Matrix Factorizations and Representations of Quivers I

Abstract

This paper introduces a mathematical definition of the category of D-branes in Landau-Ginzburg orbifolds in terms of $A_{\infty }$-categories. Our categories coincide with the categories of (graded) matrix factorizations for quasi-homogeneous polynomials. After setting up the necessary definitions, we prove that our category for the polynomial $x^{n+1}$ is equivalent to the derived category of representations of the Dynkin quiver of type $A_n$. We also construct a special stability condition for the triangulated category in the sense of T. Bridgeland, which should be the "origin" of the space of stability conditions.

20 pages, added references

References in corpus (5)

• Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities
• F-term equations near Gepner points
• Homological mirror symmetry for the quartic surface
• K. Saito's Duality for Regular Weight Systems and Duality for Orbifoldized Poincare Polynomials
• More about vanishing cycles and mutation

Cited by in corpus (15)

• Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities
• Compact generators in categories of matrix factorizations
• Spaces of stability conditions
• Triangulated surfaces in triangulated categories
• Homological Mirror Symmetry and Simple Elliptic Singularities
• Tilting and cluster tilting for quotient singularities
• On root categories of finite-dimenisonal algebras
• Global dimension function on stability conditions and Gepner equations
• Gepner point and strong Bogomolov-Gieseker inequality for quintic 3-folds
• Homological mirror symmetry for Milnor fibers of simple singularities
• Graded D-branes and skew-categories
• Homological mirror symmetry for Brieskorn-Pham singularities
• Maximally-graded matrix factorizations for an invertible polynomial of chain type
• A Remark on a Theorem of math.AG/0511155
• A Frobenius manifold for $\ell$-Kronecker quiver