D-topology

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## Idea

Given a [[diffeological space]], the _D-topology_ ("diffeological topology") is a natural [[topological space|topology]] on its underlying [[set]] of [[points]], namely the [[final topology]] which makes all its [[plots]] be [[continuous functions]].

## Definition
 {#Definition}

\begin{definition}
  Given a [[diffeological space]] $(X, plots)$, with set of $plots \,\subset\, \underset{n \in \mathbb{N}}{\coprod} Set(\mathbb{R}^n,\, X)$,
the *D-topology* $\tau_D$ on the [[underlying]] [[set]] $X$ is the [[final topology]] on $X$ with respect to the $plots$.

This means that $\tau_D \,\subset\, Sub(X)$ is the topology in which a [[subset]] $U \subset X$ is [[open subset|open]] if and only if under each $\big(\mathbb{R}^n \xrightarrow{\phi} X \big) \,\in\, plots$
its [[preimage]] $\phi^{-1}(U) \subset \mathbb{R}^n$ is open in the [[Euclidean space|Euclidean topology]] on the plot's [[domain]].
\end{definition}

([Iglesias-Zemmour 1985, Def. 1.2.3](#IglesiasZemmour85), review in [IglesiasZemmour 2013, Sec. 2.8](#IglesiasZemmour13), [CSW 2013, Sec. 3](#CSW13))

## Properties

[[!include adjunction between topological spaces and diffeological spaces]]

## Related concepts

* [[D-topological space]]

## References

The concept originates with:

* {#IglesiasZemmour85} [[Patrick Iglesias-Zemmour]], Def. 1.2.3 in: _Fibrations difféologiques et Homotopie_,  st. doct. thesis (1985) ([web](http://math.huji.ac.il/~piz/Site/The%20Articles/D9DD15EE-6993-4CA3-8B9B-4FC1DEF4A418.html), [pdf](http://math.huji.ac.il/~piz/documents/TheseEtatPI.pdf), [[IglesiasZemmourFibrationsDiffeologiques1985.pdf:file]])

* {#IglesiasZemmour13} [[Patrick Iglesias-Zemmour]], Section 2.8 of: _Diffeology_, Mathematical Surveys and Monographs, AMS (2013) ([web](http://math.huji.ac.il/~piz/Site/The%20Book.html), [ ISBN:978-0-8218-9131-5](https://bookstore.ams.org/surv-185))

The relation to [[Delta-generated topological spaces]]:

* {#SYH10} [[Kazuhisa Shimakawa]], K. Yoshida, [[Tadayuki Haraguchi]], _Homology and cohomology via enriched bifunctors_, Kyushu Journal of Mathematics, 2018 Volume 72 Issue 2 Pages 239-252 ([arXiv:1010.3336](https://arxiv.org/abs/1010.3336), [doi:10.2206/kyushujm.72.239]( https://doi.org/10.2206/kyushujm.72.239))

Discussion of the D-topology for [[mapping spaces]]:

* {#CSW13} [[J. Daniel Christensen]], [[Gord Sinnamon]], [[Enxin Wu]], _The $D$-topology for diffeological spaces_, Pacific Journal of Mathematics 272 (1), 87-110, 2014 ([arXiv:1302.2935](http://arxiv.org/abs/1302.2935), [doi:10.2140/pjm.2014.272.87](https://msp.org/pjm/2014/272-1/p04.xhtml))

[[!redirects D-topologies]]

[[!redirects diffeological topology]]
[[!redirects diffeological topologies]]

[[!redirects continuous diffeology]]
[[!redirects continuous diffeologies]]


D. Scott Alexander

* [GoogleScholar page](https://scholar.google.com/citations?user=K4ZXBIkAAAAJ&hl=en)

## Selected writings

Exposition of the general idea of [[quantum programming languages]] for [[classically controlled quantum computation]] with an eye towards the [[Quipper]]-language:

* [[Benoît Valiron]], [[Neil J. Ross]], [[Peter Selinger]], [[D. Scott Alexander]], [[Jonathan M. Smith]], *Programming the quantum future*, Communications of the ACM **58** 8 (2015) 52–61 [[doi:10.1145/2699415](https://doi.org/10.1145/2699415), [pdf](https://www.iro.umontreal.ca/~brassard/cours/6155/Langages%20de%20programmation%20quantique/Programming%20the%20Quantum%20Future.pdf),  [web](https://cacm.acm.org/magazines/2015/8/189851-programming-the-quantum-future/abstract), promo vid:[YT](https://youtu.be/mzce69oFdH0?list=PLn0nrSd4xjjbIHhktZoVlZuj2MbrBBC_f)]

category: people

[[!redirects Scott Alexander]]


D0-brane

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## Idea

The [[D-brane]] of [[dimension]] $0+1$ in [[type IIA string theory]].

## Properties

The [[worldline]] theory of a collection of D0-branes is supposed to be described by the [[BFSS matrix model]].

## Properties

### Nonperturbative dynamics and M-theory

The [[non-perturbative quantum field theory|non-perturbative]] limit of [[type IIA superstring theory]] is supposed to be [[M-theory]] compactified on a circle.

The degree-2 [[RR-field]] that the D0-brane is charged under, with local potential 1-form $A_1$ may be understood as the [[Kaluza-Klein compactification|KK-field]] induced by this compactification, hence as one part of the field of [[gravity]] in [[11-dimensional supergravity]].

Under the [[duality between M-theory and type IIA string theory]] the [[M-wave]] becomes the [[black brane|black]] [[D0-brane]] under [[double dimensional reduction]] ([Bergshoeff-Townsend 96](#BergshoeffTwonsend96)).

One aspect of the [[M-theory]] conjecture is that [[type IIA string theory]] with a [[condensate]] of [[D0-branes]] behaves like a 10-dimensional theory that develops a further circular dimension of [[radius]] scaling with the density of [[D0-branes]].  ([Banks-Fischler-Shenker-Susskind 97](#BanksFischlerShenkerSusskind97), [Polchinski 99](#Polchinski99)). See also ([FSS 13, section 4.2](#FSS13)).

### Bound states

[[bound states]]/[[brane intersections]] involving D0-branes:

* [[D0-D2 brane bound state]]

* [[D0-D4 brane bound state]]

## Relation to other branes

The [[electric-magnetic duality|electric-magnetic dual]] of the D0 is the [[D6-brane]]

[[!include electric-magnetic duality -- table]]

## Related concepts

* [[Myers effect]]

[[!include table of branes]]

## References

### Single (abelian) D0-branes

* {#BergshoeffTownsend96} E. Bergshoeff, [[Paul Townsend]], _Super D-branes_, Nucl.Phys. B490 (1997) 145-162 ([arXiv:hep-th/9611173](http://arxiv.org/abs/hep-th/9611173))

* [[Nissan Itzhaki]], [[Juan Maldacena]], [[Jacob Sonnenschein]], [[Shimon Yankielowicz]], Section 5 of: _Supergravity and The Large $N$ Limit of Theories With Sixteen Supercharges_, Phys. Rev. D 58, 046004 (1998) ([arXiv:hep-th/9802042](https://arxiv.org/abs/hep-th/9802042))

As part of the [[schreiber:brane bouquet]]:

* {#FSS13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], in section 4.2 of _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_ ([arXiv:1308.5264](http://arxiv.org/abs/1308.5264))

### Coincident (non-abelian) D0-branes

For several and (nearly) [[coincident brane|coincident]] D0-branes:

The [[BFSS matrix model]]:

* {#BanksFischlerShenkerSusskind97} [[Tom Banks]], [[Willy Fischler]], [[Stephen Shenker]], [[Leonard Susskind]], _M Theory As A Matrix Model: A Conjecture_,  Phys. Rev. D **55** (1997) [[doi:10.1103/PhysRevD.55.5112](https://doi.org/10.1103/PhysRevD.55.5112), [arXiv:hep-th/9610043](http://arxiv.org/abs/hep-th/9610043)]

* {#Polchinski99} [[Joseph Polchinski]], _M-Theory and the Light Cone_,   Prog. Theor. Phys. Suppl. **134** (1999) 158-170 [[arXiv:hep-th/9903165](http://arxiv.org/abs/hep-th/9903165)]

based on discussion of bound states of $N$ D0-branes in

* [[Edward Witten]], _Bound States of Strings and $p$-Branes_, Nucl.Phys. B 460, 1995, 335 ([arXiv:hep-th/9510135](http://arxiv.org/abs/hep-th/9510135)).

Extension to a non-abelian and supersymmetric [[DBI-action]]:

* Sudhakar Panda, [[Dmitri Sorokin]], *Supersymmetric and Kappa-invariant Coincident D0-Branes*, 	JHEP 0302 (2003) 055 [[arXivLhep-th/0301065](https://arxiv.org/abs/hep-th/0301065), [doi:10.1088/1126-6708/2003/02/055](https://doi.org/10.1088/1126-6708/2003/02/055)]

* [[Igor Bandos]], Unai D. M. Sarraga, *Complete nonlinear action for supersymmetric multiple D0-brane system*, Phys. Rev. D **106** (2022) 066004 [[doi:10.1103/PhysRevD.106.066004](https://doi.org/10.1103/PhysRevD.106.066004), [arXiv:2204.05973](https://arxiv.org/abs/2204.05973)]

* [[Igor Bandos]], Unai D. M. Sarraga, *Properties of multiple D0-brane system: 11D origin, equations of motion and their solutions* [[arXiv:2212.14829](https://arxiv.org/abs/2212.14829)]

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D0-D2 brane bound state

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## Idea

A [[bound state]]/[[brane intersection]] of [[D0-branes]] with [[D2-branes]]. Special case of [[Dp-D(p+2) brane bound states]].

## Properties

### Myers effect

What is called the _[[Myers effect]]_ ([Myers 99](#Myers99)) in [[string theory]] is the claimed phenomenon that given $N$  [[D0-branes]] in a constant background [[RR field]] $F_4$ (the [[field strength]] associated with [[D2-brane]] [[D-brane charge|charge]]) with, crucially, nonabelian effects included, then these D0-branes expand into a [[fuzzy 2-sphere]] which represents a spherical [[D0-D2 brane bound state]] of a [[D2-brane]] and $N$ [[D0-branes]] ([Myers 99, section 6, see p. 22](#Myers99), [Myers 03, section 4](#Myers03)).

## Related concepts

[[!include brane bound states -- table]]

## References

### General

(...)

### Myers effect

The effect now known as the "[[Myers effect]]" in [[D-brane]] theory was first described in:

* {#Myers99} [[Robert Myers]], _Dielectric-Branes_, JHEP 9912 (1999) 022 ([arXiv:hep-th/9910053](https://arxiv.org/abs/hep-th/9910053))

Review includes

* {#Myers03} [[Robert Myers]], _Nonabelian Phenomena on D-branes_, Class.Quant.Grav. 20 (2003) S347-S372 ([arXiv:hep-th/0303072](https://arxiv.org/abs/hep-th/0303072))

See also

* Pedro J. Silva, _Quantum Myers effect and its supergravity dual for D0/D4 systems_ ([arXiv:hep-th/0109112](https://arxiv.org/abs/hep-th/0109112))

* Yoshifumi Hyakutake, _Gravitational Dielectric Effect and Myers Effect_, Phys.Rev.D71:046007,2005 ([arXiv:hep-th/0401026](https://arxiv.org/abs/hep-th/0401026))

* Yoshinao Sato, _Dissolving D0-brane into D2-brane with background B-field_, JHEP 0512 (2005) 032 ([arXiv:hep-th/0505045](https://arxiv.org/abs/hep-th/0505045))

### In the BMN matrix model

On solutions of the [[BMN matrix model]] in relation to the [[Myers effect]] and [[D0-D2 brane bound states]]:

* Hai Lin, _The Supergravity Dual of the BMN Matrix Model_, JHEP 0412:001, 2004 ([arXiv:hep-th/0407250](https://arxiv.org/abs/hep-th/0407250))

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[[!redirects D0-D2 brane intersections]]
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[[!redirects D0-D2-brane intersections]]


D0-D4-brane bound state

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## Idea

[[bound states]] of [[D0-branes]] with [[D4-branes]]. A special case of the general phenomenon of [[Dp-D(p+4)-brane bound states]].

