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nLab Khovanov homology

Contents

Contents

Idea

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 quantum observables in D=4 TQFT in higher analogy of how the Jones polynomial arises as a quantum observable in 3-dimensional Chern-Simons D=3 TQFT. For instance for Σ:KK\Sigma \colon K \to K' a knot cobordism between two knots there is a natural morphism

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

between the Khovanov homologies associated to the two knots.

Witten 2011 argued, following indications in Gukov, Schwarz & Vafa 2005, that this 4d TQFT is related to the worldvolume theory of the image in type IIB string theory of D3-branes ending on NS5-branes in a type II supergravity 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, one interprets 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 Witten 2011, 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).

References

General

Exposition:

Original articles:

Review and lecture notes:

See also

Discussion via a cobordism category of “foams”:

Reviewed in:

A proposal to incorporate the geometry of knots to Khovanov homology is in

  • Li Shen, Jian Liu, Guo-Wei Wei. Evolutionary Khovanov homology (2024). (arXiv:2406.02821).

See also:

Knot invariants via topological strings and 5-branes

On realization of knot invariants/knot homology via topological string theory and BPS states:

Understanding this via NS5-branes/M5-branes:

Review:

An alternative approach:

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.

More on relation to topological quantum field theory:

Relation to gauge theory:

Last revised on July 26, 2024 at 10:16:52. See the history of this page for a list of all contributions to it.