### 4d QFT for Khovanov Homology

#### Posted by Urs Schreiber

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

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 an observable in 3-dimensional Chern-Simons theory. For instance for $\Sigma : K \to K'$ a cobordism between two knots there is a natural morphism

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

between the Khovanov homologies associated to the two knots.

In the recent

- Edward Witten,
*Fivebranes and knots*(arXiv:1101.3216)

it is argued, following indications in

- S. Gukov, A. S. Schwarz, and C. Vafa,
*Khovanov-Rozansky Homology And Topological Strings*, Lett. Math. Phys. 74 (2005) 53-74, (arXiv:hep-th/0412243),

that this 4d TQFT is related to the worldvolume theory of D3-branes ending on NS5-branes as they appear in the type IIA string theory spacetime. Earlier indication for this had come from the observation that Chern-Simons theory is the effective background theory for the A-model 2d TCFT (see TCFT – Worldsheet and effective background theories for details).

Posted at February 23, 2011 1:14 PM UTC
## Re: 4d QFT for Khovanov Homology

There are no D3-branes in type IIA.