Categorical Saito theory, II: Landau-Ginzburg orbifolds

Let W is an element of C[x(1), . . ., x(N)] be an invertible polynomial with an isolated singularity at origin, and let G subset of SLN boolean AND(C*)(N) be a finite diagonal and special linear symmetry group of W. In this paper, we use the category MFG(W) of G-equivariant matrix factorizations and its associated VSHS to construct a G-equivariant version of Saito's theory of primitive forms. We prove there exists a canonical categorical primitive form of MFG(W) characterized by G(W)(max)-equivariance. Conjecturally, this G-equivariant Saito theory is equivalent to the genus zero part of the FJRW theory under LG/LG mirror symmetry. In the marginal deformation direction, we verify this for the FJRW theory of (1/5(x(1)(5) + . . . + x(5)(5)), Z/5Z) with its mirror dual B-model Landau-Ginzburg orbifold (1/5( x(1)(5) + . . . + x(5)(5)), ( Z/5Z)(4)). In the case of the Quintic family W= 1/5( x(1)(5)+ . . . + x(5)(5)) - psi x(1)x(2)x(3)x(4)x(5), we also prove a comparison result of B-model VSHS's conjectured by Ganatra-Perutz-Sheridan[14]. (C) 2021 Elsevier Inc. All rights reserved.