is known that given two polynomials $W_{1}$, $W_{2}$ with the same weights and same group $G$, the corresponding A-models built with ($W_{1}$,$G$) and ($W_{2}$,$G$) are isomorphic. Though the same result cannot hold in full generality for B-models, which correspond to orbifolded Milnor rings, we provide a partial analogue. In particular, we exhibit conditions where isomorphisms between unorbifolded B-models (or Milnor rings) can extend to isomorphisms between their corresponding orbifolded B-models (or orbifolded Milnor rings).