Completed some references and rewrote the idea section.

]]>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 $Coker\phi$.

]]>Excellent. Thanks.

]]>As I updated the note above only after your reply: the reference is added also at matrix factorization, Constantin Teleman, Loop Groups and Twisted K-Theory.

]]>Yeah, that’s neat. I had gotten a preview of this last week.

I have cross-linked with *string 2-group*…

- Daniel S. Freed, Constantin Teleman,
*Dirac families for loop groups as matrix factorizations*, arxiv/1409.6051

We identify the category of integrable lowest-weight representations of the loop group LG of a compact Lie group G with the linear category of twisted, conjugation-equivariant curved Fredholm complexes on the group G: namely, the twisted, equivariant matrix factorizations of a super-potential built from the loop rotation action on LG. This lifts the isomorphism of K-groups of [FHT1,2, 3] to an equivalence of categories. The construction uses families of Dirac operators.

I added this reference at matrix factorization, Constantin Teleman, Loop Groups and Twisted K-Theory.

]]>I have reorganized the pages and put several references.

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