[Paper Review] Triangulated categories of singularities and D-branes in Landau-Ginzburg models

Dmitri Olegovich Orlov|arXiv (Cornell University)|Feb 25, 2003

Nonlinear Waves and Solitons15 references413 citations

TL;DR

This paper introduces triangulated categories of singularities as a mathematical framework to study D-branes in Landau-Ginzburg models, establishing that the category of B-branes in such models is equivalent to the triangulated category of singularities of the superpotential's fiber. The key contribution is a derived equivalence linking algebraic singularities to physical D-brane categories via matrix factorizations and Knörrer periodicity.

ABSTRACT

In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish a connection of these categories with D-branes in Landau-Ginzburg models.

Motivation & Objective

• To define and study triangulated categories of singularities for algebraic varieties, especially in the context of singularities arising from Landau-Ginzburg superpotentials.
• To establish a mathematical correspondence between B-branes in Landau-Ginzburg models and the derived categories of coherent sheaves modulo perfect complexes.
• To extend the Homological Mirror Symmetry Conjecture beyond Calabi-Yau manifolds by connecting singularities to D-brane categories in non-Calabi-Yau settings.
• To provide a derived categorical framework for understanding the physical notion of B-branes in Landau-Ginzburg models using algebraic geometry and homological algebra.

Proposed method

• Define the triangulated category of singularities $\mathbf{D}_{\text{Sg}}(X)$ as the quotient of the bounded derived category of coherent sheaves $\mathbf{D}^b(\operatorname{coh}(X))$ by the full triangulated subcategory of perfect complexes $\mathfrak{P}\mathfrak{e}\mathfrak{r}\mathfrak{f}(X)$.
• Use matrix factorizations of the superpotential $W$ to construct objects in the category of B-branes in Landau-Ginzburg models, generalizing Eisenbud's construction for maximal Cohen-Macaulay modules.
• Establish an equivalence between the category of B-branes in a Landau-Ginzburg model and the triangulated category of singularities of the fiber $W^{-1}(0)$, using Knörrer periodicity as a key tool.
• Define morphisms between objects $V_\mu$ via compositions of maps $\alpha_{\lambda}^{\nu}$, satisfying relations that encode the algebraic structure of the category.
• Construct exact triangles using the translation functor $[1]$, which sends $V_\mu$ to $V_{n-\mu}$, and verify that all exact triangles arise from specific compositions of morphisms.
• Prove that $\dim\operatorname{Hom}(V_\mu, V_\nu) = \min(\operatorname{depth}V_\mu, \operatorname{depth}V_\nu)$, where $\operatorname{depth}V_\mu = \min(\mu, n-\mu)$, and that $\operatorname{End}(V_\mu) \cong \mathbb{C}[x]/x^d$ with $d = \operatorname{depth}V_\mu$.

Experimental results

Research questions

• RQ1 How can the category of B-branes in a Landau-Ginzburg model be described in purely algebraic and homological terms?
• RQ2 What is the relationship between the triangulated category of singularities of a singular fiber $W^{-1}(0)$ and the derived category of coherent sheaves modulo perfect complexes?
• RQ3 Can Knörrer periodicity be realized and generalized within the framework of derived categories and matrix factorizations?
• RQ4 What is the structure of the morphism spaces in the category of B-branes for a Landau-Ginzburg model with a simple superpotential?
• RQ5 How do exact triangles and the translation functor $[1]$ act on the category of B-branes in such models?

Key findings

• The triangulated category of singularities $\mathbf{D}_{\text{Sg}}(X)$ is invariant under localization in the Zariski topology, as shown in Proposition 1.14.
• When $X$ is Gorenstein and the singular locus is complete, all $\operatorname{Hom}$-spaces in $\mathbf{D}_{\text{Sg}}(X)$ are finite-dimensional, as stated in Corollary 1.24.
• For the Landau-Ginzburg model with superpotential $W = x_1 + \cdots + x_n + \frac{1}{x_1 \cdots x_n}$, the category of B-branes is equivalent to the triangulated category of singularities of the fiber $W^{-1}(0)$.
• The dimension of the morphism space $\operatorname{Hom}(V_\mu, V_\nu)$ is equal to $\min(\operatorname{depth}V_\mu, \operatorname{depth}V_\nu)$, where $\operatorname{depth}V_\mu = \min(\mu, n - \mu)$.
• The endomorphism ring $\operatorname{End}(V_\mu)$ is isomorphic to $\mathbb{C}[x]/x^d$ with $d = \operatorname{depth}V_\mu$, showing a finite-dimensional structure for each object.
• All exact triangles in the category are isomorphic to those constructed from compositions of morphisms $\alpha_{\lambda}^{\nu}$, with the triangle (10) generalizing the basic triangle (9) for arbitrary morphisms.