categorification

# 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 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 $\Sigma : K \to K'$ 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 (Witten11) it is argued, following indications in (GukovSchwarzVafa05) 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 $\mathbb{R}^9 \times S^1$ with the circle transverse to both kinds of branes, under one S-duality and one T-duality operation

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

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), $Tr_{\mathcal{H}}(-1)^F e^{-\beta H}$, 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
$(D = 2n)$type IIA$\,$$\,$
D0-brane$\,$$\,$BFSS matrix model
D2-brane$\,$$\,$$\,$
D4-brane$\,$$\,$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\,$$\,$
D8-brane$\,$$\,$
$(D = 2n+1)$type IIB$\,$$\,$
D(-1)-brane$\,$$\,$$\,$
D1-brane$\,$$\,$2d CFT with BH entropy
D3-brane$\,$$\,$N=4 D=4 super Yang-Mills theory
D5-brane$\,$$\,$$\,$
D7-brane$\,$$\,$$\,$
D9-brane$\,$$\,$$\,$
(p,q)-string$\,$$\,$$\,$
(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$\,$
A-brane$\,$
B-brane$\,$
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
M-wave
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\,$C6-field on G2-manifold
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d

## References

Original sources include

• Louis Crane, Igor Frenkel, Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases, J. Math. Phys. 35 (1994) 5136-5154, hep-th/9405183

• Igor Frenkel, Mikhail Khovanov, Canonical bases in tensor products and graphical calculus for $U_q(\mathfrak{sl}_2)$, Duke Math. J. 87 (1997) 409-480, MR99a:17019, doi

• Joseph Bernstein, Igor Frenkel, Mikhail Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of $U(\mathfrak{sl}-2)$ by projective and Zuckerman functors, Selecta. Math. 5 (1999) 199-241, MR2000i:17009, doi

• Igor Frenkel, Mikhail Khovanov, Catharina Stroppel, A categorification of finite-dimensional irreducible representations of quantum $\mathfrak{sl}_2$ and their tensor products, Selecta Math. (N.S.) 12 (2006), no. 3-4, 379–431, MR2008a:17014, doi

• M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359–426 MR1740682 (2002j:57025)

• M Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002) 665–741 MR1928174 (2004d:57016)

• M Khovanov, Patterns in knot cohomology. I, Experiment. Math. 12 (2003) 365–374

• M Khovanov, An invariant of tangle cobordisms, Trans. Amer. Math. Soc. 358 (2006), no. 1, 315–327, arXiv:math.QA/0207264, MR2006g:57046, doi

• Raphaël Rouquier, Khovanov-Rozansky homology and 2-braid groups, arxiv/1203.5065

An expository reviews are

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)

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