Matrix factorizations of quadratics and K-theory

Feb 2025

talk

Summary:

I describe the category of matrix factorization from a down-to-earth perspective. I argue why the matrix factorization category only sees the zero set of functions, and why the category is boring when 0 is not a critical level. Then, I’ll talk about matrix factorizations of quadratic polynomials. I describe why this category is isomorphic to the category of Clifford modules. Finally, I relate objects in matrix factorizations to topology, extracting topological vector bundle and a relative K theory class. Finally, I discuss the two-fold Knorrer periodicity for matrix factorizations, and explain how this is a categorized manifestation of Bott periodicity.

Presented at:

• UC Berkeley geometric representation theory seminar, spring 2025

🔗 Link to file

Sources

Clifford Algebras and Matrix Factorizations, 2008, by José Bertin

• Expository note proving that the category of matrix factorizations of a quadradic is isomorphic to the category of Clifford modules.

D-BRANES IN LANDAU-GINZBURG MODELS AND ALGEBRAIC GEOMETRY, 2002, by Kapustin and Li

• Checks Konstevich’s proposal that the matrix factorization is the category of D-branes of a Landau-Ginzburg model. See section 7 for the math discussion. They discuss the case of quadratic potentials in good detail.

KNORRER PERIODICITY AND BOTT PERIODICITY ¨, 2016, Micheal Brown

• provides the relationship between Knorrer periodicity and Bott periodicty. Here, he describes the Atyiah-bott-shapio construction for getting vector bundles out of matrix factorizations, which is central to my talk.
• See also his thesis, which works things out in more detail. At the end, he discusses using K-theory as a tool to study matrix factorizations of plane singularities, which relates to ADE theory and the Mackay correspondence.
  • https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1060&context=mathstudent

CLIFFORD MODULES, 1963, by Atiyah Bott and Shapiro

• Original source for Clifford modules and their relationship with $K$-theory of spheres and Bott periodicty. The Atyiah-bott-shapio isomorphism is spelled out here.
• For a textbook account of the atiyah-bott -hapiro paper, see Section 1.9 of Lawson-Michelson spin geometry