Derived factorization categories of non-Thom--Sebastiani-type sums of potentials

2018/09/26 by Hirano, Yuki, Ouchi, Genki · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1809.09940

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 \rm HMF^L_f(f) of maximally graded matrix factorizations of an invertible polynomial f of chain type, and explicitly construct a full strong exceptional collection E₁,..., E_μ in \rm HMF^L_f(f) whose length μ is the Milnor number of the Berglund--Hübsch transpose \widetildef of f. This proves a conjecture, which postulates that for an invertible polynomial f the category \rm HMF^L_f(f) admits a tilting object, in the case when f is a chain polynomial. Moreover, by careful analysis of morphisms between the exceptional objects E_i, we explicitly determine the quiver with relations (Q,I) which represents the endomorphism ring of the associated tilting object ⊕_i=1^μE_i in \rm HMF^L_f(f), and in particular we obtain an equivalence \rm HMF^L_f(f)≅ \rm D^{\rm b}(\rm mod kQ/I).

Cited by

• Tilting theory for hypersurface singularities of dimension one

Related

• Maximally-graded matrix factorizations for an invertible polynomial of chain type
• Matrix Factorizations and Representations of Quivers I
• Categories of singularities of invertible polynomials
• Global matrix factorizations
• Thom-Sebastiani & Duality for Matrix Factorizations