## Related concepts

[[!include brane bound states -- table]]

## References

* [[Cumrun Vafa]], _Instantons on D-branes_, Nucl. Phys. B463 (1996) 435-442 ([arXiv:hep-th/9512078](https://arxiv.org/abs/hep-th/9512078))

with emphasis to the resulting [[configuration spaces of points]], as in

* [[Cumrun Vafa]], [[Edward Witten]], Section 4.1 of: _A Strong Coupling Test of S-Duality_, Nucl. Phys. B431:3-77, 1994 ([arXiv:hep-th/9408074](https://arxiv.org/abs/hep-th/9408074))

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D0-D6 brane bound state

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## Idea

A [[bound state]]/[[brane intersection]] of [[D0-branes]] with [[D6-branes]]; the special case of [[Dp-D(p+6) brane bound states]] for $p=0$.

## Related concepts

[[!include brane bound states -- table]]

## References

* [[Washington Taylor]], _Adhering 0-branes to 6-branes and 8-branes_, Nucl. Phys. B508: 122-132, 1997 ([arXiv:hep-th/9705116](https://arxiv.org/abs/hep-th/9705116))

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D1-brane

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## Idea

The [[D-brane]] of [[dimension]] $1+1$ in [[type IIB string theory]]. Also called the **D-string**, to be distinguished from the _fundamental_ [[string]].

## Properties

### S-duality with the fundamental string

Under [[S-duality]] the D1-brane mixes with the [[fundamental string]] to form the [[(p,q)-string]].

A formalization of this in terms of the [[homotopy theory]] of the [[super L-infinity algebras]] which constitute the respective [[extended super spacetimes]] is in ([FSS 13](#FSS13)).

### Bound states

[[bound states]]/[[brane intersections]] involving D1-branes:

* [[D1-D3 brane bound state]]

* [[D1-D5 brane bound state]]

* [[D1-D7 brane bound state]]

### Black hole entropy

At low string coupling [[D1-D5 brane bound states]] are described by 2d [[CFT]], which is well understood. After passage to the corresponding [[non-perturbative field theory|strongly coupled]] [[black brane]] configurations in [[type IIB supergravity]], which are [[black holes]] in the given [[Kaluza-Klein mechanism|compactification]], the [[entropy]] of these 2d CFTs matches the [[Bekenstein-Hawking entropy]] of these black holes. See at _[[black holes in string theory]]_ for more on this.

This is parts of the [[AdS/CFT correspondence]]. See ([AGMOO, chapter 5](#AGMOO)).

## Related concepts

[[!include table of branes]]

## References

The [[Green-Schwarz sigma-model]] description of $(p,q)$-1-branes via cocycles on [[extended super Minkowski spacetimes]] is obtained in

* {#Sakaguchi00} Makoto Sakaguchi, section 2 of _IIB-Branes and New Spacetime Superalgebras_, JHEP 0004 (2000) 019 ([arXiv:hep-th/9909143](https://arxiv.org/abs/hep-th/9909143))

See also

* {#AGMOO} [[Ofer Aharony]], S.S. Gubser, [[Juan Maldacena]], H. Ooguri, Y. Oz, chapter 5.3 of _Large N Field Theories, String Theory and Gravity_ ([arXiv:hep-th/9905111](http://arxiv.org/abs/hep-th/9905111))

Formulation of the [[S-duality]] with the fundamental string in terms of the [[homotopy theory]] of [[super L-infinity algebras]] of the respective [[extended super spacetimes]] is in section 4.3 of

* {#FSS13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_ ([arXiv:1308.5264](http://arxiv.org/abs/1308.5264))

[[!redirects D1-branes]]


D1-D3 intersections in AdS2-CFT1 -- references

### D1-D3 brane intersections in $AdS_2/CFT_1$

On [[D1-D3 brane intersections]] in [[AdS2/CFT1]]:

* [[Giuseppe Dibitetto]], [[Yolanda Lozano]], [[Nicolò Petri]], [[Anayeli Ramirez]], _Holographic Description of M-branes via $AdS_2$_ ([arXiv:1912.09932](https://arxiv.org/abs/1912.09932))

Via [[T-duality]] from [[D6-D8 brane intersections]]:

* [[Yolanda Lozano]], [[Carlos Nunez]], [[Anayeli Ramirez]], Stefano Speziali, _New $AdS_2$ backgrounds and $\mathcal{N}=4$ Conformal Quantum Mechanics_ ([arXiv:2011.00005](https://arxiv.org/abs/2011.00005))


D1-D3-brane bound state

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## Idea

The special case of [[Dp-D(p+2)-brane bound states]] for [[D1-brane|D1]]-[[D3-brane]] [[bound states]].

## Related concepts

* [[fuzzy sphere]]

* [[fuzzy funnel]]

\linebreak

[[!include brane bound states -- table]]

## References

For more see the references at _[[Dp-D(p+2)-brane bound state]]_:

### Parallel intersection

* Gianluca Grignani, Troels Harmark, Andrea Marini, Marta Orselli, _Thermal DBI action for the D3-brane at weak and strong coupling_, JHEP 03 (2014) 114 ([arxiv:1311.3834](https://arxiv.org/abs/1311.3834))

### Transversal intersection

#### As spikes/BIons (D3-brane perspective)

As spikes/[[BIons]] in the [[D3-brane]] [[DBI action|DBI-theory]]:

* [[Curtis Callan]], [[Juan Maldacena]], _Brane Dynamics From the Born-Infeld Action_, Nucl. Phys. B513 (1998) 198-212 ([arXiv:hep-th/9708147](https://arxiv.org/abs/hep-th/9708147))

* [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _The Self-Dual String Soliton_, Nucl. Phys. B515 (1998) 203-216 ([arXiv:hep-th/9709014](https://arxiv.org/abs/hep-th/9709014))

* [[Gary Gibbons]], _Born-Infeld particles and Dirichlet p-branes_,  	Nucl. Phys. B514: 603-639, 1998 ([arXiv:hep-th/9709027](https://arxiv.org/abs/hep-th/9709027))

from the [[M5-brane]]:

* {#HoweLambertWest97} [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _The Self-Dual String Soliton_, Nucl. Phys. B515 (1998) 203-216 ([arXiv:hep-th/9709014](https://arxiv.org/abs/hep-th/9709014))

#### As fuzzy funnels (D1-brane perspective)

As [[fuzzy funnels]] (the [[D1-brane]] [[matrix model]] perspective):

* [[Neil Constable]], [[Robert Myers]], Oyvind Tafjord, _The Noncommutative Bion Core_, Phys. Rev. D61 (2000) 106009 ([arXiv:hep-th/9911136](https://arxiv.org/abs/hep-th/9911136))

* [[Robert Myers]], Section 4 of: _Nonabelian D-branes and Noncommutative Geometry_, J. Math. Phys. 42: 2781-2797, 2001 ([arXiv:hep-th/0106178](https://arxiv.org/abs/hep-th/0106178))

* [[Neil Constable]], [[Neil Lambert]], _Calibrations, Monopoles and Fuzzy Funnels_, Phys. Rev. D66 (2002) 065016 ([arXiv:hep-th/0206243](https://arxiv.org/abs/hep-th/0206243))

* Rajsekhar Bhattacharyya, Robert de Mello Koch, _Fluctuating Fuzzy Funnels_, JHEP 0510 (2005) 036 ([arXiv:hep-th/0508131](https://arxiv.org/abs/hep-th/0508131))

#### As monopoles

Identification with [[Yang-Mills monopoles]]:

* {#Diaconescu97} Duiliu-Emanuel Diaconescu,  _D-branes, Monopoles and Nahm Equations_, Nucl. Phys. B503 (1997) 220-238 ([arxiv:hep-th/9608163](https://arxiv.org/abs/hep-th/9608163))

* Jessica K. Barrett, Peter Bowcock, _Using D-Strings to Describe Monopole Scattering_ ([arxiv:hep-th/0402163](https://arxiv.org/abs/hep-th/0402163))
* Jessica K. Barrett, Peter Bowcock, _Using D-Strings to Describe Monopole Scattering - Numerical Calculations_ ([arxiv:hep-th/0512211](https://arxiv.org/abs/hep-th/0512211))

[[!include D1-D3 intersections in AdS2-CFT1 -- references]]

### Lift to M2-M5-brane bound states
 {#ReferencesLiftToMTheory}

The lift of [[Dp-D(p+2)-brane bound states]] in [[string theory]] to [[M2-M5-brane bound states]]/[[E-strings]] in  [[M-theory]], under [[duality between M-theory and type IIA string theory]]+[[T-duality]], via generalization of [[Nahm's equation]] (this eventually motivated the [[BLG-model]]/[[ABJM model]]):

* [[Anirban Basu]], [[Jeffrey Harvey]], _The M2-M5 Brane System and a Generalized Nahm's Equation_, Nucl.Phys. B713 (2005) 136-150 ([arXiv:hep-th/0412310](https://arxiv.org/abs/hep-th/0412310))

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], Section 2.2.1 of _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

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D1-D5 brane bound state

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## Idea

The [[bound state]] of [[D1-branes]] with [[D5-branes]]. A special case of [[Dp-D(p+4)-brane bound states]].

Plays a central role in the discussion of [[black holes in string theory]].

## Related concepts

* [[black hole entropy]]

* [[black holes in string theory]]

[[!include brane bound states -- table]]

## References

Review:

* Lukas Hahn, _Is String Theory Holographic?_ ([[HahnD1D5.pdf:file]])

### Via superstrata

On [[AdS-CFT duality|AdS-CFT dual]] [[type II supergravity]]-solutions corresponding to D1-D5-P bound states ("superstrata"):

* [[Iosif Bena]], [[Stefano Giusto]], [[Rodolfo Russo]], [[Masaki Shigemori]], [[Nicholas P. Warner]], *Habemus Superstratum! A constructive proof of the existence of superstrata*, JHEP 1505 (2015) 110 [[arXiv:1503.01463](https://arxiv.org/abs/1503.01463)]

* [[Masaki Shigemori]], *Superstrata*, Gen Relativ Gravit **52** 51 (2020) [[h)arXiv:2002.01592](https://arxiv.org/abs/2002.01592), [doi:10.1007/s10714-020-02698-8](https://doi.org/10.1007/s10714-020-02698-8)]

* Daniel R. Mayerson, Masaki Shigemori, _Counting D1-D5-P Microstates in Supergravity_ [[arXiv:2010.04172](https://arxiv.org/abs/2010.04172)]

* [[Emil Martinec]], *The branes behind black holes*, talk at *[Strings 2022](https://indico.cern.ch/event/1085701/)* [[indico:4940814](https://indico.cern.ch/event/1085701/contributions/4940814/)]

And the non-supersymmetric generalization to "microstrata", via [[AdS3-CFT2 duality]]:

* [[Bogdan Ganchev]], [[Anthony Houppe]], [[Nicholas P. Warner]], *Q-Balls Meet Fuzzballs: Non-BPS Microstate Geometries*, J. High Energ. Phys. **2021** 28 (2021)
[[arXiv:2107.09677](https://arxiv.org/abs/2107.09677)]

* [[Bogdan Ganchev]], [[Stefano Giusto]], [[Anthony Houppe]], [[Rodolfo Russo]], *$AdS_3$ holography for non-BPS geometries*, European Physical Journal C **82** 217 (2022)
 [[https://arxiv.org/abs/2112.03287](https://arxiv.org/abs/2112.03287), [doi:10.1140/epjc/s10052-022-10133-2](https://doi.org/10.1140/epjc/s10052-022-10133-2)]

* [[Bogdan Ganchev]], [[Stefano Giusto]], [[Anthony Houppe]], [[Rodolfo Russo]], [[Nicholas P. Warner]], *Microstrata* [[arXiv:2307.13021](https://arxiv.org/abs/2307.13021)]

### As instantons

As [[Yang-Mills instantons]]/[[Yang-Mills monopoles]]

* [[Neil Lambert]], _D-brane Bound States and the Generalised ADHM Construction_, Nucl. Phys. B519 (1998) 214-224 ([arXiv:hep-th/9707156](https://arxiv.org/abs/hep-th/9707156))

* {#Martinec99} [[Emil Martinec]], _Manifestations of the D1-D5 system_, 1999 ([[MartinecD1D5.pdf:file]])

### As fuzzy funnels

As [[fuzzy funnels]] on [[fuzzy 4-spheres]]:

