nLab Khovanov homology

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 (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 $\mathbb{R}^9 \times S^1$ 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).

Related concepts

• categorification in representation theory

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

brane	in supergravity	charged under gauge field	has worldvolume theory
black brane	supergravity	higher gauge field	SCFT
D-brane	type II	RR-field	super Yang-Mills theory
$(D = 2n)$	type IIA	$\,$	$\,$
D(-2)-brane	$\,$	$\,$
D0-brane	$\,$	$\,$	BFSS matrix model
D2-brane	$\,$	$\,$	$\,$
D4-brane	$\,$	$\,$	D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane	$\,$	$\,$	D=7 super Yang-Mills theory
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-brane	type I, II, heterotic	circle n-connection	$\,$
string	$\,$	B2-field	2d SCFT
NS5-brane	$\,$	B6-field	little string theory
D-brane for topological string			$\,$
A-brane			$\,$
B-brane			$\,$
M-brane	11D SuGra/M-theory	circle n-connection	$\,$
M2-brane	$\,$	C3-field	ABJM theory, BLG model
M5-brane	$\,$	C6-field	6d (2,0)-superconformal QFT
M9-brane/O9-plane			heterotic string theory
M-wave
topological M2-brane	topological M-theory	C3-field on G2-manifold
topological M5-brane	$\,$	C6-field on G2-manifold
S-brane
SM2-brane, membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane	6d (2,0)-superconformal QFT
self-dual string		self-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

• Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359-426 [arXiv:math/9908171, doi:10.1215/S0012-7094-00-10131-7, 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

• Carlo Collari, The Functoriality of Khovanov Homology and the Monodromy of Knots, 2013 (pdf, pdf)

Review:

• Dror Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Alg. Geom. Topology 2 (2002) 337-370, arXiv:math.GT/0201043

• Paul Turner, Five Lectures on Khovanov Homology, math.GT/0606464

• Mikhail Khovanov, Robert Lipshitz, Categorical lifting of the Jones polynomial: a survey, Bulletin (New Series) of the American Mathematical Society (2022) [doi:10.1090/bull/1772]

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

• Edward Witten, Fivebranes and Knots, Quantum Topology, Volume 3, Issue 1, 2012, pp. 1-137 (arXiv:1101.3216, doi:10.4171/QT/26)

based on

• Sergei Gukov, Albert Schwarz, Cumrun Vafa, Khovanov-Rozansky homology and topologicals strings , Lett. Math. Phys. 74 (2005) 53-74, arXiv:hep-th/0412243, MR2007a:57014, doi

and earlier hints in

• Edward Witten, Chern-Simons gauge theory as a string theory, The Floer memorial volume, 637–678, Progr. Math. 133, Birkhäuser 1995, arXiv/hep-th/9207094, MR97j:57052

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

• Edward Witten, Khovanov homology and gauge theory, arxiv/1108.3103

• Edward Witten, Fivebranes and Knots (arXiv:1101.3216)

Related $n$Café 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.

• Dror Bar-Natan, Khovanov’s homology for tangles and cobordisms,

  Geom. Topol. 9 (2005), 1443–1499, MR2006g:57017

See also

• David Clark, Scott Morrison, Kevin Walker, Fixing the functoriality of Khovanov homology (arXiv:math/0701339 pdf slides)

Discussion via a cobordism category of “foams”:

• Mikhail Khovanov, $sl(3)$ link homology, Algebr. Geom. Topol. 4 (2004) 1045-1081 [arXiv:math/0304375, doi:10.2140/agt.2004.4.1045]

• Marco Mackaay, Pedro Vaz, The universal $sl_3$-link homology, Algebr. Geom. Topol. 7 (2007) 1135-1169 [doi:10.2140/agt.2007.7.1135]

• Marco Mackaay, Pedro Vaz, The foam and the matrix factorization $sl_3$ link homologies are equivalent, Algebr. Geom. Topol. 8 (2008) 309-342 [arXiv:0710.0771, doi:10.2140/agt.2008.8.309]

• Marco Mackaay, Pedro Vaz, The diagrammatic Soergel category and $sl(N)$-foams, for $N \gt 3$ [arXiv:0911.2485]

• Mikhail Khovanov, Louis-Hadrien Robert, Foam evaluation and Kronheimer–Mrowka theories, Advances in Mathematics 376 (2021) 107433 [arXiv:1808.09662, doi:10.1016/j.aim.2020.107433]

• Mikhail Khovanov, Louis-Hadrien Robert, Conical $SL(3)$ foams, Journal of Combinatorial Algebra 6 1/2 (2022) 79-108 [arXiv:2011.11077, doi:10.4171/jca/61]

Reviewed in:

• Mikhail Khovanov, Universal construction, foams and link homology, lecture series at QFT and Cobordism, CQTS (Mar 2023) [web, video 1:YT, 2:YT]

More on relation to topological quantum field theory:

• Nitu Kitchloo, Symmetry Breaking and Link Homologies I [arXiv:1910.07443]

• Nitu Kitchloo, Symmetry Breaking and Link Homologies II [arXiv:1910.07444]

• Nitu Kitchloo, Symmetry Breaking and Link Homologies III [arXiv:1910.07516]

Relation to gauge theory:

• Nitu Kitchloo, Symmetry breaking and homotopy types for link homologies, talk at QFT and Cobordism, CQTS (Mar 2023) [web, video:YT]

See also:

• Khaled Qazaqzeh, Nafaa Chbili, On Khovanov Homology of Quasi-Alternating Links, Mediterr. J. Math. 19 104 (2022) [arXiv:2009.08624, doi:10.1007/s00009-022-02006-5]

• Nafaa Chbili, Quasi-alternating links, Examples and obstructions, talk at QFT and Cobordism, CQTS (Mar 2023) [web]