nLab Khovanov homology




Khovanov homology is a knot invariant that is a categorification of the Jones polynomial.

Interpretation in QFT

Khovanov homology has long been expected to appear as the observables in a 4-dimensional TQFT in higher analogy of how the Jones polynomial arises as a observables in 3-dimensional Chern-Simons theory. For instance for Σ:KK\Sigma : K \to K' a cobordism between two knots there is a natural morphism

Φ Σ:𝒦(K)𝒦(K) \Phi_\Sigma : \mathcal{K}(K) \to \mathcal{K}(K')

between the Khovanov homologies associated to the two knots.

In (Witten11) it is argued, following indications in (Gukov-Schwarz-Vafa 05) that this 4d TQFT is related to the worldvolume theory of the image in type IIA string theory of D3-branes ending on NS5-branes in a type IIB background of the form 9×S 1\mathbb{R}^9 \times S^1 with the circle transverse to both kinds of branes, under one S-duality and one T-duality operation

(D3NS5)S(D3D5)T(D4D6). (D3-NS5) \stackrel{S}{\mapsto} (D3-D5) \stackrel{T}{\mapsto} (D4-D6) \,.

To go from the Jones polynomial to Khovanov homology, we interpret the circle as Euclidean time. The path integral with the circle is the partition function (Witten index), Tr (1) Fe βHTr_{\mathcal{H}}(-1)^F e^{-\beta H}, of a 5D theory. Khovanov homology is \mathcal{H} itself, rather than the index.

See (Witten11, p. 14).

Earlier indication for this had come from the observation Witten92 that Chern-Simons theory is the effective background theory for the A-model 2d TCFT (see TCFT – Worldsheet and effective background theories for details).

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
(D=2n)(D = 2n)type IIA\,\,
D0-brane\,\,BFSS matrix model
D4-brane\,\,D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane\,\,D=7 super Yang-Mills theory
(D=2n+1)(D = 2n+1)type IIB\,\,
D1-brane\,\,2d CFT with BH entropy
D3-brane\,\,N=4 D=4 super Yang-Mills theory
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection\,
string\,B2-field2d SCFT
NS5-brane\,B6-fieldlittle string theory
D-brane for topological string\,
M-brane11D SuGra/M-theorycircle n-connection\,
M2-brane\,C3-fieldABJM theory, BLG model
M5-brane\,C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane\,C6-field on G2-manifold
membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d


Original sources include

Review and lecture notes:

A proposal for a 4-dimensional quantum field theory whose observables are given by Khovanov homology is discussed in

based on

and earlier hints in

Lecture notes on this and its relation to the Jones polynomial are in

  • Edward Witten, A New Look At The Jones Polynomial of a Knot, Clay Conference, Oxford, October 1, 2013 (pdf)

  • Edward Witten, Khovanov Homology And Gauge Theory, Clay Conference, Oxford, October 1, 2013 (pdf)

See also

Related nnCafé discussions: categorification in Glasgow, Kamnitzer on categorifying tangles, link homology in Paris, 4d QFT and Khovanov homology

Parts of the above remarks on the QFT interpretation makes use of comments provided by Jacques Distler in this blog discussion.

See also

Discussion via a cobordism category of “foams”:

Reviewed in:

More on relation to topological quantum field theory:

Relation to gauge theory:

See also:

Last revised on November 7, 2023 at 10:17:20. See the history of this page for a list of all contributions to it.