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1. • CommentRowNumber1.
   • CommentAuthorKevin Lin
   • CommentTimeJun 25th 2010
   • PermaLink
   Author: Kevin Lin
   Format: TextAdded a page on [[matrix factorization]]s.

   Added a page on matrix factorizations.
2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeSep 22nd 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownI have reorganized the pages and put several references.

   I have reorganized the pages and put several references.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeSep 23rd 2014
   • (edited Sep 23rd 2014)
   • PermaLink
   Author: zskoda
   Format: MarkdownItex* [[Daniel S. Freed]], Constantin Teleman, _Dirac families for loop groups as matrix factorizations_, [arxiv/1409.6051](http://arxiv.org/abs/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]].

   • 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.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 23rd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexYeah, that's neat. I had gotten a preview of this last week. I have cross-linked with _[[string 2-group]]_...

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

   I have cross-linked with string 2-group…

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeSep 23rd 2014
   • (edited Sep 23rd 2014)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexAs 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]].

   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.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeSep 23rd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexExcellent. Thanks.

   Excellent. Thanks.

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeJun 23rd 2023
   • PermaLink
   Author: zskoda
   Format: MarkdownItexDefinition (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 $Coker\phi$. <a href="https://ncatlab.org/nlab/revision/diff/matrix+factorization/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/matrix+factorization/13">v13</a>, <a href="https://ncatlab.org/nlab/show/matrix+factorization">current</a>

   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 $Coker\phi$.

   diff, v13, current

8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeJun 23rd 2023
   • PermaLink
   Author: zskoda
   Format: MarkdownItexCompleted some references and rewrote the idea section. <a href="https://ncatlab.org/nlab/revision/diff/matrix+factorization/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/matrix+factorization/13">v13</a>, <a href="https://ncatlab.org/nlab/show/matrix+factorization">current</a>

   Completed some references and rewrote the idea section.

   diff, v13, current

9. • CommentRowNumber9.
   • CommentAuthorperezl.alonso
   • CommentTimeMay 28th 2024
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexpointer * 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](https://arxiv.org/abs/1211.2446)). <a href="https://ncatlab.org/nlab/revision/diff/matrix+factorization/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/matrix+factorization/19">v19</a>, <a href="https://ncatlab.org/nlab/show/matrix+factorization">current</a>

   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