[Paper Review] Threefold Flops via Matrix Factorization

Carina Curto, David R. Morrison|ArXiv.org|Nov 1, 2006

Advanced Topics in Algebra4 citations

TL;DR

This paper establishes a matrix factorization approach to classify threefold flops via the McKay correspondence, proving that Pinkham's deformation theory of partial resolutions corresponds to deformation theory of matrix factorizations for A_n-1 and D_n rational double points. The key contribution is a systematic construction of matrix factorizations for universal flops of length 1 and 2, with conjectures for E_6, E_7, and E_8 singularities.

ABSTRACT

The explicit McKay correspondence, as formulated by Gonzalez-Sprinberg and Verdier, associates to each exceptional divisor in the minimal resolution of a rational double point a matrix factorization of the equation of the rational double point. We study deformations of these matrix factorizations, and show that they exist over an appropriate "partially resolved" deformation space for rational double points of types A and D. As a consequence, all simple flops of lengths 1 and 2 can be described in terms of blowups defined from matrix factorizations. We also formulate conjectures which would extend these results to rational double points of type E and simple flops of length greater than 2.

Motivation & Objective

• To address the lack of detailed geometric information about neighborhoods of the center curve in threefold simple flops, which is essential for string theory models involving D-branes.
• To bridge the gap between birational geometry and string theory by applying matrix factorization techniques, inspired by Landau–Ginzburg models.
• To prove that Pinkham’s deformation theory of partial resolutions of rational double points is equivalent to deformation theory of associated matrix factorizations for A_n-1 and D_n singularities.
• To construct explicit matrix factorizations for universal flops of length 1 and 2, and to conjecture their existence for higher-length flops.
• To provide a geometric realization of the McKay correspondence via matrix factorizations for rational double points and their deformations.

Proposed method

• Use the McKay correspondence to associate maximal Cohen–Macaulay modules (via matrix factorizations) to partial resolutions of rational double points.
• Prove that deformation theory of partial resolutions (Pinkham's theory) corresponds to deformation theory of matrix factorizations for A_n-1 and D_n singularities.
• Construct explicit matrix factorizations (Φ, Ψ) such that ΦΨ = fI_k for the defining polynomials f of rational double points.
• For each singularity type, compute the matrices Φ and Ψ explicitly using recursive patterns and Dynkin diagram labeling (e.g., ℓ, ℓ⁺, ℓ⁻).
• Present universal flops as hypersurfaces in affine space defined by f = x₁² + g(x₂,…,xₘ), with g encoding the A_n-1 or D_n singularity.
• Use the relation (XI_{2ℓ} − Ξₗ^•)(XI_{2ℓ} + Ξₗ^•) = (X² + g(Y,Z))I_{2ℓ} to generate matrix factorizations for higher-length flops.

Experimental results

Research questions

• RQ1 Can matrix factorizations provide a geometric description of the neighborhood of the center curve in a threefold simple flop, sufficient for physical models in string theory?
• RQ2 Is Pinkham’s deformation theory of partial resolutions of rational double points equivalent to deformation theory of the corresponding matrix factorizations?
• RQ3 What are the explicit matrix factorizations for universal flops of length 1 and 2, and can they be generalized to higher-length flops?
• RQ4 How do the Dynkin diagrams of A_n-1 and D_n singularities encode the structure of the associated matrix factorizations?
• RQ5 Can the construction of matrix factorizations for E_6, E_7, and E_8 singularities be completed, and what would be their physical and geometric significance?

Key findings

• The paper proves that for A_n-1 and D_n rational double points, the deformation theory of partial resolutions is equivalent to the deformation theory of their associated matrix factorizations.
• Explicit matrix factorizations are constructed for the universal flop of length 1 (defined by xy − z² + t² = 0) and length 2, with matrices Φ and Ψ satisfying ΦΨ = fI_k.
• For the E_6 singularity (g(Y,Z) = Y³ + Z⁴), matrix factorizations are explicitly computed for all labels ℓ = 1⁺, 2⁺, 3, 2⁻, 1⁻, including Ξ₁^± and Ξ₂^± matrices.
• For E_7 (g(Y,Z) = Y³ + YZ³), the paper provides matrix factorizations for labels 2′, 3′, 4, 2′′, 3, 2, 1, with φ and ψ matrices of increasing size.
• For E_8 (g(Y,Z) = Y³ + Z⁵), the paper constructs matrix factorizations for all labels 2′, 4′, 6, 3′′, 5, 4, 3, 2, including 6×6 matrices φ₆ and ψ₆ satisfying the factorization condition.
• The construction reveals a recursive structure in the matrices, with smaller factorizations φₗ^•, ψₗ^• existing in most cases, though some are defined directly as Ξₗ^•.