# Contents

## Idea

The D-brane of dimension $4+1$ in type IIA string theory.

## Properties

### Worldvolume theory

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 $\Sigma :K\to K\prime$ a cobordism between two knots there is a natural morphism

${\Phi }_{\Sigma }:𝒦\left(K\right)\to 𝒦\left(K\prime \right)$\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 (GukovSchwarzVafa) that this 4d TQFT 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 one T-duality operation:

$\left(D3-\mathrm{NS}4\right)\stackrel{S}{↦}\left(D3-D5\right)\stackrel{T}{↦}\left(D4-D6\right)\phantom{\rule{thinmathspace}{0ex}}.$(D3 - NS4) \stackrel{S}{\mapsto} (D3 - D5) \stackrel{T}{\mapsto} (D4-D6) \,.

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).

Notice that after the above T-duality operation the $\left(D4-D6\right)$-system wraps the ${S}^{1}$ (circle) along which the T-duality takes place.

Lifting that configuration to 11-dimensional supergravity gives M5-branes (the erstwhile D4-branes) on Taub-NUT? ($×{S}^{1}$). 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”). The low-energy theory, on a stack of M5-branes, is the 6d (2,0)-susy QFT.

Table of branes appearing in supergravity/string theory

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
$\left(D=2n\right)$type IIA$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D0-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$BFSS matrix model
D2-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D4-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$D=5 super Yang-Mills theory with Khovanov homology observables
$\left(D=2n+1\right)$type IIB$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D1-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$2d CFT with BH entropy
D3-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$N=4 D=4 super Yang-Mills theory
D5-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D7-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
NS-branetype I, II, heteroticcircle n-connection$\phantom{\rule{thinmathspace}{0ex}}$
string$\phantom{\rule{thinmathspace}{0ex}}$B2-field2d SCFT
NS5-brane$\phantom{\rule{thinmathspace}{0ex}}$B6-fieldlittle string theory
M-brane11D SuGra/M-theorycircle n-connection$\phantom{\rule{thinmathspace}{0ex}}$
M2-brane$\phantom{\rule{thinmathspace}{0ex}}$C3-fieldABJM theory, BLG model
M5-brane$\phantom{\rule{thinmathspace}{0ex}}$C6-field6d (2,0)-superconformal QFT
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\phantom{\rule{thinmathspace}{0ex}}$C6-field on G2-manifold

## References

The relation to Khovanov homology is discussed in

Created on October 14, 2011 02:36:23 by Urs Schreiber (131.211.235.86)