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Fivebranes and Knots

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arxiv 1101.3216 v2 pith:5U4RRWFQ submitted 2011-01-17 hep-th math.GTmath.RT

Fivebranes and Knots

Edward Witten This is my paper
classification hep-th math.GTmath.RT
keywords knotsdescriptiondualitygaugehandhomologyinvariantkhovanov
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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We develop an approach to Khovanov homology of knots via gauge theory (previous physics-based approches involved other descriptions of the relevant spaces of BPS states). The starting point is a system of D3-branes ending on an NS5-brane with a nonzero theta-angle. On the one hand, this system can be related to a Chern-Simons gauge theory on the boundary of the D3-brane worldvolume; on the other hand, it can be studied by standard techniques of $S$-duality and $T$-duality. Combining the two approaches leads to a new and manifestly invariant description of the Jones polynomial of knots, and its generalizations, and to a manifestly invariant description of Khovanov homology, in terms of certain elliptic partial differential equations in four and five dimensions.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A supergroup series for knot complements

    math.GT 2025-08 unverdicted novelty 7.0

    Defines the three-variable superalgebra series F_K(y,z,q) for knot complements, derives its surgery relation to hat Z(q), and computes examples for torus knots.

  2. 3d-3d correspondence for knot complements with finite and large $N$

    hep-th 2026-07 conditional novelty 6.0

    The SU(N) homological block in inverted-Habiro form equals a half-index of an explicit 3d N=2 theory for the figure-eight and the two trefoil knots, with a conjectural all-knot extension.

  3. Refined Vafa-Witten invariants for toric surfaces from supersymmetric localization in 5D gauge theory

    hep-th 2026-07 conditional novelty 6.0

    5D N=1* supersymmetric localization on toric surfaces equals refined Vafa–Witten invariants for odd first Chern class; the even-c1 case fails without a hand-added constant.

  4. 3d-3d correspondence and abelian flat connection

    hep-th 2026-03 conditional novelty 6.0

    The homological block of a knot complement is realized as a half-index of a 3d N=2 theory via a contour enclosing z=q^k poles, and the same integral at z=q^-k poles gives the colored Jones polynomial.