nLab matrix factorization

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

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

• decomposition of an element of a ring, understood as a multiple of an identity matrix, as a product of matrices over that 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.

Overview

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.

Definition

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

Note that if $(\phi,\psi)$ is a matrix factorization of $x$, then $x$ annihilates $Coker\phi$. Often instead of the assumption that $A$-modules $F,G$ are free, one assumes that they are finitely generated projective.

Related concepts

• Landau-Ginzburg model

Literature

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

• Wikipedia, Matrix factorization of a polynomial

• 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

• Alexander Polishchuk, Arkady Vaintrob, Matrix factorizations and singularity categories for stacks, Annales de l’Institut Fourier 61:7 (2011) 2609-2642 doi numdam

• A. Polishchuk, Arkady Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016) 1–122

• Alexander Polishchuk, Homogeneity of cohomology classes associated with Koszul matrix factorizations, Compositio Math. 152 (2016) 2071–2112

  doi arXiv:1409.7115

• Alexander Efimov, Leonid Positselski, Coherent analogues of matrix factorizations and relative singularity categories, Algebra & Number Theory 9 (2015) 1159–1292 doi

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

• Daniel S. Freed, Constantin Teleman, Dirac families for loop groups as matrix factorizations, Comptes Rendus Mathematique 353:5 (2015) 415-419 doi arXiv:1409.6051

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

• Nils Carqueville, Daniel Murfet, Adjunctions and defects in Landau-Ginzburg models, Adv. Math. 289 (2016) 480-566 (doi:10.1016/j.aim.2015.03.033, arXiv:1208.1481)

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

• Daniel Murfet, The cut operation on matrix factorisations, J. Pure Appl. Alg. 222 7 (2018) 1911-1955 (arXiv:1402.4541, doi:10.1016/j.jpaa.2017.08.014)

Constructing $A_\infty$-category of matrix factorizations

• Daniel Murfet, Constructing $A_\inf$-categories of matrix factorisations, arXiv:1903.07211

Generalization of matrix factorizations to higher matrix factorizations:

• David Eisenbud, Irena Peeva?, Matrix factorizations for complete intersections and minimal free resolutions, arXiv:1306.2615

Generalization to factorization categories

• Matthew Ballard, Dragos Deliu, David Favero, M. Umut Isik, Ludmil Katzarkov, Resolutions in factorization categories, arXiv:1212.3264

On their relation to noncommutative resolutions:

• Nicolas M. Addington, Ed Segal, Eric Sharpe. D-brane probes, branched double covers, and noncommutative resolutions. Advances in Theoretical and Mathematical Physics 18, no. 6 (2014): 1369-1436. (arXiv:1211.2446).