# Contents

## Idea

In the context of string theory the NS5-brane is a certain extended physical objects – a brane – that appears in/is predicted by the theory.

There are different incarnations of this object:

For instance the effective background QFT of the type II stringtype II supergravity – admits solutions to its generalized Einstein equations which describe higher dimensional analogs of charged black holes in ordinary gravity. Among them is a 5+1-dimensional “black brane” which is magnetically charged under the Kalb-Ramond field. Since the KR field and the field of gravity constituting this solution of type II supergravity have as quanta the worldsheet excitations of the spinning string sigma-model that sit in what is called the Neveu-Schwarz sector? one calls this the NS5-brane.

This is to distinguish it from the D5-brane which is instead charged under the RR-field whose quanta come from the Ramond-Ramond sector? of the superstring.

There are other incarnations of the NS 5-brane:

by the general logic of higher electromagnetism the (1+1)-dimensional string has under electric-magnetic duality a magnetic dual . By dimension counting this is a 5-brane. If we think of the string this way as the structure that supports the sigma-model that defines perturbative string theory, we also call it the F1-brane (the fundamental 1-brane). In this sense the the corresponding magnetic dual is the F5-brane – the fundamental fivebrane.

One can understand the NS5-“black brane” solution to type II supergravity as being the solitonic incarnation of the fundamental 5-brane in much the same way as an ordinary black hole in ordinary gravity is a solitonic incarnation of the fundamental particle: as the particle, the black hole it is characterized just by mass, charge and angular momentum.

Similarly, the “black” NS5-brane is characterizes by mass, B-field charge and angular momentum.

## Properties

### Relation to Khovanov homology

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 (for classification see at brane scan).

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}}$$\phantom{\rule{thinmathspace}{0ex}}$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
D8-brane$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
$\left(D=2n+1\right)$type IIB$\phantom{\rule{thinmathspace}{0ex}}$$\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

Actions for the NS5-brane are discussed in

• Igor Bandos, Alexei Nurmagambetov, Dmitri Sorokin, The type IIA NS5–Brane (arXiv:hep-th/0003169)

• Marco Cariglia, Kurt Lechner, NS5-branes in IIA supergravity and gravitational anomalies (arXiv:hep-th/0203238)

The fact that the worldvolume theory of the M5-brane should support fields that are self-dual connections on a 2-bundle ($\sim$ a gerbe) is discussed in

as well as sections 3 and 4 of

A review of some aspects is in

The relation to Khovanov homology is discussed in

• Greg Moore, On the role of six‐dimensional $\left(2,0\right)$-theories in recent developments in Physical Mathematics , talk at Strings2011 (pdf slides)