D=11 N=1 supergravity

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#Contents#
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## Idea

$N=1$ [[supergravity]] in $d = 11$.

> for the moment see the respective section at [[D'Auria-Fre formulation of supergravity]]

## The action functional

(...)

### Kinetic terms

(...)

### The higher Chern-Simons term

> under construction

$$
  \int_X
  \left(
  \frac{1}{6}
  \left(
    C \wedge G \wedge G
    -
    C \wedge
    \frac{1}{8}
    \left(
      p_2 + (\frac{1}{2}p_1)^2
    \right)
  \right)
  \right)
$$

where $p_i$ is the $i$th [[Pontryagin class]].

$$
  \lambda \coloneqq \frac{1}{2}p_1
  \,.
$$

Concerning the integrality of the [[I8]]-term

$$
  I_8 \coloneqq \frac{1}{48}(p_2 + (\lambda)^2)
$$

on a [[spin structure|spin manifold]] $X$.
([Witten96, p.9](#Witten96))

First, the [[index]] of a [[Dirac operator]] on $X$ is

$$
  I = \frac{1}{1440}(7 (\frac{1}{2}p_1)^2 - p_2) \in \mathbb{Z}
  \,.
$$

Notice that $1440 = 6 \times 8 \times 30$. So

$$
  p_2 - (\frac{1}{2}p_2)^2 = 6 ( (\frac{1}{2}p_1)^2 - 30 \times 8 I)
$$

is divisible by 6.

Assume that $(\frac{1}{2}p_1)$ is further divisible by 2 (see the relevant discussion at _[[M5-brane]]_).

$$
  (\frac{1}{2}p_1) = 2 x
  \,.
$$

Then the above becomes

$$
  p_2 - (\frac{1}{2}p_2)^2 = 24 ( x^2 - 30 \times 2 I)
$$

and hence then $p_2 + (\frac{1}{2}p_1)^2$ is divisible at least by 24.

But moreover, on a [[spin structure|Spin manifold]]
the [[first fractional Pontryagin class]] $\frac{1}{2}p_1$ is
the [[Wu class]] $\nu_4$ (see there). By definition this means that

$$
  x^2 = x (\frac{1}{2}p_1) \; mod \; 2
$$

and so when $(\frac{1}{2}p_1)^2$ is further divisible by 2 we have that
$p_2 - (\frac{1}{2}p_1)^2$ is divisible by 48. Hence $I_8$ is integral.

### Higher curvature corrections
 {#HigherCurvatureCorrection}

Possible [[higher curvature corrections]] to 11-dimensional supergravity are discussed in the references listed [below](#ReferencesOnHigherCurvatureCorrectionsIn11dSupergravity).

The first correction is an $R^4$-term at order $\ell^3_{P}$ (11d [[Planck length]]). In [Tsimpis 04](#Tsimpis04) it is shown that part of this is a topological term (total derivative) which relates to the [[shifted C-field flux quantization]].

For effects of higher curvature corrections in a [[Starobinsky model of cosmic inflation]] see [there](Starobinsky+model+of+cosmic+inflation#EmbeddingIntoSupergravity).

### The hidden deformation

There is in fact a hidden 1-parameter deformation of the Lagrangian of 11d sugra.
Mathematically this was maybe first noticed in ([D'Auria-Fre 82 ](#DAuriaFre)) around equation (4.25). This shows that there is a topological term which may be expressed as

$$
  \propto
  \;
  \textstyle{\int}_{X_11} G_4 \wedge G_7
$$

where $G_4$ is the [[curvature]] 3-form of the [[supergravity C-field]] and $G_7$ that of the [[electric-magnetic duality|magnetically dual]] [[C6-field]]. However, ([D'Auria-Fre 82 ](#DAuriaFre)) consider only topologically trivial (trivial [[instanton sector]]) configurations of the  [[supergravity C-field]], and since on them this term is a total derivative, the authors "drop" it.

The term then re-appears in the literatur in ([Bandos-Berkovits-Sorokin 97, equation (4.13)](#BandosBerkovitsSorokin97)).
And it seems that this is the same term later also redicovered around equation (4.2) in ([Tsimpis 04](#Tsimpis04)).

