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    • CommentRowNumber1.
    • CommentAuthorKevin Lin
    • CommentTimeJun 25th 2010
    Added a page on matrix factorizations.
    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeSep 22nd 2010

    I have reorganized the pages and put several references.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeSep 23rd 2014
    • (edited Sep 23rd 2014)

    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.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 23rd 2014

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

    I have cross-linked with string 2-group

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeSep 23rd 2014
    • (edited Sep 23rd 2014)

    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.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 23rd 2014

    Excellent. Thanks.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeJun 23rd 2023

    Definition

    (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.

    diff, v13, current

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeJun 23rd 2023

    Completed some references and rewrote the idea section.

    diff, v13, current

    • CommentRowNumber9.
    • CommentAuthorperezl.alonso
    • CommentTimeMay 28th 2024

    pointer

    • 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).

    diff, v19, current