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I have heard during some seminar talks that there are applications of the theory of matrix factorizations in string theory. A quick search shows mostly papers written by physicists. Are there any survey type papers aimed at an algebraic audience on this topic, especially with current state/open questions motivated by physics?

Background: A matrix factorization of an element $x$ in a ring $R$ is a pair of square matrices $A,B$ of size $n$ such that $AB=BA=xId_n$. For more see, for example, Section 3 of this paper .

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Great question! I hope you get good answers. –  Graham Leuschke Dec 25 '09 at 13:44
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up vote 17 down vote accepted

Indeed matrix factorizations come up in string theory. I don't know if there are good survey articles on this stuff, but here is what I can say about it. There might be an outline in the big Mirror Symmetry book by Hori-Katz-Klemm-etc., but I am not sure.

When we are considering the B-model of a manifold, for example a compact Calabi-Yau, the D-branes (boundary states of open strings) are given by coherent sheaves on the manifold (or to be more precise, objects of the derived category of coherent sheaves). Matrix factorizations come up in a different situation, namely, they are the D-branes in the B-model of a Landau-Ginzburg model. Mathematically, a Landau-Ginzburg model is just a manifold (or variety) $X$, typically non-compact, plus the data of a holomorphic function $W: X \to \mathbb{C}$ called the superpotential. In this general situation, a matrix factorization is defined to be a pair of coherent sheaves $P_0, P_1$ with maps $d : P_0 \to P_1$, $d : P_1 \to P_0$ such that $d^2 = W$. I guess you could call this a "twisted (or maybe it's 'curved'? I forget the terminology) 2-periodic complex of coherent sheaves". When $X = \text{Spec}R$ is affine, and when the coherent sheaves are free $R$-modules, this is the same as the definition that you gave.

The relationship between matrix factorization categories and derived categories of coherent sheaves was worked out by Orlov: http://arxiv.org/abs/math/0503630 http://arxiv.org/abs/math/0503632 http://arxiv.org/abs/math/0302304

I believe that the suggestion to look at matrix factorizations was first proposed by Kontsevich. I think the first paper that explained Kontsevich's proposal was this paper by Kapustin-Li: http://arXiv.org/abs/hep-th/0210296v2

There are some interesting recent papers regarding the relationship between the open-string B-model of a Landau Ginzburg model (which is, again, mathematically given by the matrix factorizations category) and the closed-string B-model, which I haven't described, but an important ingredient is the Hochschild (co)homology of the matrix factorizations category. Take a look at Katzarkov-Kontsevich-Pantev http://arxiv.org/abs/0806.0107 section 3.2. There is a paper of Tobias Dyckerhoff http://arxiv.org/abs/0904.4713 and a paper of Ed Segal http://arxiv.org/abs/0904.1339 which work out in particular the Hochschild (co)homology of some matrix factorization categories. The answer is it's the Jacobian ring of the superpotential. This is the correct answer in terms of physics: the Jacobian ring is the closed state space of the theory.

Katzarkov-Kontsevich-Pantev also has some interesting stuff about viewing matrix factorization categories as "non-commutative spaces" or "non-commutative schemes".

Edit 1: I forgot to mention: Kontsevich's original homological mirror symmetry conjecture stated that the Fukaya category of a Calabi-Yau is equivalent to the derived category of coherent sheaves of the mirror Calabi-Yau. Homological mirror symmetry has since been generalized to non-Calabi-Yaus. The rough expectation is that given any compact symplectic manifold, there is a mirror Landau-Ginzburg model such that, among other things, the Fukaya category of the symplectic manifold should be equivalent to the matrix factorizations category of the Landau-Ginzburg model. For example, if your symplectic manifold is $\mathbb{CP}^n$, the mirror Landau-Ginzburg model is given by the function $x_1+\cdots+x_n + \frac{1}{x_1\cdots x_n}$ on $(\mathbb{C}^\ast)^n$. This is sometimes referred to as the Hori-Vafa mirror http://arxiv.org/abs/hep-th/0002222

I think that various experts probably know how to prove this form of homological mirror symmetry, at least when the symplectic manifold is, for example, a toric manifold or toric Fano manifold, but it seems that very little of this has been published. There may be some hints in this direction in Fukaya-Ohta-Oh-Ono http://arxiv.org/abs/0802.1703 http://arxiv.org/abs/0810.5654, but I'm not sure. There is an exposition of the case of $\mathbb{CP}^1$ in this paper of Matthew Ballard http://arxiv.org/abs/0801.2014 -- this case is already non-trivial and very interesting, and the answer is very nice: the categories in this case are equivalent to the derived category of modules over a Clifford algebra. I quite like Ballard's paper; you might be interested in taking a look at it anyway.

Edit 2: Seidel also has a proof of this form of homological mirror symmetry for the case of the genus two curve. Here is the paper http://arxiv.org/abs/0812.1171 and here is a video http://www.maths.ed.ac.uk/~aar/atiyah80.htm of a talk he gave on this stuff at the Atiyah 80 conference.

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I forgot to mention that matrix factorizations are also related to knot theory! e.g. arxiv.org/abs/math/0401268 –  Kevin H. Lin Dec 25 '09 at 16:19
    
Very interesting answer Kevin, thanks! –  Hailong Dao Dec 25 '09 at 17:22
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Here's a video of a talk by Kentaro Hori at MSRI: Matrix Factorizations and Complexes of Vector Bundle ---- An Approach from 2d QFT with Boundary. It's not strictly what you're asking for, but does seem to be mostly expository, and does explain a little bit why matrix factorizations come into the equation; they're standins for a certain derived category, of course.

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"Derived categories of coherent sheaves and triangulated categories of singularities" by D. Orlov. "Landau-Ginzburg to Calabi-Yau dictionary for D-branes" by Aspinwall."Triangulated categories of matrix factorizations" by Kajiura Saito and Takahashi. Also, I personally have many open questions in this area. Perhaps we should talk!

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I would also be very interested in hearing about the open questions you have :-) –  Kevin H. Lin Dec 25 '09 at 15:50
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