> (hm, check)

### BPS states

The basic [[BPS states]] of 11d SuGra are

* the [[M2-brane]]

* the [[M5-brane]]

* the [[M-wave]]

* the [[Kaluza-Klein monopole]]

(e.g. [EHKNT 07](#EHKNT07))

## Related concepts

* [[supergravity]]

* [[M-theory]], [[F-theory]]

* [[D=12 supergravity]], [[D=14 supersymmetry]]

* 10-dimensional [[type II supergravity]], [[heterotic supergravity]]

* [[7-dimensional supergravity]]

* [[5-dimensional supergravity]]

* [[4-dimensional supergravity]]

  * [[M-theory on G2-manifolds]], [[G2-MSSM]]

  * [[Freund-Rubin compactification]]

* [[3-dimensional supergravity]]

* [[supergravity C-field]], [[supergravity Lie 3-algebra]], [[supergravity Lie 6-algebra]]

* [[Horava-Witten theory]]

* [string theory FAQ -- Does string theory predict supersymmetry?](string+theory+FAQ#DoesSTPredictSupersymmetry)

[[!include table of branes]]

## References

### General

That there is a maximal dimension $d = 11$ in which supergravity may exist was found in

* {#Nahm78} [[Werner Nahm]], _[[Supersymmetries and their Representations]]_, Nucl. Phys. B **135** (1978) 149 [[spire](https://inspirehep.net/record/120988/), [pdf](http://cds.cern.ch/record/132743/files/197709213.pdf)]

The theory was then actually constructed (as a [[Lagrangian field theory]]) in

* {#CremmerJuliaScherk78} [[Eugene Cremmer]], [[Bernard Julia]], [[Joël Scherk]], _Supergravity in theory in 11 dimensions_, Phys. Lett. B **76** (1978) 409 [doi:10.1016/0370-2693(78)90894-8]

Formulation in terms of [[supergeometry]] ("[[superspace]]") is in

* {#CremmerFerrara80} [[Eugene Cremmer]], [[Sergio Ferrara]], *Formulation of Eleven-Dimensional Supergravity in Superspace*, Phys.Lett. B **91** (1980) 61 [doi:10.1016/0370-2693(80)90662-0]

* [[Lars Brink]], [[Paul Howe]], *Eleven-Dimensional Supergravity on the Mass-Shell in Superspace*, Phys.Lett. B **91** (1980) 384 [doi:10.1016/0370-2693(80)91002-3]

The history as of 1990s with an eye towards the development to [[M-theory]] is survey in

* {#Duff99} [[Mike Duff]], chapter I of: _[[The World in Eleven Dimensions]]: Supergravity, Supermembranes and M-theory_, IoP 1999 ([publisher](https://www.crcpress.com/The-World-in-Eleven-Dimensions-Supergravity-supermembranes-and-M-theory/Duff/9780750306720))

The description of 11d supergravity in terms of the [[D'Auria-Fre formulation of supergravity]] originates in

* {#DAuriaFré82} [[Riccardo D'Auria]], [[Pietro Fré]], _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B **201** (1982) 101-140 [doi:10.1016/0550-3213(82)90376-5,  [[GeometricSupergravityErrata.pdf:file]]]

of which a textbook account is in

* {#CastellaniDAuriaFré91} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], chapters III.8 and V.4-V.11 in vol 2 of: _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991) [[doi:10.1142/0224](https://doi.org/10.1142/0224), [epdf](https://epdf.pub/supergravity-and-superstrings-a-geometric-perspective-vol-2-supergravity.html), ch III.8: [[CastellaniDAuriaFre-ChIII8.pdf:file]]]

reviewed again in

* [[Pietro Fré]], §6.4 in: *Gravity, a Geometrical Course*, Volume 2: *Black Holes, Cosmology and Introduction to Supergravity*, Springer (2013) [[doi:10.1007/978-94-007-5443-0](https://doi.org/10.1007/978-94-007-5443-0)]

The topological deformation (almost) noticed in equation (4.25) of [D'Auria-Fre 82](#DAuriaFre82) later reappears in (4.13) of

* {#BandosBerkovitsSorokin97} [[Igor Bandos]], [[Nathan Berkovits]], [[Dmitri Sorokin]], _Duality-Symmetric Eleven-Dimensional Supergravity and its Coupling to M-Branes_, Nucl. Phys. B522 (1998) 214-233 ([arXiv:hep-th/9711055](http://arxiv.org/abs/hep-th/9711055))

and around (4.2) of [Tsimpis 04](#Tsimpis04)

The [[D'Auria-Fre formulation of supergravity|D'Auria-Fré formulation]] is a [[first-order formulation of gravity|first-order formulation]] OF $D=11$ supergravity; for more on this see:

* [[Bernard Julia]], S. Silva, §6 of: *On first order formulations of supergravities*, JHEP 0001 (2000) 026 [[arXiv:hep-th/9911035](https://arxiv.org/abs/hep-th/9911035), [doi:10.1088/1126-6708/2000/01/026](https://doi.org/10.1088/1126-6708/2000/01/026)]

More recent textbook accounts include

* {#VanProeyenFreedman12} [[Antoine Van Proeyen]], [[Daniel Freedman]], Section 10 of: *Supergravity*, Cambridge University Press (2012) [[doi:10.1017/CBO9781139026833]( https://doi.org/10.1017/CBO9781139026833)]

Discussion of the equivalence of the 11d SuGra [[equations of motion]] with the [[supergravity torsion constraints]] is in

* {#Howe97} [[Paul Howe]], _Weyl Superspace_, Physics Letters B Volume 415, Issue 2, 11 December 1997, Pages 149–155 ([arXiv:hep-th/9707184](http://arxiv.org/abs/hep-th/9707184))

following

*  A. Candiello, [[Kurt Lechner]], _Duality in Supergravity Theories_, Nucl.Phys. B412 (1994) 479-501 ([arXiv:hep-th/9309143](http://arxiv.org/abs/hep-th/9309143))

Much computational detail is displayed in

* {#MiemiecSchnakenburg06} [[André Miemiec]], [[Igor Schnakenburg]], *Basics of M-Theory*, Fortsch. Phys. **54** (2006) 5-72 [[arXiv:hep-th/0509137](http://arxiv.org/abs/hep-th/0509137), [doi:10.1002/prop.200510256]( https://doi.org/10.1002/prop.200510256)]

In terms of [[pure spinors]]:

* [[Max Guillen]], _Pure spinors and $D=11$ supergravity_ ([arXiv:2006.06014](https://arxiv.org/abs/2006.06014))

### Duality-symmetric formulation

[[!include self-duality for pregeometric C-field -- references]]

[[!include C-field gauge algebra -- references]]

### Classical solutions and BPS states

Bosonic solutions of eleven-dimensional supergravity were studied in the 1980s in the context of [[Kaluza-Klein mechanism|Kaluza-Klein]] supergravity. The topic received renewed attention in the mid-to-late 1990s as a result of the [[branes]] and duality paradigm and the [[AdS/CFT correspondence]].

One of the earliest solutions of eleven-dimensional supergravity is the maximally supersymmetric [[Freund-Rubin compactification]] with geometry $AdS_4 \times S^7$ and 4-form [[flux]] proportional to the [[volume form]] on $AdS_4$.

* [[Peter Freund]], Mark Rubin, _Dynamics of Dimensional Reduction_ Phys.Lett. B97 (1980) 233-235 ([inSpire](http://inspirehep.net/record/154579?ln=en))

The radii of curvatures of the two factors are furthermore in a ratio of 1:2. The modern avatar of this solution is as the [[near-horizon geometry]] of coincident [[M2-branes]].

* {#DuffStelle91} [[Mike Duff]], [[Kellogg Stelle]], _Multi-membrane solutions of $D = 11$ supergravity_, Phys. Lett. B 253, 113 (1991) ([spire:299386](http://inspirehep.net/record/299386), doi:10.1016/0370-2693(91)91371-2)

Shortly after the original Freund-Rubin solution was discovered, Englert discovered a deformation of this solution where one could turn on [[flux]] on the $S^7$; namely, singling out one of the [[Killing spinors]] of the solution, a suitable multiple of the 4-form one constructs by squaring the [[spinor]] can be added to the [[volume form]] in $AdS_4$ and the resulting 4-form still obeys the supergravity [[Euler-Lagrange equations|field equations]], albeit with a different relation between the radii of [[curvature]] of the two factors. The flux breaks the [[special orthogonal group|SO(8)]] symmetry of the [[sphere]] to an $SO(7)$ subgroup.

* [[Francois Englert]], _Spontaneous Compactification of Eleven-Dimensional Supergravity_ Phys.Lett. B119 (1982) 339 ([inSPIRE](http://inspirehep.net/record/180130))

Some of the above is taken from [this TP.SE thread](http://theoreticalphysics.stackexchange.com/questions/329/modern-avatar-of-englerts-solution).

