nLab matrix factorization

Matrix factorizations

Matrix factorizations

There are two different meanings of phrase “matrix factorization” which are closely related

  • (typically canonical) matrix decomposition?s of matrices into products of matrices of specific kind, like Gauss decomposition, polar decomposition, LU-decomposition etc.

  • decomposition of a ring element, understood as a diagonal matrix, as a product of matrices over a ring, in the sense of Eisenbud and followers.

This entry is dedicated to the latter as it concerns appearance of certain categories in mathematical physics.


Matrix factorizations were introduced by David Eisenbud, and they were originally studied in the context of commutative algebra.

Matrix factorizations arise in string theory as categories of D-branes for Landau-Ginzburg B-models. This was proposed by Kontsevich and elaborated in the paper of Kapustin-Li.

Matrix factorizations have also been used by Khovanov-Rozansky in knot theory math/0401268.

Matrix factorization categories are examples of Calabi-Yau categories, hence correspond to TCFTs by the Costello/Kontsevich/Hopkins-Lurie theorem. The Calabi-Yau structure is elucidated in recent work of Dyckerhoff-Murfet and Polishchuk-Vaintrob.

Dyckerhoff has proved that the Hochschild homology of the matrix factorizations category of an isolated singularity is the Jacobian ring of the singularity. See also the work of E. Segal and Caldararu-Tu. In light of the Costello/Kontsevich/Hopkins-Lurie theorem, this result has been anticipated for some time, as the closed state space of a Landau-Ginzburg B-model is the Jacobian ring.

There is also the Calabi-Yau/Landau-Ginzburg correspondence. In some cases, categories of matrix factorizations turn out to be equivalent to categories of coherent sheaves.

For general theory and properties of matrix factorizations, see work of Orlov. For example, matrix factorization categories are related to derived categories of singularities.


(Eisenbud 1980) A matrix factorization of an element xx in a commutative ring AA is an ordered pair of maps of free AA-modules (ϕ:FG,ψ:GF)(\phi:F\to G,\psi: G\to F) such that ϕψ=x1 G\phi\circ\psi = x\cdot 1_G and ψϕ=x1 F\psi\circ\phi = x\cdot 1_F.

Note that if (ϕ,ψ)(\phi,\psi) is a matrix factorization of xx, then xx annihilates CokerϕCoker\phi. Often instead of the assumption that AA-modules F,GF,G are free, one assumes that they are finitely generated projective.


  • David Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260:35-64 (1980) doi

  • Tobias Dyckerhoff, Compact generators in the categories of matrix factorizations, Duke Math. J. 159(2): 223-274 (2011) doi arXiv:0904.4713

The definition of a triangulated category of B-branes for the Landau-Ginzburg model via matrix factorization was proposed by Maxim Kontsevich and is written out in

  • Anton Kapustin, Yi Li, D-Branes in Landau-Ginzburg models and algebraic geometry (arXiv:hep-th/0210296)

  • Dmitri Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Proc. Steklov Inst. Math. 2004, no. 3 (246), 227–248 (arXiv:math/0302304)

  • Dmitri Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 503–531, Progr. Math. 270, Birkhäuser 2009 arXiv:math.AG/0503632

See also

  • Junwu Tu, Matrix factorizations via Koszul duality, Compositio Mathematica 150:9 (2014) 1549–1578 doi arXiv:1009.4151

  • J. Burke, M. E. Walker, Matrix factorizations over projective schemes, Homology, Homotopy and Applications 14(2) (2012) 37–61 arXiv:1110.2918

  • Matthew Ballard, David Favero, Ludmil Katzarkov, A category of kernels for graded matrix factorizations and its implications for Hodge theory, Publ.math.IHES 120, 1–111 (2014) doi arXiv:1105.3177

  • Alexander I. Efimov, Cyclic homology of categories of matrix factorizations, Intern. Math. Res. Notices 12 (2018) 3834–3869 doi arXiv:1212.2859

A matrix factorization of a potential W which is a section of a line bundle on an algebraic stack is studied in

On a connection to the representation theory of loop groups and families of Dirac operators (and the string 2-group)

On a bicategory of Landau-Ginzburg models with matrix factorizations as 1-morphisms:

On a category of matrix factorizations in the context of Landau-Ginzburg models, described in terms of linear logic and the geometry of interactions:

Constructing A A_\infty-category of matrix factorizations

Generalization of matrix factorizations to higher matrix factorizations:

Generalization to factorization categories

Last revised on July 31, 2023 at 15:22:49. See the history of this page for a list of all contributions to it.