Derived factorization categories of non‐Thom–Sebastiani‐type sums of potentials

Abstract

We first prove semi-orthogonal decompositions of derived factorization categories arising from sums of potentials of gauged Landau–Ginzburg models, where the sums are not necessarily Thom–Sebastiani type. We then apply the result to the category HMF L f ( f ) $\operatorname{HMF}^{L_f}(f)$ of maximally graded matrix factorizations of an invertible polynomial f $f$ of chain type, and explicitly construct a full strong exceptional collection E 1 , ⋯ , E μ $E_1,\hdots ,E_{\mu }$ in HMF L f ( f ) $\operatorname{HMF}^{L_f}(f)$ whose length μ $\mu$ is the Milnor number of the Berglund–Hübsch transpose f ∼ $\widetilde{f}$ of f $f$ . This proves a conjecture, which postulates that for an invertible polynomial f $f$ the category HMF L f ( f ) $\operatorname{HMF}^{L_f}(f)$ admits a tilting object, in the case when f $f$ is a chain polynomial. Moreover, by careful analysis of morphisms between the exceptional objects E i $E_i$ , we explicitly determine the quiver with relations ( Q , I ) $(Q,I)$ which represents the endomorphism ring of the associated tilting object ⊕ i = 1 μ E i $\oplus _{i=1}^{\mu }E_i$ in HMF L f ( f ) $\operatorname{HMF}^{L_f}(f)$ , and in particular we obtain an equivalence HMF L f ( f ) ≅ D b ( mod k Q / I ) $\operatorname{HMF}^{L_f}(f)\cong \operatorname{D}^{\operatorname{b}}(\operatorname{mod}kQ/I)$ .