•연구자: 수학과 이상욱
•발표일: 2024.04
•DOI: https://doi.org/10.48550/arXiv.2007.11732
•Cheol-Hyun Cho and Sangwook Lee, Asian Journal of Mathematics (Q3), Volume 28, Issue 2, 2024
•Abstract
A version of mirror symmetry predicts a ring isomorphism between quantum cohomology of a symplectic manifold and Jacobian algebra of the Landau-Ginzburg mirror, and for toric manifolds Fukaya-Oh-Ohta-Ono constructed such a map called Kodaira-Spencer map using Lagrangian Floer theory. We discuss a general construction of Kodaira-Spencer ring homomorphism when LG mirror potential W is given by J-holomorphic discs with boundary on a Lagrangian L: we find an A∞-algebra B whose m1-complex is a Koszul complex for W under mild assumptions on L. Closed-open map gives a ring homomorphism from quantum cohomology to cohomology algebra of B which is Jacobian algebra of W.
We also construct an equivariant version for orbifold LG mirror (W,H). We construct a Kodaira-Spencer map from quantum cohomology to another A∞-algebra (B⋊H)H whose cohomology algebra is isomorphic to the orbifold Jacobian algebra of (W,H) under an assumption. For the 2-torus whose mirror is an orbifold LG model given by Fermat cubic with a Z/3-action, we compute an explicit Kodaira-Spencer isomorphism.