Knot homology via derived categories of coherent sheaves, I: The sl(2)-case

Cautis, Sabin; Kamnitzer, Joel

doi:10.1215/00127094-2008-012

Cited by 110 publications

(174 citation statements)

References 23 publications

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“…(Such affine braid group actions have played a central role in constructions of knot homology, both in mathematics and physics, cf. [54][55][56][57][58][59].) In the 2d reductions of 3d gauge theories that we study in section 7, two commuting braid-group actions will appear.…”

Section: Jhep10(2016)108mentioning

confidence: 99%

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Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

Bullimore

Dimofte

Gaiotto

et al. 2016

J. High Energ. Phys.

106

229

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Abstract:We introduce several families of N = (2, 2) UV boundary conditions in 3d N = 4 gauge theories and study their IR images in sigma-models to the Higgs and Coulomb branches. In the presence of Omega deformations, a UV boundary condition defines a pair of modules for quantized algebras of chiral Higgs-and Coulomb-branch operators, respectively, whose structure we derive. In the case of abelian theories, we use the formalism of hyperplane arrangements to make our constructions very explicit, and construct a half-BPS interface that implements the action of 3d mirror symmetry on gauge theories and boundary conditions. Finally, by studying two-dimensional compactifications of 3d N = 4 gauge theories and their boundary conditions, we propose a physical origin for symplectic duality -an equivalence of categories of modules associated to families of Higgs and Coulomb branches that has recently appeared in the mathematics literature, and generalizes classic results on Koszul duality in geometric representation theory. We make several predictions about the structure of symplectic duality, and identify Koszul duality as a special case of wall crossing.

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Section: Jhep10(2016)108mentioning

confidence: 99%

“…In the mathematics literature, braid actions on categories D b O H and D b O C have also been used to construct knot homology [57,58], cf. the related [54][55][56]. One expects these various braid actions to all be equivalent.…”

Section: Jhep10(2016)108mentioning

confidence: 99%

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

Bullimore

Dimofte

Gaiotto

et al. 2016

J. High Energ. Phys.

106

229

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“…Cautis and Kamnitzer [7,8] constructed a link homology using the derived category of coherent sheaves on certain flag-like varieties. Their homology is conjectured to be isomorphic to the sl(N ) Khovanov-Rozansky homology in [19].…”

Section: 3mentioning

confidence: 99%

A colored sl(N) homology for links in S³

2014

Dissertationes Math.

121

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Abstract. Fix an integer N ≥ 2. To each diagram of a link colored by 1, . . . , N , we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain complex defined by Khovanov and Rozansky in [19]. The homology of this chain complex decategorifies to the Reshetikhin-Turaev sl(N ) polynomial of links colored by exterior powers of the defining representation.

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“…To compare with the ideal in [25], we observe that the ideal I n of Z[q ⌋ is a subset of the ideal generated by p and [2] p − [2] by the strong integrality of the quantum link invariant [21]. Furthermore, if n is odd, the ideal I n is a subset of the ideal generated by p and [3] p − [3]. To compare the ideal generated by p and To compare with the ideal in [6], we use only fundamental representations of the quantum Lie algebras sl(n) which are finite but Chen and Le used all representations of the quantum sl(n) which are obviously infinite.…”

Section: Discussionmentioning

confidence: 99%

THE QUANTUM sl(n, ℂ) REPRESENTATION THEORY AND ITS APPLICATIONS

Jeong¹,

Kim²

2012

Journal of the Korean Mathematical Society

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Abstract. In this paper, we study the quantum sl(n) representation category using the web space. Specially, we extend sl(n) web space for n ≥ 4 as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial Pn(q) specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of sl(n). Moreover, we correct the false conjecture [30] given by Chbili, which addresses the relation between some link polynomials of a periodic link and its factor link such as Alexander polynomial (n = 0) and Jones polynomial (n = 2) and prove the corrected conjecture not only for HOMFLY polynomial but also for the colored HOMFLY polynomial specialized to a one variable polynomial.

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Knot homology via derived categories of coherent sheaves, I: The sl(2)-case

Cited by 110 publications

References 23 publications

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

A colored sl(N) homology for links in S³

THE QUANTUM sl(n, ℂ) REPRESENTATION THEORY AND ITS APPLICATIONS

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Knot homology via derived categories of coherent sheaves, I: The sl(2)-case

Cited by 110 publications

References 23 publications

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

A colored sl(N) homology for links in S3

THE QUANTUM sl(n, ℂ) REPRESENTATION THEORY AND ITS APPLICATIONS

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Product

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A colored sl(N) homology for links in S³