* [[Neil Constable]], [[Robert Myers]], Oyvind Tafjord, _Non-abelian Brane Intersections_, JHEP 0106:023, 2001 ([arXiv:hep-th/0102080](https://arxiv.org/abs/hep-th/0102080))

* Rajsekhar Bhattacharyya, Robert de Mello Koch, _Fluctuating Fuzzy Funnels_, JHEP 0510 (2005) 036 ([arXiv:hep-th/0508131](https://arxiv.org/abs/hep-th/0508131))

### KK-reduction  to JT gravity

Realization of [[JT-gravity]] as [[Kaluza-Klein reduction]] of [[D=6 supergravity]] on the [[worldvolume]] of [[D1-D5 brane bound states]] or [[M2-M5 brane bound states]]:

* Yue-Zhou Li, Shou-Long Li, H. Lu, _Exact Embeddings of JT Gravity in Strings and M-theory_, Eur. Phys. J. C (2018) 78: 791 ([arXiv:1804.09742](https://arxiv.org/abs/1804.09742))

* Iosif Bena, Pierre Heidmann, David Turton, _$AdS_2$ Holography: Mind the Cap_, JHEP 1812 (2018) 028 ([arXiv:1806.02834](https://arxiv.org/abs/1806.02834))

[[!include AdS3-CFT2 on D1-D5 branes -- references]]

[[!redirects D1-D5 brane bound states]]

[[!redirects D1/D5 brane bound state]]
[[!redirects D1/D5 brane bound states]]

[[!redirects D1-D5 brane system]]

[[!redirects D1-D5-brane bound state]]
[[!redirects D1-D5-brane bound states]]

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[[!redirects D1-D5-brane intersections]]

[[!redirects D1-D5 brane intersection]]
[[!redirects D1-D5 brane intersections]]

[[!redirects D1-D5-P bound state]]
[[!redirects D1-D5-P bound states]]
[[!redirects superstratum]]
[[!redirects superstrata]]
[[!redirects microstratum]]
[[!redirects microstrata]]


D1-D7 brane bound state

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## Idea

A [[bound state]]/[[brane intersection]] of [[D1-branes]] with [[D7-branes]].

## Related concepts

[[!include brane bound states -- table]]

## References

On [[D1-D7 brane intersections]] as [[fuzzy funnels]] on [[fuzzy 6-spheres]]:

* [[Paul Cook]], Robert de Mello Koch, Jeff Murugan, _Non-Abelian BIonic Brane Intersections_,  Phys. Rev. D68:126007, 2003 ([arXiv:hep-th/0306250](https://arxiv.org/abs/hep-th/0306250))

[[!redirects D1-D7 brane bound states]]

[[!redirects D1-D7-brane bound state]]
[[!redirects D1-D7-brane bound states]]

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[[!redirects D1-D7 brane intersections]]
[[!redirects D1-D7-brane intersection]]
[[!redirects D1-D7-brane intersections]]


D2-brane

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## Idea

The [[D-brane]] in [[type IIA supergravity]] of [[dimension]] 2+1.

## Properties

### Bound states

[[bound states]]/[[brane intersections]] involving D2-branes:

* [[D0-D2 brane bound state]]

* [[D2-D4 brane bound state]]

* [[D2-D6 brane bound state]]

## Related concepts

* [[Myers effect]]

\linebreak

[[!include table of branes]]

## References

* M.P. Garcia del Moral, C. Las Heras, _D2-brane with RR and NSNS fluxes and its dual M-theory origin_ ([arXiv:2012.14069](https://arxiv.org/abs/2012.14069))

[[!redirects D2-branes]]


D2-D4-brane bound state

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## Idea

The special case of [[Dp-D(p+2)-brane bound state]] for [[D2-brane|D2]]-[[D4-brane]] [[bound states]].

## Related concepts

[[!include brane bound states -- table]]

## References

For more see the references at _[[Dp-D(p+2)-brane bound state]]_:

### General

Discussion of [[D2-branes]] [[D-brane polarization|polarizing]] to [[D2-D4 brane bound states]]:

* [[Iosif Bena]], Aleksey Nudelman, _Warping and vacua of $(S)YM_{3+1}$_, Phys. Rev. D62 (2000) 086008 ([arXiv:hep-th/0005163](https://arxiv.org/abs/hep-th/0005163))

* [[Iosif Bena]], Aleksey Nudelman, _Exotic polarizations of D2 branes and oblique vacua of $(S)YM_{2+1}$_, Phys. Rev. D62 (2000) 126007 ([arXiv:hep-th/0006102](https://arxiv.org/abs/hep-th/0006102))

With emphasis on the [[s-rule]]:

* Minoru Eto, Youichi Isozumi, Muneto Nitta, Keisuke Ohashi, Kazutoshi Ohta, Norisuke Sakai, Section 3 of: _D-brane Construction for Non-Abelian Walls_, Phys. Rev. D71 (2005) 125006 ([arXiv:hep-th/0412024](https://arxiv.org/abs/hep-th/0412024))

### In holographic QCD

For transversal [[D2-D4-brane bound states]] (with an eye towards [[AdS/QCD]]):

* Alexander Gorsky, Valentin Zakharov, Ariel Zhitnitsky, _On Classification of QCD defects via holography_, Phys. Rev. D79:106003, 2009 ([arxiv:0902.1842](https://arxiv.org/abs/0902.1842))

[[!include AdS3-CFT2 on D2-D4-D6-D8 branes -- references]]

### Lift to M2-M5-brane bound states
 {#ReferencesLiftToMTheory}

The lift of [[Dp-D(p+2)-brane bound states]] in [[string theory]] to [[M2-M5-brane bound states]]/[[E-strings]] in  [[M-theory]], under [[duality between M-theory and type IIA string theory]]+[[T-duality]], via generalization of [[Nahm's equation]] (this eventually motivated the [[BLG-model]]/[[ABJM model]]):

* [[Anirban Basu]], [[Jeffrey Harvey]], _The M2-M5 Brane System and a Generalized Nahm's Equation_, Nucl.Phys. B713 (2005) 136-150 ([arXiv:hep-th/0412310](https://arxiv.org/abs/hep-th/0412310))

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], Section 2.2.1 of _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

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[[!redirects D2-D4 brane intersections]]
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D2-D6 brane bound state

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## Idea

[[brane intersection]] of [[D2-branes]] with [[D6-branes]]; special case of [[Dp-D(p+4)-brane bound state]]

## Related concepts

## References

[[!include brane bound states -- table]]

### As black branes

As [[black branes]] ([[D2-D6-NS5 brane bound state]]):

#### General

* [[Konstadinos Sfetsos]], [[Kostas Skenderis]], Section 2.2 of: _Microscopic derivation of the Bekenstein-Hawking entropy formula for non-extremal black holes_, Nucl. Phys. B517 (1998) 179-204 ([arXiv:hep-th/9711138](https://arxiv.org/abs/hep-th/9711138))

* [[Oskar Pelc]], Ruud Siebelink, *The D2-D6 System and a Fibered AdS Geometry*, Nucl. Phys. B **558** (1999) 127-158 [[arXiv:hep-th/9902045](https://arxiv.org/abs/hep-th/9902045), doi:10.1016/S0550-3213(99)00332-6]

* Federico Faedo, [[Yolanda Lozano]], [[Nicolo Petri]], _Searching for surface defect CFTs within $AdS_3$_, JHEP 11 (2020) 052 ([arXiv:2007.16167](https://arxiv.org/abs/2007.16167))

#### As defects in D4/D8 realizing $AdS_3/CFT_2$

[[black brane|black]]$\;$[[D4-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D6-brane bound states]] inside them realizing [[AdS3-CFT2]] "inside" [[AdS7-CFT6]]:

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _Surface defects in the D4 − D8 brane system_, JHEP 01 (2019) 193 ([arxiv:1807.07768](https://arxiv.org/abs/1807.07768))

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_3$ vacua and surface defects in massive IIA_ ([arxiv:1904.02455](https://arxiv.org/abs/1904.02455))

### As black holes

Discussion of [[black hole entropy]] of [[D2-D6 brane bound states]] as [[black holes in string theory]]:

* [[Juan Maldacena]], [[Andrew Strominger]], _Statistical Entropy of Four-Dimensional Extremal Black Holes_, Phys. Rev. Lett. 77:428-429, 1996 ([arXiv:hep-th/9603060](https://arxiv.org/abs/hep-th/9603060))

* [[Konstadinos Sfetsos]], [[Kostas Skenderis]], _Microscopic derivation of the Bekenstein-Hawking entropy formula for non-extremal black holes_, Nucl. Phys. B517 (1998) 179-204 ([arXiv:hep-th/9711138](https://arxiv.org/abs/hep-th/9711138))

[[!redirects D2-D6 brane bound states]]

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[[!redirects D2-D6-NS5-brane bound states]]


D2-D8 brane bound state

## Idea

[[bound state]]/[[brane intersection]] of [[D2-branes]] with [[D8-branes]]

## Related concepts

[[!include brane bound states -- table]]

## References

* Luca Grisa, Section 2 of: _Delocalization from Anomaly Inflow and Intersecting Brane Dynamics_, JHEP 0703:017, 2007 ([arxiv:hep-th/0611331](https://arxiv.org/abs/hep-th/0611331))

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D210


D25-brane

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## Idea

The D25 is the space-filling [[D-brane]] in [[open string|open]] [[bosonic string theory]].

The decay of the D25 is exhibited by the [[tachyon]] of the [[perturbative string theory|perturbative]] [[bosonic string]], described by _[[Sen's conjecture]]_.

For the moment see the pointers at [[bosonic string field theory]] in the sections _[achievements](string+field+theory#Achievements)_ and at _[references -- Tachyon dynamics, decaying D-branes and Sen's conjecture](string+field+theory#BosonicStringVacuumAndSenConjecture).

See also

* Wikipedia, _[Tachyon condensation](http://en.wikipedia.org/wiki/Tachyon_condensation)_

## Related concepts

[[!include table of branes]]

[[!redirects D25]]


D3-brane

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## Idea

The [[D-brane]] of [[dimension]] $3+1$ in [[type IIB string theory]].

## Properties

### Worldvolume theory

For small string coupling its [[worldvolume]] theory is [[super Yang-Mills theory]]. At strong coupling it forms a [[black brane]] solution of [[type II supergravity]] with near horizon [[anti de Sitter spacetime]] geometry $AdS_5$. The relation between the two description in the [[AdS/CFT correspondence|AdS5/CFT4-correspondence]].

[[!include electric-magnetic duality -- table]]

### Bound states

[[bound states]]/[[brane intersections]] involving D3-branes:

* [[D1-D3 brane bound state]]

* [[D3-D5 brane bound state]]

* [[D3-D7 brane bound state]]

## Related concepts

* [[D3-brane instanton]]

* [[3-brane in 6d]]

[[!include table of branes]]

## References

The [[Green-Schwarz sigma-model]] description of D3-branes via cocycles on [[extended super Minkowski spacetimes]] is obtained in

* {#Sakaguchi00} Makoto Sakaguchi, section 2 of _IIB-Branes and New Spacetime Superalgebras_, JHEP 0004 (2000) 019 ([arXiv:hep-th/9909143](https://arxiv.org/abs/hep-th/9909143))

[[!redirects D3-branes]]

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[[!redirects D3 branes]]


D3-brane instanton

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## Idea

D3-brane [[instantons]] are constituted by [[D3-branes]]  [[wrapped brane|wrapped]] on [[spacelike]] [[cycles]].

