There are two different meanings of phrase “matrix factorization” which are closely related, one generic for various matrix decomposition?s, like Gauss, LU etc. decompositions of matrices into products, and another rather specific, 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.
David Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc., 260:3564, 1980.
T. Dyckerhoff, Compact generators in the categories of matrix factorizations, 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 Boston, Inc., Boston, MA, 2009 (arXiv:math.ag/0503632)
See also
Junwu Tu, Matrix factorizations via Koszul duality, arxiv/1009.4151
J. Burke, M. E. Walker, Matrix factorizations over projective schemes, arxiv/1110.2918
Matthew Ballard, David Favero, Ludmil Katzarkov, A category of kernels for graded matrix factorizations and its implications for Hodge theory, arxiv/1105.3177
Alexander I. Efimov, Cyclic homology of categories of matrix factorizations, arxiv/1212.2859
Alexander Polishchuk, Homogeneity of cohomology classes associated with Koszul matrix factorizations, arxiv/1409.7115
A formulation in terms of linear logic and the geometry of interactions is in
A connection to the representation theory of loop groups and families of Dirac operators (and the string 2-group) is in