See also

* {#Page83} [[Don Page]], _Classical stability of round and squashed seven-spheres in eleven-dimensional supergravity_, Phys. Rev. D 28, 2976 (1983) ([spire:14480](http://inspirehep.net/record/14480) [doi:10.1103/PhysRevD.28.2976](https://doi.org/10.1103/PhysRevD.28.2976))

* [[Ergin Sezgin]], _11D Supergravity on $AdS_4 \times S^7$ versus $AdS_7 \times S^4$_ ([arXiv:2003.01135](https://arxiv.org/abs/2003.01135))

A classification of symmetric solutions is discussed in

* [[José Figueroa-O'Farrill]], _Symmetric M-Theory Backgrounds_ ([arXiv:1112.4967](http://arxiv.org/abs/1112.4967))

* Linus Wulff, _All symmetric space solutions of eleven-dimensional supergravity_ ([arXiv:1611.06139](https://arxiv.org/abs/1611.06139))

Discussion of [[black branes]] and [[BPS states]] includes

* {#Stelle98} [[Kellogg Stelle]], section 3 of _BPS Branes in Supergravity_ ([arXiv:hep-th/9803116](http://arxiv.org/abs/hep-th/9803116))

* {#EHKNT07} [[Francois Englert]], Laurent Houart, [[Axel Kleinschmidt]], [[Hermann Nicolai]], Nassiba Tabti, _An $E_9$ multiplet of BPS states_, JHEP 0705:065,2007 ([arXiv:hep-th/0703285](http://arxiv.org/abs/hep-th/0703285))

* Andrew Callister, [[Douglas Smith]], _Topological BPS charges in 10 and 11-dimensional supergravity_, Phys. Rev. D78:065042,2008 ([arXiv:0712.3235](http://arxiv.org/abs/0712.3235))

* Andrew Callister, [[Douglas Smith]], _Topological charges in $SL(2,\mathbb{R})$ covariant massive 11-dimensional and Type IIB SUGRA_, Phys.Rev.D80:125035,2009 ([arXiv:0907.3614](http://arxiv.org/abs/0907.3614))

* Andrew Callister, _Topological BPS charges in 10- and 11-dimensional supergravity_, thesis 2010 ([spire](http://inspirehep.net/record/1221591?ln=en))

* A. A. Golubtsova, V.D. Ivashchuk, _BPS branes in 10 and 11 dimensional supergravity_, talk at DIAS 2013 ([pdf slides](http://theor.jinr.ru/~diastp/summer13/lectures/Golubtsova.pdf))

* Cristine N. Ferreira, _BPS solution for eleven-dimensional supergravity with a conical defect configuration_ ([arXiv:1312.0578](http://arxiv.org/abs/1312.0578))

Discussion of [[black hole]] horizons includes

* [[Jan Gutowski]], [[George Papadopoulos]], _Static M-horizons_ ([arXiv:1106.3085](http://arxiv.org/abs/1106.3085))

See also

* Teng Fei, Bin Guo, Duong H. Phong, _A Geometric Construction of Solutions to 11D Supergravity_ ([arXiv:1805.07506](https://arxiv.org/abs/1805.07506))

Resolution of [[scalar field]]-dressed [[Schwarzschild black holes]] in [[D=11 supergravity]]:

* [[Ibrahima Bah]], Pierre Heidmann, *Geometric Resolution of Schwarzschild Horizon* [[arXiv:2303.10186](https://arxiv.org/abs/2303.10186)]

[[!include higher curvature corrections to D=11 supergravity -- references]]

### Scattering amplitudes and Effective action

Computation of [[Feynman amplitudes]]/[[scattering amplitudes]] and [[effective action]] in 11d supergravity:

* [[Stanley Deser]], [[Domenico Seminara]], _Counterterms/M-theory Corrections to D=11 Supergravity_, Phys.Rev.Lett.82:2435-2438, 1999 ([arXiv:hep-th/9812136](https://arxiv.org/abs/hep-th/9812136))

* [[Stanley Deser]], [[Domenico Seminara]], _Tree Amplitudes and Two-loop Counterterms in D=11 Supergravity_, Phys.Rev.D62:084010, 2000 ([arXiv:hep-th/0002241](https://arxiv.org/abs/hep-th/0002241))

* L. Anguelova, P. A. Grassi, P. Vanhove, _Covariant One-Loop Amplitudes in $D=11$_, Nucl. Phys. B702 (2004) 269-306 ([arXiv:hep-th/0408171](https://arxiv.org/abs/hep-th/0408171))

* [[Kasper Peeters]], [[Jan Plefka]], Steffen Stern, _Higher-derivative gauge field terms in the M-theory action_, JHEP 0508 (2005) 095 ([arXiv:hep-th/0507178](https://arxiv.org/abs/hep-th/0507178))

* Hamid R. Bakhtiarizadeh, _Gauge field corrections to eleven dimensional supergravity via dimensional reduction_ ([arXiv:1711.11313](https://arxiv.org/abs/1711.11313))