## Related concepts

* [[D(-1)-brane]]

* [[non-perturbative effect]]

* [[membrane instanton]]

* [[M5-brane instanton]]

[[!include table of branes]]

## References

* [[Katrin Becker]], [[Melanie Becker]], [[Andrew Strominger]], section 2.1 of _Five-branes, membranes and nonperturbative string theory_, Nucl. Phys. B 456, 130 (1995) ([hep-th/9507158](http://arxiv.org/abs/hep-th/9507158))

* Sergei Alexandrov, Jan Manschot, [[Boris Pioline]], _D3-instantons, Mock Theta Series and Twistors_, JHEP 1304 (2013) 002 ([arXiv:1207.1109](http://arxiv.org/abs/1207.1109))

On D-brane instantons and [[moduli stabilization]] in [[type II string theory]]:

* Sergei Alexandrov, Atakan Hilmi Fırat, Manki Kim, [[Ashoke Sen]], [[Bogdan Stefański]], *D-instanton Induced Superpotential*, J. High Energ. Phys. **2022** 90 (2022) [[arXiv:2204.02981](https://arxiv.org/abs/2204.02981), doi:10.1007/JHEP07(2022)090]

and in 0B string theory using methods from [[string field theory]]:

* Joydeep Chakravarty, [[Ashoke Sen]], *Normalization of D instanton amplitudes in two dimensional type 0B string theory* [[arXiv:2207.07138](https://arxiv.org/abs/2207.07138)]

reviewed in

* [[Ashoke Sen]], *D-instanton amplitudes in string theory*, talk at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (2023) [[web](/nlab/show/M-Theory+and+Mathematics#Sen2023)]

[[!redirects D3-brane instanton]]

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[[!redirects D-brane instantons]]


D3-D5 brane bound state

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## Idea

The [[bound states]]/[[brane intersections]] of [[D3-branes]] with [[D5-branes]]. The special case $p=3$ of [[Dp-D(p+2)-brane bound states]]

## Properties

### Fuzzy funnel geometry

The [[fuzzy funnel]] [[noncommutative geometry]] of transveral $D3 \bot D5$-brane intersections ([Gaiotto-Witten 08, Section 4.3](#GaiottoWitten08), [Fazzi 17, Section 3.2.3](#Fazzi17)):

<img src="https://ncatlab.org/nlab/files/D3D5FuzzyFunnel.jpg" width="600">

> graphics grabbed from [Fazzi 17](#Fazzi17)

## Related concepts

[[!include brane bound states -- table]]

## References

With emphasis on the [[worldvolume]] [[D=3 N=4 super Yang-Mills theory]]:

* [[Amihay Hanany]], [[Edward Witten]], _Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics_, Nucl. Phys. B492:152-190, 1997 ([arXiv:hep-th/9611230](https://arxiv.org/abs/hep-th/9611230))

With emphasis on the relation to [[Nahm's equation]] and [[fuzzy funnel]] [[noncommutative geometry]]:

* {#GaiottoWitten08} [[Davide Gaiotto]], [[Edward Witten]], _Supersymmetric Boundary Conditions in $\mathcal{N} = 4$ Super Yang-Mills Theory_, J Stat Phys (2009) 135: 789 ([arXiv:0804.2902](https://arxiv.org/abs/0804.2902))

With derivation of the [[s-rule]]:

* [[Constantin Bachas]], [[Michael Green]], Adam Schwimmer, Section 2.3 of: _$(8,0)$ Quantum mechanics and symmetry enhancement in type I' superstrings_, JHEP 9801 :006, 1998 ([arXiv:hep-th/9712086](https://arxiv.org/abs/hep-th/9712086))

* Akikazu Hashimoto, Peter Ouyang, Masahito Yamazaki, Section 3.6 of: _Boundaries and Defects of N=4 SYM with 4 Supercharges, Part II: Brane Constructions and 3d N=2 Field Theories_, JHEP 1410 : 108, 2014 ([arXiv:1406.5501](https://arxiv.org/abs/1406.5501))

* Federico Carta, Hirotaka Hayashi, _Hilbert series and mixed branches of $T[SU(N)]$ theory_, JHEP 02 (2017) 037 ([arXiv:1609.08034](https://arxiv.org/abs/1609.08034))

Review and highlighting of the [[fuzzy funnel]] geometry with [[fuzzy 2-sphere]] sections:

* {#Fazzi17} [[Marco Fazzi]], Section 3.2.3 of: _Higher-dimensional field theories from type II supergravity_ ([arxiv:1712.04447](https://arxiv.org/abs/1712.04447))

With emphasis on the [[Hanany-Witten effect]]:

* Benjamin Assel, _Hanany-Witten effect and $SL(2,\mathbb{Z})$ dualities in matrix models_, JHEP 10 (2014) 117 ([arXiv:1406.5194](https://arxiv.org/abs/1406.5194))

* Naotaka Kubo, Sanefumi Moriyama, _Hanany-Witten Transition in Quantum Curves_ ([arXiv:1907.04971](https://arxiv.org/abs/1907.04971))

See also

* Curtis G. Callan, Alberto Guijosa, Konstantin G. Savvidy,
_Baryons and String Creation from the Fivebrane Worldvolume Action_, Nucl.Phys. B547 (1999) 127-142 ([arXiv:hep-th/9810092](https://arxiv.org/abs/hep-th/9810092))

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[[!redirects D3-D5-brane bound states]]

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D3-D7 brane bound state

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## Idea

The [[bound states]]/[[brane intersections]] of [[D3-branes]] with [[D7-branes]]. The special case $p=3$ of [[Dp-D(p+4)-brane bound states]].

# Related concepts

[[!include brane bound states -- table]]

## References

Relation to [[spin chain]] for [[single trace operators]]:

* Marius de Leeuw, Tamás Gombor, Charlotte Kristjansen, Georgios Linardopoulos, Balázs Pozsgay, _Spin Chain Overlaps and the Twisted Yangian_ ([arXiv:1912.09338](https://arxiv.org/abs/1912.09338))

Relation to the [[K-theory classification of topological phases of matter]] via [[AdS/CFT duality in solid state physics]]:

* Charlotte Kristjansen, Gordon W. Semenoff, _The D3-probe-D7 brane holographic fractional topological insulator_, JHEP10 (2016) 079 ([arXiv:1604.08548](https://arxiv.org/abs/1604.08548))

[[!redirects D3-D7 brane bound states]]

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D4

> This entry is about items in the [[ADE-classification]] labeled by $D4$. For the [[D4-brane]], see there.

***

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## Idea

In the [[ADE-classification]], the items labeled $D4$ include the following:

1. as [[finite subgroups of SO(3)]]:

   the [[Klein four-group]]  (the smallest [[dihedral group]])

   $\mathbb{Z}/2 \times \mathbb{Z}/2$

1. as [[finite subgroups of SU(2)]]:

   the [[quaternion group]] of [[order of a group|order]] 8 (the smallest [[binary dihedral group]]):

   $Q_8 \simeq 2 D_4$

1. as [[simple Lie groups]]: the [[special orthogonal group]] in 8 dimensions

   [[SO(8)]], [[Spin(8)]]

1. as a [[Dynkin diagram]]/[[Dynkin quiver]]:

\begin{tikzpicture}
  \node (center) at (0,0) {};
  \node (topright) at (60:1) {};
  \node (botright) at (-60:1) {};
  \node (left) at (180:1) {};

  \draw[fill=black] (center) circle (.1);
  \draw[fill=black] (topright) circle (.1);
  \draw[fill=black] (botright) circle (.1);
  \draw[fill=black] (left) circle (.1);

  \draw (center) to (topright);
  \draw (center) to (botright);
  \draw (center) to (left);
\end{tikzpicture}

## Properties

### Triality

The [[symmetric group|S3]]-[[symmetry group]] of the D4-diagram translates into interesting 3-fold symmetries of structures associated with the corresponding objects in the above list. This is known as _[[triality]]_.

## Related concepts

[[!include ADE -- table]]


D4-brane

> This entry is about a [[D-brane]] species in [[string theory]]. For the items in the [[ADE-classification]] of name [[D4]], see there.

***

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## 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-[[dimension]]al [[TQFT]] in higher analogy of how the [[Jones polynomial]] arises as an observable in 3-dimensional [[Chern-Simons theory]]. For instance for $\Sigma : K \to K'$ 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](#Witten11)) it is argued, following indications in ([GukovSchwarzVafa](#GukovSchwarzVafa))
that this 4d TQFT is related to the [[worldvolume]] theory of the _image_ in [[type IIA string theory|type IIA]] of [[D3-branes]] ending on NS5-branes in [[type IIB string theory|type IIB]] after one [[S-duality]] and one [[T-duality]] operation:

$$
  (D3 - NS5)
    \stackrel{S}{\mapsto}
  (D3 - D5)
    \stackrel{T}{\mapsto}
  (D4-D6)
    \stackrel{IIA/M}{\mapsto}
  (M5 - MK6)
  \,.
$$

Earlier indication for this had come from the observation that [[Chern-Simons theory]] is the [[effective QFT|effective background theory]] for the [[A-model]] 2d [[TCFT]] (see [TCFT – Worldsheet and effective background theories](http://ncatlab.org/nlab/show/TCFT#ActionFunctionals) for details).

Notice that after the above [[T-duality]] operation the $(D4-D6)$-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-brane]]s) on [[Taub-NUT spacetime|Taub-NUT]] ($\times S^1$). The [[M5-branes]] wrap the circle-fiber of [[Taub-NUT space]], which shrinks to zero size at the origin (the [[MK6]], 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]].

### Bound states

[[bound states]]/[[brane intersections]] involving D4-branes:

* [[D0-D4 brane bound state]]

* [[D2-D4 brane bound state]]

* [[D4-D6 brane bound state]]

* [[D4-D8 brane bound state]]

## Related concepts

[[!include superconformal symmetry -- table]]

[[!include table of branes]]

## References

### General

As a [[Green-Schwarz sigma-model]]:

* {#APPS97b} [[Mina Aganagic]], [[Jaemo Park]], Costin Popescu, [[John Schwarz]], Section 6 of _Dual D-Brane Actions_, Nucl. Phys. B496 (1997) 215-230 ([arXiv:hep-th/9702133](https://arxiv.org/abs/hep-th/9702133))

* {#Sakaguchi00} Makoto Sakaguchi, _IIB-Branes and New Spacetime Superalgebras_, JHEP 0004 (2000) 019 ([arXiv:hep-th/9909143](https://arxiv.org/abs/hep-th/9909143))

### Double dimensional reduction from M5-brane

The relation of the [[M5-brane]] to the [[D4-brane]] and the [[D=5 super Yang-Mills theory]] in its [[worldvolume]] [[physical theory|theory]] by [[double dimensional reduction]]:

* [[Eric Bergshoeff]], [[Mees de Roo]], [[Tomas Ortin]], *The Eleven-dimensional Five-brane*, Phys. Lett. B **386** (1996) 85-90 [[arXiv:hep-th/9606118](https://arxiv.org/abs/hep-th/9606118), doi:10.1016/0370-2693(96)00913-6]

* {#APPS97a} [[Mina Aganagic]], [[Jaemo Park]], Costin Popescu, [[John Schwarz]], Section 6 of  _World-Volume Action of the M Theory Five-Brane_, Nucl.Phys. B496 (1997) 191-214 ([arXiv:hep-th/9701166](http://arxiv.org/abs/hep-th/9701166))

* [APPS 97b, Section 6](#APPS97b)

* [[Neil Lambert]], Constantinos Papageorgakis, Maximilian Schmidt-Sommerfeld, _M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills_, JHEP 1101:083 (2011) ([arXiv:1012.2882](http://arxiv.org/abs/1012.2882))

* [[Chong-Sun Chu]], Sheng-Lan Ko, _Non-abelian Action for Multiple Five-Branes with Self-Dual Tensors_, ([arXiv:1203.4224](http://arxiv.org/abs/1203.4224)) JHEP05(2012)028

* [[Neil Lambert]], Miles Owen, _Charged Chiral Fermions from M5-Branes_ ([arXiv:1802.07766](https://arxiv.org/abs/1802.07766))

### D4-D8 intersection

On [[D4-D8 bound states]]:

With an eye towards [[holographic QCD]]:

* {#Nastase03} [[Horatiu Nastase]], _On Dp-Dp+4 systems, QCD dual and phenomenology_ ([arXiv:hep-th/0305069](https://arxiv.org/abs/hep-th/0305069))

### String phenomenology / intersecting D4-brane models

[[intersecting D-brane models]] with intersecting D4-branes:

Only [[D4-branes]] (possibly on [[O4-plane]] [[orientifolds]]):

* {#BKL01} D. Bailin, G. V. Kraniotis, A. Love, _Standard-like models from intersecting D4-branes_, Phys. Lett. B530 (2002) 202-209 ([arXiv:hep-th/0108131](https://arxiv.org/abs/hep-th/0108131))

* {#KataokaShimojo01} H. Kataoka, M. Shimojo, _$SU(3) \times SU(2) \times U(1)$ Chiral Models from Intersecting D4-/D5-branes_, Progress of Theoretical Physics, Volume 107, Issue 6, June 2002, Pages 1291–1296 ([arXiv:hep-th/0112247](https://arxiv.org/abs/hep-th/0112247), [doi:10.1143/PTP.107.1291](https://doi.org/10.1143/PTP.107.1291))

