Examples/classes:
Types
Related concepts:
categorification
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 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
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
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).
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:
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
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)
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.
Geom. Topol. 9 (2005), 1443–1499, MR2006g:57017
See also
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:
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:
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]
Last revised on May 6, 2023 at 08:27:32. See the history of this page for a list of all contributions to it.