The geometric Langlands program
:* geometric Satake?
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 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 (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 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), , of a 5D theory. Khovanov homology is 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).
|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|
|D0-brane||BFSS matrix model|
|D4-brane||D=5 super Yang-Mills theory with Khovanov homology observables|
|D1-brane||2d CFT with BH entropy|
|D3-brane||N=4 D=4 super Yang-Mills theory|
|(D25-brane)||(bosonic string theory)|
|NS-brane||type I, II, heterotic||circle n-connection|
|NS5-brane||B6-field||little string theory|
|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|
|topological M2-brane||topological M-theory||C3-field on G2-manifold|
|topological M5-brane||C6-field on G2-manifold|
|solitons on M5-brane||6d (2,0)-superconformal QFT|
|self-dual string||self-dual B-field|
|3-brane in 6d|
Original sources include
Joseph Bernstein, Igor Frenkel, Mikhail Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of 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 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
Raphaël Rouquier, Khovanov-Rozansky homology and 2-braid groups, arxiv/1203.5065
An expository reviews are
and earlier hints in
Lecture notes on this and its relation to the Jones polynomial are in