* {#Bailin02} D. Bailin, _Standard-like models from D-branes_, J Phys (2003) 60: 199 ([arXiv:hep-th/0210227](https://arxiv.org/abs/hep-th/0210227))

* D. Bailin, G. V. Kraniotis, A. Love, _New Standard-like Models from Intersecting D4-Branes_, Phys. Lett. B547 (2002) 43-50 ([arXiv:hep-th/0208103](https://arxiv.org/abs/hep-th/0208103))

[[D4-branes]] that [[brane intersection|intersect]] [[D8-branes]]:

* [[Witten-Sakai-Sugimoto model]] for [[non-perturbative effect|non-perturbative]] [[quantum chromodynamics]]

### Matrix model

The [[nuclear matrix]] model describes [[D4-branes]] bound to [[flavour brane|flavour]] [[D8-branes]] in the [[WSS-model]]:

* {#HashimotoIizukaYi10} [[Koji Hashimoto]], Norihiro Iizuka, Piljin Yi, _A Matrix Model for Baryons and Nuclear Forces_, JHEP 1010:003, 2010 ([arXiv:1003.4988](https://arxiv.org/abs/1003.4988))

* Si-wen Li, Tuo Jia, _Matrix model and Holographic Baryons in the D0-D4 background_, Phys. Rev. D 92, 046007 (2015) ([arXiv:1506.00068](https://arxiv.org/abs/1506.00068))

* {#HashimotoMatsuoMorita19} [[Koji Hashimoto]], [[Yoshinori Matsuo]], [[Takeshi Morita]], _Nuclear states and spectra in holographic QCD_, JHEP12 (2019) 001 ([arXiv:1902.07444](https://arxiv.org/abs/1902.07444))

* Yasuhiro Hayashi, Takahiro Ogino, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Stringy excited baryons in holographic QCD_, Prog Theor Exp Phys (2020) ([arXiv:2001.01461](https://arxiv.org/abs/2001.01461))

### Relation to Khovanov homology

The relation to [[Khovanov homology]]

* [[Edward Witten]], _Fivebranes and knots_ ([arXiv:1101.3216](http://arxiv.org/abs/1101.3216))
 {#Witten11}

[[!redirects D4-branes]]


D4-D6 brane bound state

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## Idea

A [[bound state]]/[[brane intersection]] of [[D4-branes]] and [[D6-branes]]. The special case $p=4$ of [[Dp-D(p+2)-brane bound states]].

## Properties

### s-Rule

By the [[s-rule]] a [[supersymmetry|supersymmetric]] configurations of [[D4-branes]] stretching between [[NS5-branes]] and [[D6-branes]] has at most one [[D4-brane]] ending on any [[D6-brane]]:

> graphics grabbed from [Fazzi 17](#Fazzi17)

## Related concepts

[[!include brane bound states -- table]]

## References

With emphasis on the [[s-rule]]:

* {#OoguriVafa97} [[Hirosi Ooguri]], [[Cumrun Vafa]], _Geometry of $N = 1$ Dualities in Four Dimensions_, Nucl. Phys. B 500:62-74, 1997 ([arXiv:hep-th/9702180](https://arxiv.org/abs/hep-th/9702180))

* [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], Section 3 of: _Strong Coupling Dynamics of Four-Dimensional N=1 Gauge Theories from M Theory Fivebrane_, Adv. Theor. Math. Phys.1:1-52, 1998 ([arXiv:hep-th/9706082](https://arxiv.org/abs/hep-th/9706082))

* {#Fazzi17} [[Marco Fazzi]], around Figure 3.5 in: _Higher-dimensional field theories from type II supergravity_ ([arxiv:1712.04447](https://arxiv.org/abs/1712.04447))

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[[!redirects D4-D6-brane bound state]]
[[!redirects D4-D6-brane bound states]]

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[[!redirects D4-D6 brane intersections]]
[[!redirects D4-D6-brane intersection]]
[[!redirects D4-D6-brane intersections]]


D4-D8 brane bound state

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## Idea

[[bound states]] of [[D4-branes]] with [[D8-branes]]. A special case of the general phenomenon of [[Dp-D(p+4)-brane bound states]].

## Properties

### As a black brane

As a [[black brane]] [[supergravity]] solution:  [Brandhuber-Oz 99](#BrandhuberOz99), [Nastase 03, Sec. 2 and 3](#Nastase03), [BSU 09](#BSU09),  [Van Pol 15, Sec 5](#VanPol15), [Dibitetto-Petri 18](#DibitettoPetri18)

### As geometric engineering of QCD
 {#AsGeometricEngineeringOfQCD}

The D4-D8 system is the [[intersecting D-brane model]] underlying the [[Witten-Sakai-Sugimoto model]] for [[holographic QCD]]

> graphics grabbed from [Erlich 09, section 1.1](#Erlich09)

> graphics grabbed from [Rebhan 14](#Rebhan14)

### As instantons on the D8-brane
  {#AsInstantonsOnTheD8Brane}

Generally, due to the [[higher WZW term]] $\propto \int_{D_{p+4}} C_{p+1} \wedge \langle F \wedge F \rangle$ in the [[Green-Schwarz sigma model]] for [[D-brane|D(p+4)-branes]], [[Yang-Mills instantons]] in the [[Chan-Paton gauge field]] on $D (p+4)$-branes are equivalently [[Dp-D(p+4)-brane bound states]] (see e.g. [Polchinski 96, 5.4](#Polchinski96), [Tong 05, 1.4](#Tong05)).

In terms of [[holographic QCD]] as [above](#AsGeometricEngineeringOfQCD) these instantons correspond to [[baryon]]-[[bound states]] ([Sakai-Sugimoto 04, 5.7](#SakaiSugimoto04)).

> graphics grabbed from [Sugimoto 16](AdS/QCD#Sugimoto16)

## Related entries

[[!include brane bound states -- table]]

## References

### As black brane solutions

#### General

As [[black brane]] [[supergravity]]-solutions:

* {#BrandhuberOz99} [[Andreas Brandhuber]], [[Yaron Oz]], _The D4-D8 Brane System and Five Dimensional Fixed Points_, Phys.Lett. B460:307-312, 1999 ([arXiv:hep-th/9905148](https://arxiv.org/abs/hep-th/9905148))

* {#Nastase03} [[Horatiu Nastase]], Sections 2, 3 of: _On Dp-Dp+4 systems, QCD dual and phenomenology_ ([arXiv:hep-th/0305069](https://arxiv.org/abs/hep-th/0305069))

* {#BSU09} Pierre Binetruy, Misao Sasaki, Kunihito Uzawam, _Dynamical D4-D8 and D3-D7 branes in supergravity_, Phys. Rev. D80:026001, 2009 ([arXiv:0712.3615](https://arxiv.org/abs/0712.3615))

* [[Oren Bergman]], Diego Rodriguez-Gomez, Section 2.3 of: _5d quivers and their $AdS_6$ duals_, J. High Energ. Phys. (2012) 2012: 171 ([arXiv:1206.3503](https://arxiv.org/abs/1206.3503))

* {#VanPol15} Bert Van Pol, Section 5 of: _Aspects of D-branes in supergravity_, thesis 2015 ([pdf](http://inspirehep.net/record/1404388/files/bertvanpol-phdthesis.pdf), [[VanPolThesis15.pdf:file]])

* {#DibitettoPetri18} [[Giuseppe Dibitetto]], [[Nicolò Petri]], _Surface defects in the D4-D8 brane system_, J. High Energ. Phys. (2019) 2019: 193 ([arXiv:1807.07768](https://arxiv.org/abs/1807.07768))

* Minwoo Suh, *D4-D8-branes wrapped on a manifold with non-constant curvature* ([arXiv:2108.08326](https://arxiv.org/abs/2108.08326))

#### With D2/D6-defects realizing $AdS_3/CFT_2$

[[black brane|black]]$\;$[[D4-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D6-brane bound states]] inside them realizing [[AdS3-CFT2]] "inside" [[AdS7-CFT6]]:

* Giuseppe Dibitetto, Nicolò Petri, _Surface defects in the D4 − D8 brane system_, JHEP 01 (2019) 193 ([arxiv:1807.07768](https://arxiv.org/abs/1807.07768))

* Giuseppe Dibitetto, Nicolò Petri, _$AdS_3$ vacua and surface defects in massive IIA_ ([arxiv:1904.02455](https://arxiv.org/abs/1904.02455))

### As geometric engineering of QCD

With application to [[geometric engineering of QFTs|geometric engineering]] of [[QCD]] ("[[holographic QCD]]", such as the [[Witten-Sakai-Sugimoto model]]):

* [[David Morrison]], [[Nathan Seiberg]], _Extremal Transitions and Five-Dimensional Supersymmetric Field Theories_, Nucl. Phys.B483:229-247, 1997 ([arXiv:hep-th/9609070](https://arxiv.org/abs/hep-th/9609070))

* [Nastase 03, Sec. 5](#Nastase03)

* Hee-Cheol Kim, Sung-Soo Kim, Kimyeong Lee, _5-dim Superconformal Index with Enhanced En Global Symmetry_, JHEP 1210 (2012) 142 ([arXiv:1206.6781](https://arxiv.org/abs/1206.6781))

* Chiung Hwang, Joonho Kim, Seok Kim, [[Jaemo Park]], _General instanton counting and 5d SCFT_ ([arxiv:1406.6793](https://arxiv.org/abs/1406.6793))

Review:

* Yolanda Lozano, Section 2 of: _Hints on 5d Fixed Point Theories from Non-Abelian T-duality_, IFT 2014 ([slides pdf](https://www.unioviedo.es/hepth/people/Yolanda/talks/IFT2014.pdf), [[Lozano5d.pdf:file]])

As a model for [[baryons]]:

* {#SakaiSugimoto04} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Low energy hadron physics in holographic QCD_, Prog. Theor. Phys.113:843-882, 2005 ([arXiv:hep-th/0412141](https://arxiv.org/abs/hep-th/0412141))

Among review of the [[Witten-Sakai-Sugimoto model]]:

* {#Erlich09} Joshua Erlich, _How Well Does AdS/QCD Describe QCD?_,  	Int. J. Mod. Phys.A25:411-421,2010 ([arXiv:0908.0312](https://arxiv.org/abs/0908.0312))

* {#Rebhan14} [[Anton Rebhan]], _The Witten-Sakai-Sugimoto model: A brief review and some recent results_, 3rd International Conference on New Frontiers in Physics, Kolymbari, Crete, 2014 ([arXiv:1410.8858](https://arxiv.org/abs/1410.8858))

### General $D p$-$D(p+4)$-brane bound states

The argument that [[Yang-Mills instantons]] in the [[Chan-Paton gauge field]] on a [[D-brane|D(p+4)-brane]] are equivalent to [[Dp-D(p+4) brane bound states]] goes back to

* [[Edward Witten]], _Small Instantons in String Theory_,  	Nucl. Phys. B460:541-559, 1996 ([arXiv:hep-th/9511030](https://arxiv.org/abs/hep-th/9511030))

* [[Michael Douglas]], _Gauge Fields and D-branes_, J. Geom. Phys. 28 (1998) 255-262 ([arXiv:hep-th/9604198](https://arxiv.org/abs/hep-th/9604198))

Review is in:

* {#Polchinski96} [[Joseph Polchinski]], Section 5.4 of: _TASI Lectures on D-Branes_ ([arXiv:hep-th/9611050](https://arxiv.org/abs/hep-th/9611050))

* {#Tong05} [[David Tong]], Section 1.4 of _TASI Lectures on Solitons_ ([hep-th/0509216](https://arxiv.org/abs/hep-th/0509216))

Discussion specifically of [[D0-D4-brane bound states]]:

* [[Cumrun Vafa]], _Instantons on D-branes_, Nucl. Phys. B463 (1996) 435-442 ([arXiv:hep-th/9512078](https://arxiv.org/abs/hep-th/9512078))

with emphasis to the resulting [[configuration spaces of points]], as in

* [[Cumrun Vafa]], [[Edward Witten]], Section 4.1 of: _A Strong Coupling Test of S-Duality_, Nucl. Phys. B431:3-77, 1994 ([arXiv:hep-th/9408074](https://arxiv.org/abs/hep-th/9408074))

Discussion specifically of [[D1-D5-brane bound states]]

* [[Neil Lambert]], _D-brane Bound States and the Generalised ADHM Construction_, Nucl. Phys. B519 (1998) 214-224 ([arXiv:hep-th/9707156](https://arxiv.org/abs/hep-th/9707156))

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[[!redirects D4/D8-brane intersections]]


D5

> This entry is about items in the [[ADE-classification]] labeled by $D5$. For the [[D5-brane]], see there.

