Fivebranes and Knots
By E. Witten. 2011
307 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics
View Paper: https://www.semanticscholar.org/paper/ba8cbabc326dda0a0c44569efa9103ac554fda18
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Citations in the corpus
- Homological algebra of knots and BPS states (2011, background): “Similarly, in the gauge theory framework [64] the moduli space of solutions on R(2) with a single defect operator in the adjoint representation of the gauge group G D SU.”
- Quantum Geometry and Quiver Gauge Theories (2013, background): “The three dimensional Chern-Simons theory has been related to N = 4 SYM on a four-manifold with a boundary in [55], and the parameter q in the normalization (1.”
- A new look at the path integral of quantum mechanics (2010, methodology): “These brane constructions will be used in [14], but are not needed for our present purposes.”
- Vortex Counting and Lagrangian 3-Manifolds (2010, background): “Surface operators in N = 2 supersymmetric gauge theories play a key role in the gauge theory approach to homological knot invariants [11,12].”
- Elliptic stable envelopes (2016, background): “The relation of this theory to knot theory and invariants of quantum groups has long been predicted by physicists [22, 53, 60].”
- Knot Invariants from Four-Dimensional Gauge Theory (2011, methodology): “Once φy and A1 are set to zero, the equations can be put in the above-described form, as shown in detail in [14], section 3.”
- Knot invariants and higher representation theory II: the categorification of quantum knot invariants (2010, background): “s cobordism formalism for Khovanov homology [?]. On the other hand, the work of physicists suggests that categorifications for all representations exist; one schema for defining them is given by Witten [Wit]. The relationship betweenthese invariants arising from gauge theories andthose presented in th”
- Wall-crossing in coupled 2d-4d systems (2011, methodology): “Recently, Witten has introduced an approach to knot homology based on the Hilbert space associated to surface defects in six-dimensional (2, 0) theory [74].”
- Knot Homology from Refined Chern-Simons Theory (2011, methodology): “Relation to the work of [16](20) In [16] a physical approach to knot homology was proposed, based on studying gauge theory on D4-branes wrapping a four-manifolds with a boundary on the three-manifold M , where the knots live, times a thermal S(1) (there were other duality frames studied in [16] as w”