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February 23, 2011

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 Σ:K→K′\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 the recent

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

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Re: 4d QFT for Khovanov Homology

… is related to the worldvolume theory of D3-branes ending on NS5-branes as they appear in the type IIA string theory spacetime.

There are no D3-branes in type IIA.

Posted by: Jacques Distler on February 23, 2011 3:11 PM | Permalink | PGP Sig | Reply to this

Re: 4d QFT for Khovanov Homology

…is related to the worldvolume theory of D3-branes ending on NS5-branes as they appear in the type IIA string theory spacetime.

There are no D3-branes in type IIA.

… is related to the worldvolume theory of the image in type IIA of D3-branes ending on NS5-branes in type IIB after one S-duality and then one T-duality operation.

(D3−NS5)↦S(D3−D5)↦T(D4−D6) (D3-N S 5) \stackrel{S}{\mapsto} (D3-D5) \stackrel{T}{\mapsto} (D4-D6)

Posted by: Urs Schreiber on February 23, 2011 4:16 PM | Permalink | Reply to this

Re: 4d QFT for Khovanov Homology

… is related to the worldvolume theory of the image in type IIA of D3-branes ending on NS5-branes in type IIB after one S-duality and then one T-duality operation.

Which is to say that we are not considering D3-branes ending on NS5-branes in ℝ 10\mathbb{R}^{10}, but rather in ℝ 9×S 1\mathbb{R}^9\times S^1, where the circle is transverse to both the D3s and the NS5s.

This circle is important.

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

Posted by: Jacques Distler on February 23, 2011 5:21 PM | Permalink | PGP Sig | Reply to this

Re: 4d QFT for Khovanov Homology

Is there any scope to bring your work, e.g., Fivebrane Structures or Differential twisted String and Fivebrane structures, to bear on Witten’s thinking?

Posted by: David Corfield on February 23, 2011 5:14 PM | Permalink | Reply to this

Re: 4d QFT for Khovanov Homology

Is there any scope to bring your work, e.g., Fivebrane Structures or Differential twisted String and Fivebrane structures, to bear on Witten’s thinking?

That is effectively asking to which extent the fivebrane’s quantum anomaly is relevant for the discussion of 4d QFTs related to Khovanov homology. I don’t know.

By the way, I have been busy working out and writing up the mathematical structure behind the articles with Jim and Hisham, a pdf is linked to at differential cohomology in a cohesive topos.

Posted by: Urs Schreiber on February 25, 2011 9:21 AM | Permalink | Reply to this

Re: 4d QFT for Khovanov Homology

It’s worth noting that Witten’s paper gives another construction, which is to me (and one gets a sense perhaps for Witten as well) conceptually much more satisfying than the brane construction: he constructs a 4d TQFT as a topological twist of a particular reduction of the (admittedly mysterious) 6d (0,2) theory. Namely we reduce this CFT on a “cigar” geometry. Witten explains the (0,2) theory has a twist which depends holomorphically on two dimensions and topologically on four, and so this reduction (modulo the existence of the (0,2) theory!) gives a complete topological field theory (rather than the brane constructions which are special to type A and to certain geometries I think?). This also makes clearer possible relations to geometric Langlands (which comes from reducing the (0,2) theory on a two-torus - giving N=4 SYM - and then twisting).

Posted by: David Ben-Zvi on February 25, 2011 4:35 AM | Permalink | Reply to this

Re: 4d QFT for Khovanov Homology

Thanks, I was hoping you would say something like this. I’ve heard that you are looking into this currently.

Jacques, David, I hear that you’ve been running a seminar on this at UTA. Do you by any chance have online notes or similar to share? I image there is some insight I’d care about that’s not in Witten’s writeup.

Posted by: Urs Schreiber on February 25, 2011 8:11 AM | Permalink | Reply to this

Re: 4d QFT for Khovanov Homology

It’s worth noting that Witten’s paper gives another construction, which is to me (and one gets a sense perhaps for Witten as well) conceptually much more satisfying than the brane construction ….

It (or, at least one version thereof) follows from the brane constructions.

After you follow Urs’s chain of dualities, you end up with D4-branes ending on a D6-brane (both wrapping an S 1S^1 direction).

Lifting that to M-theory gives M5-branes (the erstwhile D4s) on Taub-NUT (×S 1\times S^1). The M5-branes wrap the circle-fiber of Taub-NUT, which shrinks to zero size at the origin (the location of the erstwhile D6, which is where the D4s “end”). Hence your “cigar”. The low-energy theory, on a stack of M5-branes, is the 6D (2,0) theory.

So that give your picture of the (2,0) theory on the cigar.

What’s satisfying about this is that the final configuration is a UV-complete field theory, rather than the full string theory.

Of course, no one could possibly have arrived at this end-point, without the string theory intuition.

Posted by: Jacques Distler on February 25, 2011 2:36 PM | Permalink | PGP Sig | Reply to this

Re: 4d QFT for Khovanov Homology

How I can read this paper?
I mean what amount of math and physics needed for reading this paper.
(I reviewed Witten and Kapustin Paper On Langlands program For my M.s, Does This Prepare me for this paper?!!! )

Posted by: QGravity on June 6, 2011 7:31 PM | Permalink | Reply to this

Re: 4d QFT for Khovanov Homology

I am not sure if I can be of help, but I’ll try to say something in reply to this:

I mean what amount of math and physics needed for reading this paper.

There is a whole lot of physics-motivated math ideas in this article, some of which have been around for a while, some of which are fairly new. You’ll need to be familiar with a fair bit of the string theory folklore to follow the storyline, I’d think.

In a better world one would be able to say: if you’ve read the standard string theory textbook xyz, you should be able to get something out of this article. But I am not sure if xyz with this property exists. Though those that do exist probably approximate the idea.

(I reviewed Witten and Kapustin Paper On Langlands program For my M.s,

In that case, though, I’d say you should definitely just give it a try and see what happens to you.

Posted by: Urs Schreiber on June 7, 2011 4:56 PM | Permalink | Reply to this

Re: 4d QFT for Khovanov Homology

Thanks for guidance,
I am reading “a new look at path interal of QM, for understanding the relation between Path integral of chern-simons and path interal of N=4 SUSY thories.

Posted by: QGravity on June 18, 2011 3:48 PM | Permalink | Reply to this

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