***

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## Idea

In the [[ADE-classification]], the items labeled $D5$ include the following:

1. as [[finite subgroups of SO(3)]]:

   the [[dihedral group of order 6]]

   $\mathbb{D}_6$

1. as [[finite subgroups of SU(2)]]:

   the [[binary dihedral group of order 12]]

   $2 D_6$

1. as [[simple Lie groups]]: the [[special orthogonal group]]/[[spin group]] in 10 dimensions

   [[SO(10)]], [[Spin(10)]]

1. as a [[Dynkin diagram]]/[[Dynkin quiver]]:

\begin{tikzpicture}
  \node (center) at (0,0) {};
  \node (topright) at (60:1) {};
  \node (botright) at (-60:1) {};
  \node (left) at (-1,0) {};
  \node (D5) at (-2,0) {};

  \draw[fill=black] (center) circle (.1);
  \draw[fill=black] (topright) circle (.1);
  \draw[fill=black] (botright) circle (.1);
  \draw[fill=black] (left) circle (.1);

  \draw[fill=black] (D5) circle (.1);

  \draw (center) to (topright);
  \draw (center) to (botright);
  \draw (center) to (left);

  \draw (D5) to (left);
\end{tikzpicture}

## Related concepts

[[!include ADE -- table]]


D5-brane

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## Idea

The [[D-brane]] of [[dimension]] $5+1$ in [[type IIB string theory]].

## Properties

### Bound states

[[bound states]]/[[brane intersections]] involving D5-branes

* [[D1-D5 brane bound state]]

* [[D3-D5 brane bound state]]

* [[D5-D9 brane bound state]]

### Black hole entropy

At low [[string coupling]] [[D1-D5 brane bound states]] are described by 2d [[CFT]], which is well understood. After passage to the corresponding strongly coupled [[black brane]] configurations in [[type IIB supergravity]], which are [[black hole]]s in the given [[Kaluza-Klein mechanism|compactification]], the [[entropy]] of these 2d CFTs matches the [[Bekenstein-Hawking entropy]] of these black holes.

This is parts of the [[AdS/CFT correspondence]]. See ([AGMOO, chapter 5](#AGMOO)).

## Related concepts

[[!include table of branes]]

## References

The [[Green-Schwarz sigma-model]] description of $(p,q)$-5-branes via cocycles on [[extended super Minkowski spacetimes]] is obtained in

* {#Sakaguchi00} Makoto Sakaguchi, section 2 of _IIB-Branes and New Spacetime Superalgebras_, JHEP 0004 (2000) 019 ([arXiv:hep-th/9909143](https://arxiv.org/abs/hep-th/9909143))

See also

* {#AGMOO} [[Ofer Aharony]], S.S. Gubser, [[Juan Maldacena]], H. Ooguri, Y. Oz, chapter 5.3 of _Large N Field Theories, String Theory and Gravity_ ([arXiv:hep-th/9905111](http://arxiv.org/abs/hep-th/9905111))

[[!redirects D5-branes]]


D5-D9-brane bound state

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## Idea

A [[bound state]] of [[D5-branes]] and [[D9-branes]], a special case of [[Dp-D(p+4)-brane bound states]].

## Properties

### Relation to small instantons

Under [[duality between heterotic and type I string theory]] $D5-D9$ brane bound states translate to [[small instantons]] in [[heterotic string theory]] given by [[wrapped brane|wrapped]] [[NS5-branes]], or rather wrapped [[M5-branes]] in [[heterotic M-theory]] ([Witten 96](#Witten96), [Gimon-Polchinski 96](#GimonPolchinski96))

## Related concepts

[[!include brane bound states -- table]]

## References

* {#Witten96} [[Edward Witten]], _Small Instantons in String Theory_, Nucl. Phys. B460:541-559, 1996 ([arXiv:hep-th/9511030](https://arxiv.org/abs/hep-th/9511030))

* {#GimonPolchinski96} [[Eric Gimon]], [[Joseph Polchinski]], _Consistency Conditions for Orientifolds and D-Manifolds_, Phys. Rev. D54:1667-1676, 1996 ([arxiv:hep-th/9601038](https://arxiv.org/abs/hep-th/9601038))

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D6

> This entry is about items in the [[ADE-classification]] labeled by $D6$. For the [[D6-brane]], see there.

***

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## Idea

In the [[ADE-classification]], the items labeled $D6$ include the following:

1. as [[finite subgroups of SO(3)]]:

   the [[dihedral group of order 8]]

   $\mathbb{D}_8$

1. as [[finite subgroups of SU(2)]]:

   the [[binary dihedral group of order 16]]

   $2 D_8$

1. as [[simple Lie groups]]: the [[special orthogonal group]]/[[spin group]] in 12 dimensions

   [[SO(12)]], [[Spin(12)]]

1. as a [[Dynkin diagram]]/[[Dynkin quiver]]:

\begin{tikzpicture}
  \node (center) at (0,0) {};
  \node (topright) at (60:1) {};
  \node (botright) at (-60:1) {};
  \node (left) at (-1,0) {};
  \node (D5) at (-2,0) {};
  \node (D6) at (-3,0) {};

  \draw[fill=black] (center) circle (.1);
  \draw[fill=black] (topright) circle (.1);
  \draw[fill=black] (botright) circle (.1);
  \draw[fill=black] (left) circle (.1);

  \draw[fill=black] (D5) circle (.1);
  \draw[fill=black] (D6) circle (.1);

  \draw (center) to (topright);
  \draw (center) to (botright);
  \draw (center) to (left);

  \draw (D5) to (left);
  \draw (D6) to (D5);
\end{tikzpicture}

## Related concepts

[[!include ADE -- table]]


D6-brane

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## Idea

The [[D-brane]] of [[dimension]] 6+1 in [[type IIA string theory]].

## Relation to other branes
 {#RelationToOtherBranes}

### To the KK-monopole in M-theory

The lift of the D6 to [[M-theory]] is the [[Kaluza-Klein monopole]] there ([Townsend 95](#Townsend95)).

One way to understand this is as follows: the [[first Chern class]] of the M-theory [[circle bundle]] is the [[D0-brane]] degree-2 [[RR-field|RR-]][[field strength]] $R_0$. By the [[self-dual higher gauge field|self-duality]] of the [[RR-fields]] this is the [[Hodge duality|Hodge dual]] of the D6-brane degree-8 field strength. Now given a  [[Kaluza-Klein monopole]] after removing the locus of the monopole where the circle-[[fiber]] degenerates (as in the original [[Dirac charge quantization]] argument) it is a non-degenerated circle bundle such that the [[integration of differential forms|integral]] of the 2-form $R_0$ over any [[2-sphere]] surrounding the singular locus is non-trivial. By the usual yoga of [[electric-magnetic duality]] this integral measures the [[magnetic flux]] of a D6-brane sitting at the singular locus.
For more on this see at _[Kaluza-Klein monopole -- Relation to the D6-brane](Kaluza-Klein+monopole#RelationToTheD6Brane)_, and see at _[[M-theory lift of gauge enhancement on D6-branes]]_.

### To D7-branes

The [[T-duality|T-dual]] of the D6 branes is the [[D7-brane]] in [[type IIB string theory]].

### Bound states

[[bound states]]/[[brane intersections]] involving D6-branes:

* [[D6-D8 brane bound state]]

* [[D2-D6 brane bound state]]

* [[D4-D6 brane bound state]]

### Table of branes

[[!include F-branes -- table]]

## Phenomenology -- Intersecting D-brane models

D6-branes are the main ingredient in type IIA [[string phenomenology]] in the guise of [[intersecting D-brane models]], see also at [[string phenomenology]] the section _[Models in type II with intersecting branes](string+phenomenology#ModelsInTypeIIWithIntersectingBranes)_.

<img src="https://ncatlab.org/nlab/files/D6BranesIntersectingAtAnAngle.png" width="650"/>

> graphics grabbed from [Ibáñez-Uranga 12](#IbanezUranga12)

## Related concept

[[!include table of branes]]

## References

The relation to the [[Kaluza-Klein monopole]] in [[M-theory]] is due to

* {#Townsend95} [[Paul Townsend]], _The eleven-dimensional supermembrane revisited_, Phys.Lett.B350:184-187,1995 ([arXiv:hep-th/9501068](http://arxiv.org/abs/hep-th/9501068))

Further discussion and derivation of the [[Chan-Paton gauge field]] content on the D6 from the KK-monopole is discussed in

* [[César Gómez]], [[Juan José Manjarín]], _A note on the dyonic D6-brane_, 6th International Workshop on Conformal Field Theory and Integrable Models, Landau Institute, Sept. 2002 ([arXiv:hep-th/0302096](http://arxiv.org/abs/hep-th/0302096))

* [[Juan José Manjarín]], _Topics on D-brane charges with B-fields_, Int. J. Geom. Meth. Mod. Phys. 1 (2004) ([arXiv:hep-th/0405074](http://arxiv.org/abs/hep-th/0405074))

Discussion of D6 [[intersecting branes]] for [[intersecting D-brane models]]:

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge 2012

* [[Angel Uranga]], _Model building in IIA: Intersecting brane worlds_, 2012 ([pdf](http://scgp.stonybrook.edu/wp-content/uploads/2012/05/Uranga-Lecture-Notes-1.pdf))

[[!redirects D6-branes]]


D6-D8-brane bound state

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## Idea

The special case of [[Dp-D(p+2)-brane bound state]] for [[D6-brane|D6]]-[[D8-brane]] [[bound states]].

## Properties

### Flux density and Hanany Witten effect

See [this section](Hanany-Witten+effect#D6StretchingBetweenNS5AndD8) at *[[Hanany-Witten effect]]*.

### Fuzzy funnel geometry
 {#FuzzyFunnelGeometry}

The [[fuzzy funnel]]/[[fuzzy 2-sphere]] [[noncommutative geometry]] of $D6\BOT d8$-brane intersections, from the point of the of the [[D6-brane]] non-abelian [[DBI-action]] ([Gaiotto-Tomasiello 14](#GaiottoTomasiello14), [Fazzi 17, p. 41-46](#Fazzi17)):

> graphics grabbed from [Fazzi 17, Fig. 3.14](#Fazzi17), taken in turn from [Gaiotto-Tomasiello 14, Figure 5](#GaiottoTomasiello14)

> graphics grabbed from [Gaiotto-Tomasiello 14](#GaiottoTomasiello14)

### The s-rule

By the [[s-rule]] a [[supersymmetry|supersymmetric]] configurations of [[D6-branes]] stretching between [[NS5-branes]] and [[D8-branes]] has at most one [[D6-brane]] ending on any [[D8-brane]]:

> graphics grabbed from [Fazzi 17](#Fazzi17)

> graphics grabbed from [Gaiotto-Tomasiello 14](#GaiottoTomasiello14)

> graphics grabbed from [GKSTY 02](NS5-brane#GKSTY02)

### As QCD Monopoles in the Witten-Sakai-Sugimoto model
 {#AsQCDMnopoles}

Generally [[Dp-D(p+2)-brane intersections]] [[geometric engineering of QFT|geometrically engineer]] [[Yang-Mills monopoles]] in the [[worldvolume]] of the higher dimensional D$(p+2)$-branes.

Specifically for $p = 6$, i.e. for [[D6-D8-brane intersections]], this fits with the [[Witten-Sakai-Sugimoto model]] [[geometric engineering of QFT|geometrically engineering]] [[quantum chromodynamics]], and then gives a [[geometric engineering of QFT|geometric engineering]] of the [[Yang-Mills monopoles]] in actual [[QCD]] ([HLPY 08, p. 16](#HLPY08)).

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

Here we are showing

1. the [[color charge|color]] [[D4-branes]];

1. the [[flavour (particle physics)|flavor]] [[D8-branes]];

   with

   1. the [[5d Chern-Simons theory]] on their [[worldvolume]]

   1. the corresponding [[4d WZW model]] on the boundary

   both exhibiting the [[meson fields]]

1. the [[baryon]] [[D4-branes]]

   (see below at _[WSS -- Baryons](Witten-Sakai-Sugimoto+model#Baryons)_)

1. the [[Yang-Mills monopole]] [[D6-branes]]

   (see at _[[D6-D8-brane bound state]]_)

1. the [[NS5-branes]].

