Matrix Factorizations and Representations of Quivers I

By A. Takahashi. 2005

93 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8870b3db907fba1412c7e2a1153b6d61d3893d2c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. H. Lenzing
2. Dmitri Orlov
3. A. Takahashi
4. H. Kajiura
5. Kyoji Saito
6. D. Kussin
7. H. Meltzer
8. Kazushi Ueda

In Geometry and Topology

1. H. Lenzing
2. D. Kussin
3. H. Meltzer
4. A. Takahashi
5. Dmitri Orlov
6. Kazushi Ueda
7. M. Futaki
8. H. Kajiura

In Mathematical Physics

1. H. Lenzing
2. D. Kussin
3. H. Meltzer
4. Dmitri Orlov
5. Kazushi Ueda
6. M. Futaki
7. H. Kajiura
8. Kyoji Saito

Citations in the corpus

• Triangle singularities, ADE-chains, and weighted projective lines (2012, methodology): “Let us finally point out that the authors of [26,27] take a different approach to analyze the singularity category of a Kleinian or Fuchsian singularity; their approach is based on graded maximal Cohen-Macaulay modules and matrix factorizations, not on stable categories of vector bundles, as propose”
• Weighted Projective Lines Associated to Regular Systems of Weights of Dual Type (2007, methodology): “full strongly exceptional collection (see [ T2 ]).”
• Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities (2005, methodology): “This example is considered in detail in the paper [ 26 ].”
• Homological mirror symmetry for Brieskorn–Pham singularities (2009, background): “evich [7] for Calabi-Yau manifolds, and later generalized to more general classes of manifolds [8, 5, 14]. Homological mirror symmetry for singularities are studied by Takahashi and his collaborators [19, 3, 4, 18]. Okada [9] also discusses homological mirror symmetry for Brieskorn-Pham singularitie”
• Triangulated categories of matrix factorizations for regular systems of weights with $\epsilon=-1$ (2007, methodology): “[Sa3, T1] and the (homological) mirror symmetry [Ko], the third author [ T2 ] proposed to use”
• Nilpotent operators and weighted projective lines (2010, background): “Stable categories of (graded) Cohen-Macaulay modules play an important role in the analysis of singularities, see [3, 9, 10, 17].”
• Matrix factorizations and representations of quivers II: Type ADE case (2005, methodology): “Dynkin quiver of type Al. He conjectured ([ T2 ], Conjecture 1.3) further that the same type”