Derived Knörrer periodicity and Orlov’s theorem for gauged Landau–Ginzburg models

Abstract

Abstract. We prove a Knörrer periodicity type equivalence between derived factorization categories of gauged LG models, which is an analogy of a theorem proved by Shipman and Isik independently. As an application, we obtain a gauged LG version of Orlov's theorem describing a relationship between categories of graded matrix factorizations and derived categories of hypersurfaces in projective spaces, by combining the above Knörrer periodicity type equivalence and the theory of variations of GIT quotients due to Ballard, Favero and Katzarkov.

Equivalences of derived factorization categories of gauged Landau–Ginzburg models

“…Since Φ P ⊣ Φ P R by Proposition 4.47, this isomorphism implies that the functor Φ P : Dcoh(X 1 , W 1 ) → Dcoh(X 2 , W 2 ) is fully faithful by [13,Lemma 4.6]. If the integral functor Φ j * (P ) : D b (cohX 1 ) → D b (cohX 2 ) is an equivalence, its left adjoint functor Φ j * (P ) L is fully faithful.…”

Fractional Calabi�Yau categories from Landau�Ginzburg models

“…, we may just use objects of the form det(E) ⊗ L ⊗n | Z(taut) . Finally, under the equivalence (see Theorem 3.6 of [Hir16]),…”

“…The following theorem is originally due to Isik [Isi13] and Shipman [Shi12] and due to Hirano [Hir16] in the G-equivariant case, which is the case we will use. Theorem 3.5 (Proposition 4.8 of [Hir16]). Assume that w is a regular section of E. There is an equivalence of categories,…”

Derived categories of BHK mirrors

“…The theorem is originally due to independently to Isik [Isi13] and Shipman [Shi12]. With the G-action, it is a special of Theorem 1.2 of [Hir16].…”