## Related concepts

[[!include brane bound states -- table]]

## References

### As black brane solutions

On [[D6-D8-brane bound states]] as [[black brane]]-solutions in [[massive type IIA string theory]]:

* [[Harvendra Singh]], _Duality symmetric massive type II theories in $D=8$ and $D=6$_, JHEP 0204:017, 2002 ([arXiv:hep-th/0109147](https://arxiv.org/abs/hep-th/0109147))

* [[Harvendra Singh]], _Note on (D6,D8) Bound State, Massive Duality and Non-commutativity_, Nucl. Phys. B661 (2003) 394-408 ([arXiv:hep-th/0212103](https://arxiv.org/abs/hep-th/0212103))

* Fabio Apruzzi, [[Marco Fazzi]], Dario Rosa, [[Alessandro Tomasiello]],  _All $AdS_7$ solutions of type II supergravity_, JHEP 04 (2014) 064 ([arxiv:1309.2949](https://arxiv.org/abs/1309.2949))

With emphasis on [[charge quantization]] of the [[RR-field]] [[flux]] forms via [[bundle gerbes]]:

* [[Jussi Kalkkinen]], _Gerbes and Massive Type II Configurations_, JHEP 9907  :002, 1999 ([arXiv:hep-th/9905018](https://arxiv.org/abs/hep-th/9905018))

With emphases of the [[fuzzy funnel]] [[noncommutative geometry]]:

* {#GaiottoTomasiello14} [[Davide Gaiotto]], [[Alessandro Tomasiello]], _Holography for $(1,0)$ theories in six dimensions_, JHEP 12 (2014) 003 ([arXiv:1404.0711](https://arxiv.org/abs/1404.0711))

* {#Fazzi17} [[Marco Fazzi]], p. 41-46 of: _Higher-dimensional field theories from type II supergravity_ ([arxiv:1712.04447](https://arxiv.org/abs/1712.04447))

In view of [[AdS/CFT]]:

* Fabio Apruzzi, Giuseppe Dibitetto, Luigi Tizzano, _A new 6d fixed point from holography_, JHEP 11 (2016) 126 ([arxiv:1603.06576](https://arxiv.org/abs/1603.06576))

* Nikolay Bobev, [[Giuseppe Dibitetto]], Fridrik Freyr Gautason, Brecht Truijen, _Holography, Brane Intersections and Six-dimensional SCFTs_, JHEP 1702 (2017) 116 ([arxiv:1612.06324](https://arxiv.org/abs/1612.06324))

[[!include AdS3-CFT2 on D2-D4-D6-D8 branes -- references]]

The [[T-duality]] to [[AdS2/CFT1]]:

* [[Yolanda Lozano]], [[Carlos Nunez]], [[Anayeli Ramirez]], Stefano Speziali, _New $AdS_2$ backgrounds and $\mathcal{N}=4$ Conformal Quantum Mechanics_ ([arXiv:2011.00005](https://arxiv.org/abs/2011.00005))

### Hanany-Witten effect

On the [[Hanany-Witten effect]] for NS5/D6/D8-branes:

* [[Amihay Hanany]], [[Alberto Zaffaroni]], §2.4 *Chiral Symmetry from Type IIA Branes*, Nucl. Phys. B **509** (1998) 145-168 [[arXiv:hep-th/9706047](https://arxiv.org/abs/hep-th/9706047), doi:10.1016/S0550-3213(97)00595-6]

* {#BergshoeffLozanoOrtin98} [[Eric Bergshoeff]], [[Yolanda Lozano]], [[Tomas Ortin]], p. 60 of: *Massive Branes*, Nucl. Phys. B **518** (1998) 363-423 [[arXiv:hep-th/9712115](https://arxiv.org/abs/hep-th/9712115), doi:10.1016/S0550-3213(98)00045-5]

### In geometric engineering of monopoles

For transversal [[D6-D8-brane bound states]] with an eye towards describing [[Yang-Mills monopoles]]:

with an emphasis of [[half NS5-brane]] in [[type I' string theory]]:

* {#HananyZaffaroni99} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Monopoles in String Theory_, JHEP 9912 (1999) 014 ([arxiv:hep-th/9911113](https://arxiv.org/abs/hep-th/9911113))

with an emphasis on [[holographic QCD]]:

* {#HLPY08} Deog Ki Hong, Ki-Myeong Lee, Cheonsoo Park, Ho-Ung Yee, Section V of: _Holographic Monopole Catalysis of Baryon Decay_, JHEP 0808:018, 2008 ([arXiv:0804.1326](https://arxiv.org/abs/0804.1326))

For more see the references at _[[Dp-D(p+2)-brane bound states]]_.

### Relation to Seiberg-Witten curves

On [[quantum Seiberg-Witten curves]] of [[E-string]]-theories in relation to [[D6-D8-brane bound states]]:

* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], [[Xin Wang]], *E-string Quantum Curve*, Nuclear Physics B **973** (2021) 115602 [[arXiv:2103.16996](https://arxiv.org/abs/2103.16996), [doi:10.1016/j.nuclphysb.2021.115602](https://doi.org/10.1016/j.nuclphysb.2021.115602)]

[[!redirects D6-D8-brane bound states]]

[[!redirects D6-D8 brane intersection]]
[[!redirects D6-D8 brane intersections]]
[[!redirects D6-D8-brane intersection]]
[[!redirects D6-D8-brane intersections]]

[[!redirects D6-D8 brane bound state]]
[[!redirects D6-D8 brane bound states]]
[[!redirects D6-D8 brane intersection]]
[[!redirects D6-D8 brane intersections]]

[[!redirects D6/D8-brane bound state]]
[[!redirects D6/D8-brane bound states]]
[[!redirects D6/D8-brane intersection]]
[[!redirects D6/D8-brane intersections]]

[[!redirects D6/D8 brane bound state]]
[[!redirects D6/D8 brane bound states]]
[[!redirects D6/D8 brane intersection]]
[[!redirects D6/D8 brane intersections]]

[[!redirects D8-D6-brane bound state]]
[[!redirects D8-D6-brane bound states]]
[[!redirects D8-D6-brane intersection]]
[[!redirects D8-D6-brane intersections]]

[[!redirects D8-D6 brane bound state]]
[[!redirects D8-D6 brane bound states]]
[[!redirects D8-D6 brane intersection]]
[[!redirects D8-D6 brane intersections]]

[[!redirects D8/D6-brane bound state]]
[[!redirects D8/D6-brane bound states]]
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[[!redirects D8/D6-brane intersections]]

[[!redirects D8/D6 brane bound state]]
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[[!redirects D8/D6 brane intersections]]


D7-brane

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## Idea

The _D7-brane_ is the [[D-brane]] in [[type IIB string theory]] with [[worldvolume]] of [[dimension]] $7+1$, hence of [[codimension]] $10 - (7+1) = 2$. This is the prototypical _[[defect brane]]_.

In [[F-theory]] these are the degenration loci of the [[axio-dilaton]] [[elliptic fibration]], where the [[discriminant]] $\Delta$ of the [[elliptic curve]] fibers vanishes and the fiber degenrates to the [[nodal curve]] (e.g. [Sen 97](#Sen97), [Blumenhagen 10, (11)](#Blumenhagen10)).

## Properties

### Bound states

* [[D1-D7 brane bound state]]

* [[D3-D7 brane bound state]]

### Relation to other branes

[[!include F-branes -- table]]

## Related concepts

[[!include table of branes]]

See also _[[F-theory]]_.

## References

### General

As a [[black brane]] the D7 is identified in

* {#BergshoeffRooGreenPapadopoulosTownsend96} [[Eric Bergshoeff]], [[Mees de Roo]], [[Michael Green]], [[George Papadopoulos]], [[Paul Townsend]], _Duality of Type II 7-branes and 8-branes_, Nucl. Phys. B **470**  (1996) 113-135 [[arXiv:hep-th/9601150](https://arxiv.org/abs/hep-th/9601150)]

The [[Green-Schwarz sigma-model]] description of $(p,q)$-7-branes via cocycles on [[extended super Minkowski spacetimes]] is obtained in

* {#Sakaguchi00} Makoto Sakaguchi, section 2 of _IIB-Branes and New Spacetime Superalgebras_, JHEP 0004 (2000) 019 ([arXiv:hep-th/9909143](https://arxiv.org/abs/hep-th/9909143))

See also

* Oliver DeWolfe, Tamas Hauer, Amer Iqbal, [[Barton Zwiebach]], _Uncovering Infinite Symmetries on [p,q] 7-branes: Kac-Moody Algebras and Beyond_  ([arXiv:hep-th/9812209](http://arxiv.org/abs/hep-th/9812209))

* {#Sen97} [[Ashoke Sen]], _Orientifold Limit of F-theory Vacua_ ([arXiv:hep-th/9702165](http://arxiv.org/abs/hep-th/9702165))

* {#Blumenhagen10} Blumenhagen, _Basics of F-theory from the Type IIB Perspective_ ([arXiv:1002.2836](http://arxiv.org/abs/1002.2836))

In

* P. Meessen, T. Ortin, _An $Sl(2,\mathbb{Z})$ Multiplet of Nine-Dimensional Type II Supergravity Theories_ ([arXiv:hep-th/9806120](http://arxiv.org/abs/hep-th/9806120))

it is argued that 7-branes in [[type II string theory]] form not just a doublet, but a triplet under [[S-duality]].

### In M-theory

A lift of [[D8-branes]] to [[M-theory]] [[M-branes]], by [[generalized Scherk-Schwarz reduction]], relating to [[D7-branes]] in [[F-theory]], is proposed in:

* {#Hull98} [[Chris Hull]], _Massive String Theories From M-Theory and F-Theory_, JHEP 9811:027, 1998 ([arXiv:hep-th/9811021](https://arxiv.org/abs/hep-th/9811021))

[[!redirects D7-branes]]


D8-brane

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## Idea

The [[D-brane]] in [[type II string theory]] of [[dimension]] $8+1$.

## Properties

### As black branes in massive IIA supergravity

As [[black brane]]-solutions, D8-branes appear in [[massive type IIA supergravity]] ("[[Romans supergravity]]")/[[massive type IIA string theory]]

([Bergshoess-de Roo-Green-Papadopoulos-Townsend 96](#BergshoessdeRooGreenPapadopoulosTownsend96), [Chamblin-Perry 97](#ChamblinPerry97), [Janssen-Meessen-Ortin 99](#JanssenMeessenOrtin99))

Since the [[normal bundle|normal]] [[n-sphere]] around a [[D8-brane]] (with its 9-dimensional [[worldvolume]]) in 10-dimensional [[spacetime]] is a [[0-sphere]], [[D-brane charge|D8-brane charge]] is measured by the [[RR-field]] 0-[[flux form]] $F_0$. This behaves like a [[cosmological constant]] in the corresponding [[D=10 supergravity]] ("[[Romans supergravity]]"). Its presences changes ordinary [[type IIA string theory]] to what is called _[[massive type IIA string theory]]_, whose relation to the web of [[dualities in string theory]] remains not well understood. In particular the [[duality between type IIA string theory and M-theory]] remains ill-understood for [[massive type IIA string theory]].

### Bound states

For the [[bound states]] and [[brane intersections]] involving D8-branes, see at

* _[[D4-D8-brane bound states]]_

* _[[D6-D8-brane bound states]]_

### As geometric engineering of flavour physics

In [[geometric engineering of QFT|geometric engineering]] of [[flavor physics]] in [[intersecting D-brane models]], the flavour degrees of freedom come from [[open strings]] ending on [[spacetime]]-filling [[D-branes]] ([Karch-Katz 02](#KarchKatz02)).

