Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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