Specifically in the [[Sakai-Sugimoto model]] [[geometric engineering of QFT|geometrically engineering]] something close to actual [[quantum chromodynamics]] ([Sakai-Sugimoto 04](#SakaiSugimoto04), [Sakai-Sugimoto 05](#SakaiSugimoto05)), flavour is encoded in the presence of [[D8-branes]] in the model:

Here we are showing

1. the [[color charge|color]] [[D4-branes]];

1. the [[flavour (particle physics)|flavor]] [[D8-branes]];

   with

   1. the [[5d Chern-Simons theory]] on their [[worldvolume]]

   1. the corresponding [[4d WZW model]] on the boundary

   both exhibiting the [[meson fields]];

1. the [[baryon]] [[D4-branes]]

   (see at _[WSS-model -- Baryons](WSS+model#Baryons)_);

1. the [[Yang-Mills monopole]] [[D6-branes]]

   (see at _[[D6-D8-brane bound state]]_);

1. the [[NS5-branes]] (often not considered here).

## Related concepts

* [[D4-D8 brane bound state]]

[[!include table of branes]]

[[!redirects D8-branes]]

## References
 {#References}

### As black brane solutions

As a [[black brane]] the D8 was identified as a solution to [[Romans supergravity]]/[[massive type IIA string theory]] in

* {#BergshoeffdeBooGreenPapadopoulosTownsend96} [[Eric Bergshoeff]], [[Mees de Roo]], [[Michael Green]], [[George Papadopoulos]], [[Paul Townsend]], _Duality of Type II 7-branes and 8-branes_, Nucl. Phys. B **470** (1996) 113-135 [[arXiv:hep-th/9601150](https://arxiv.org/abs/hep-th/9601150)]

See also

* {#ChamblinPerry97} A. Chamblin, [[Malcolm Perry]], _Dynamic D8-branes in IIA string theory_ ([arXiv:hep-th/9712112](https://arxiv.org/abs/hep-th/9712112))

* {#JanssenMeessenOrtin99} [[Bert Janssen]], [[Patrick Meessen]], [[Tomas Ortin]], _The D8-brane tied up: String and brane solutions in massive type IIA supergravity_, Phys. Lett. B453 (1999) 229-236 ([spire:494174](http://inspirehep.net/record/494174), doi:10.1016/S0370-2693(99)00315-9)

On [[D6-D8-brane bound states]] in [[massive type IIA string theory]]:

* [[Harvendra Singh]], _Duality symmetric massive type II theories in $D=8$ and $D=6$_, JHEP 0204:017, 2002 ([arXiv:hep-th/0109147](https://arxiv.org/abs/hep-th/0109147))

* [[Harvendra Singh]], _Note on (D6,D8) Bound State, Massive Duality and Non-commutativity_, Nucl. Phys. B661 (2003) 394-408 ([arXiv:hep-th/0212103](https://arxiv.org/abs/hep-th/0212103))

* Fabio Apruzzi, Marco Fazzi, Dario Rosa, [[Alessandro Tomasiello]],  _All $AdS_7$ solutions of type II supergravity_, JHEP 04 (2014) 064 ([arxiv:1309.2949](https://arxiv.org/abs/1309.2949))

### As geometric engineering of flavor physics
 {#ReferencesAsGeometricEngineeringOfFlavorPhysics}

[[geometric engineering of QFT|geometric engineering]] of [[flavour physics]] on spacetime-filling [[D-branes]] in [[intersecting D-brane models]] ([[AdS/QCD]]) was originally understood in

* {#KarchKatz02} [[Andreas Karch]], [[Emanuel Katz]],  _Adding flavor to AdS/CFT_, JHEP 0206:043, 2002 ([arxiv:hep-th/0205236](https://arxiv.org/abs/hep-th/0205236))

and then developed in detail for [[QCD]] via [[D8-branes]] in the [[Sakai-Sugimoto model]]:

* {#SakaiSugimoto04} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Low energy hadron physics in holographic QCD_, Prog.Theor.Phys.113:843-882, 2005 ([arXiv:hep-th/0412141](https://arxiv.org/abs/hep-th/0412141))

* {#SakaiSugimoto05} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _More on a holographic dual of QCD_, Prog.Theor.Phys.114:1083-1118, 2005 ([arXiv:hep-th/0507073](https://arxiv.org/abs/hep-th/0507073))

### Relation to K-theory

The role of the D8 in the [[K-theory]]-classification of [[D-branes]] is discussed in

* [[John Schwarz]], sections 6.2 and 6.3 of _TASI Lectures on Non-BPS D-Brane Systems_ ([arXiv:hep-th/9908144](https://arxiv.org/abs/hep-th/9908144))

* Ion V. Vancea, _On the Algebraic K-theory of The Massive D8 and M9 Branes_,   Int. J. Mod. Phys. A16 (2001) 4429-4452 ([arXiv:hep-th/9905034](https://arxiv.org/abs/hep-th/9905034))

### Lift to M-theory

A lift of [[D8-branes]] to [[M-theory]] [[M-branes]] by [[generalized Scherk-Schwarz reduction]], relating to [[D7-branes]] in [[F-theory]], is proposed in

* {#Hull98} [[Chris Hull]], _Massive String Theories From M-Theory and F-Theory_, JHEP 9811:027, 1998 ([arXiv:hep-th/9811021](https://arxiv.org/abs/hep-th/9811021))

Relation of [[heterotic M-theory]] to [[type I' string theory]] with [[D8-branes]] at [[O8-planes]], via a sequence of [[dualities in string theory]]:

* [[Ofer Aharony]], Zohar Komargodski, Assaf Patir, _The Moduli Space and M(atrix) Theory of 9d $\mathcal{N} =1$ Backgrounds of M/String Theory_ ([arXiv:hep-th/0702195](https://arxiv.org/abs/hep-th/0702195))


D9-brane

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## Idea

The [[D-brane]] in [[type IIB superstring theory]] of [[dimension]] 9+1.

## Properties

### Bound states

[[bound states]]:

* [[D5-D9 brane bound state]]

## Related concepts

[[!include table of branes]]

## References

In

* {#Hull97} [[Chris Hull]], _Gravitational Duality, Branes and Charges_ ([arXiv:hep-th/9705162](https://arxiv.org/abs/hep-th/9705162))

the existence of D9-branes is argued from [[BPS state]] considerations. The [[S-duality]] doublet 9-brane in [[type II string theory]] "$(p,q)$-9-brane" is mentioned in section 6.

The [[Green-Schwarz sigma-model]] description of $(p,q)$-9-branes via cocycles on [[extended super Minkowski spacetimes]] is obtained in

* {#Sakaguchi00} Makoto Sakaguchi, section 2 of _IIB-Branes and New Spacetime Superalgebras_, JHEP 0004 (2000) 019 ([arXiv:hep-th/9909143](https://arxiv.org/abs/hep-th/9909143))

Cohomological discussion (maybe mostly of [[M9-branes]]) is in

* {#Sati14} [[Hisham Sati]], _Ninebrane structures_ ([arXiv:1405.7686](http://arxiv.org/abs/1405.7686))

[[!redirects D9-branes]]

[[!redirects D9]]


D=10 super Yang-Mills theory

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## Idea

[[super Yang-Mills theory]] in [[spacetime]] [[dimension]] 10.

## Properties

### Equations of motion from the Bianchi identity
 {#EquationsOfMotionFromTheBianchiIdentity}

The [[equations of motion]] of 10d super Yang-Mills happen to be equivalent in [[superspace]] to the [[Bianchi identity]]

$$
  D F = 0
$$

subject to the constraint that the bispinorial part of the [[curvature]] 2-form vanishes

$$
  F_{\alpha \beta}
   = 0
$$

(e.g. [Witten 86](#Witten86), [Atick-Dhar-Ratra 86, (4.14)](#AtickDharRatra86), [Bonora-Bregola-Lechner-Pasti-Tonin 87, above (2.7)](#BonoraBregolaLechnerPastiTonin87)).

After embedding into [[heterotic supergravity]] this becomes parts of the [[torsion constraints of supergravity]]. See [there](torsion+constraints+in+supergravity#HeteroticSupergravity).

In this [[superspace]] formulation the [[gaugino]] $\chi$ appears as the even-odd component of the super-[[curvature]] form

$$
  \label{EvenOddFieldStrengthViaGaugino}
  F_{(1,1)}
    \;\propto\;
  \left(\overline{\psi} \Gamma_a \chi\right) \wedge e^a
$$

(where $(e^a, \psi^\alpha)$ is the [[super vielbein]]). This is ([Witten 86 (8)](#Witten86), [Bonora-Pasti-Tonin 87, below (11)](#BonoraPastiTonin87), [Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.27)](#BonoraBregolaLechnerPastiTonin87)).

### Compactification to the point

The [[KK-compactification]] of $D=10$ super-Yang-Mills to the point is a [[theory (physics)|theory]] whose [[field (physics)|fields]] are simply elements of the gauge [[Lie algebra]], hence [[matrices]] if we have a [[matrix Lie algebra]]. The theory defined by this reduction is called the _[[IKKT matrix model]]_.

## References

* {#Nilsson81} [[Bengt Nilsson]], _Off-shell fields for the 10-dimensional supersymmetric Yang–Mills theory_, Feb 1981 ([Spire](http://inspirehep.net/record/164407/))

* {#AtickDharRatra86} [[Joseph Atick]], Avinash Dhar, and Bharat Ratra, _Superspace formulation of ten-dimensional $N=1$ supergravity coupled to $N=1$ super Yang-Mills theory_, Phys. Rev. D 33, 2824, 1986 ([doi.org/10.1103/PhysRevD.33.2824](https://doi.org/10.1103/PhysRevD.33.2824))

* {#Witten86} [[Edward Witten]], _Twistor-like transform in ten dimensions_, Nuclear Physics B Volume 266, Issue 2, 17 March 1986 ([spire:214192](http://inspirehep.net/record/214192), doi:10.1016/0550-3213(86)90090-8)

* {#BonoraPastiTonin87} [[Loriano Bonora]], [[Paolo Pasti]], [[Mario Tonin]], _Superspace formulation of 10D SUGRA+SYM theory a la Green-Schwarz_, Physics Letters B Volume 188, Issue 3, 16 April 1987, Pages 335–339 (doi:10.1016/0370-2693(87)91392-X)

* {#BonoraBregolaLechnerPastiTonin87} [[Loriano Bonora]], M. Bregola, [[Kurt Lechner]], [[Paolo Pasti]], [[Mario Tonin]], _Anomaly-free supergravity and super-Yang-Mills theories in ten dimensions_, Nuclear Physics B Volume 296, Issue 4, 25 January 1988 (doi:10.1016/0550-3213(88)90402-6)

* [[Martin Cederwall]], [[Bengt Nilsson]], [[Dimitrios Tsimpis]], _The structure of maximally supersymmetric Yang-Mills theory: constraining higher-order corrections_, JHEP 0106:034,2001 ([arXiv:hep-th/0102009](http://arxiv.org/abs/hep-th/0102009))

* [[Martin Cederwall]], [[Bengt Nilsson]], [[Dimitrios Tsimpis]], _$D=10$ Super-Yang-Mills at $O(\alpha'^2)$_, JHEP 0107:042,2001 ([arXiv:hep-th/0104236](http://arxiv.org/abs/hep-th/0104236))

* [[Mikhail Movshev]], [[Albert Schwarz]], _On maximally supersymmetric Yang-Mills theories_, Nucl. Phys. B681 (2004) 324-350 ([arXiv:hep-th/0311132](https://arxiv.org/abs/hep-th/0311132))

On [[scattering amplitudes]] in [[pure spinor]]-formalism:

* Maor Ben-Shahar, [[Max Guillen]], *10D Super-Yang-Mills Scattering Amplitudes From Its Pure Spinor Action* ([arXiv:2108.11708](https://arxiv.org/abs/2108.11708))

[[!redirects 10d super Yang-Mills]]

[[!redirects 10d super Yang-Mills theory]]

[[!redirects 10d SYM]]


D=10 supergravity

[[!redirects 10-dimensional supergravity]]

10-[[dimension|dimensional]] [[supergravity]]

* [[type II supergravity]]

  * [[type IIA supergravity]]

    * [[massive type IIA supergravity]]

  * [[type IIB supergravity]]

* [[heterotic supergravity]]

## Related concepts

* [[12-dimensional supergravity]]

* [[11-dimensional supergravity]]

* [[9-dimensional supergravity]]

* [[8-dimensional supergravity]]

* [[7-dimensional supergravity]]

* [[6-dimensional supergravity]]

* [[5-dimensional supergravity]]

* [[4-dimensional supergravity]]

* [[3-dimensional supergravity]]

* [[2-dimensional supergravity]]

[[!redirects 10-dimensional supergravity theories]]