### Truncations and compactifications

[[Kaluza-Klein compactifications]] of supergravity and its consistent truncations:

* {#DuffNilssonPope86} [[Mike Duff]], [[Bengt Nilsson]], [[Christopher Pope]], _Kaluza-Klein supergravity_, Physics Reports Volume 130, Issues 1–2, January 1986, Pages 1-142 ([spire:229417](https://inspirehep.net/record/229417), doi:10.1016/0370-1573(86)90163-8)

Discussion of [[Freund-Rubin compactifications]]:

* [[Hermann Nicolai]], [[Krzysztof Pilch]], _Consistent truncation of $d = 11$ supergravity on $AdS_4 \times S^7$_, JHEP 03 (2012) 099 ([arXiv:1112.6131](http://arxiv.org/abs/1112.6131))

### Topology and anomaly cancellation

Discussion of [[quantum anomaly]] cancellation and [[Green-Schwarz mechanism]] in 11D supergravity includes the following articles. (For more see at _[M5-brane -- anomaly cancellation](M5-brane#AnomalyCancellation)_).

* {#Witten96} [[Edward Witten]], _On Flux Quantization In M-Theory And The Effective Action_ ([arXiv:hep-th/9609122](http://arxiv.org/abs/hep-th/9609122))

* {#Witten96} [[Edward Witten]], _Five-Brane Effective Action In M-Theory_, J.Geom.Phys.22:103-133, 1997 ([arXiv:hep-th/9610234](https://arxiv.org/abs/hep-th/9610234))

* {#FreedHarveyMinasianMoore98} [[Dan Freed]], [[Jeff Harvey]], [[Ruben Minasian]], [[Greg Moore]], _Gravitational Anomaly Cancellation for M-Theory Fivebranes_, Adv.Theor.Math.Phys.2:601-618, 1998 ([arXiv:hep-th/9803205](https://arxiv.org/abs/hep-th/9803205))

* {#BilalMetzger03} [[Adel Bilal]], Steffen Metzger, _Anomaly cancellation in M-theory: a critical review_, Nucl.Phys. B675 (2003) 416-446 ([arXiv:hep-th/0307152](https://arxiv.org/abs/hep-th/0307152))

* {#BahBonettiMinasianNardoni18} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Class $\mathcal{S}$ Anomalies from M-theory Inflow_ ([arXiv:1812.04016](https://arxiv.org/abs/1812.04016))

* [[Daniel Freed]], _Two nontrivial index theorems in odd dimensions_ ([arXiv:dg-ga/9601005](http://arxiv.org/abs/dg-ga/9601005))

* [[Adel Bilal]], Steffen Metzger, _Anomaly cancellation in M-theory: a critical review_ ([arXiv:hep-th/0307152](http://arxiv.org/abs/hep-th/0307152))

* {#FreedHopkins21} [[Daniel S. Freed]], [[Michael J. Hopkins]], *Consistency of M-Theory on nonorientable manifolds*, The Quarterly Journal of Mathematics **72** 1-2 (2021) 603–671 [[arXiv:1908.09916](https://arxiv.org/abs/1908.09916), [doi:10.1093/qmath/haab007]( https://doi.org/10.1093/qmath/haab007)]

* {#HanHuangLiuZhang20} [[Fei Han]], [[Ruizhi Huang]], [[Kefeng Liu]], [[Weiping Zhang]], *Cubic forms, anomaly cancellation and modularity*, Advances in Mathematics **394** (2022) 108023 [[arXiv:2005.02344](https://arxiv.org/abs/2005.02344), [doi:10.1016/j.aim.2021.108023](https://doi.org/10.1016/j.aim.2021.108023)]

### Description by exceptional generalized geometry

* Paulo Pires Pacheco, [[Daniel Waldram]], _M-theory, exceptional generalised geometry and superpotentials_ ([arXiv:0804.1362](http://arxiv.org/abs/0804.1362))

Review of [[U-duality]] and [[exceptional generalized geometry]]  in [[KK-compactification]] of [[D=11 supergravity]]:

* [[Henning Samtleben]], *11D Supergravity and Hidden Symmetries*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2303.12682](https://arxiv.org/abs/2303.12682)]

[[!redirects 11d supergravity]]

[[!redirects D=11 supergravity]]

[[!redirects d=11 supergravity]]

[[!redirects 11-dimensional supergravity]]
[[!redirects 11-dimensional supergravity]]


D=12 supergravity

[[!redirects 12-dimensional supergravity]]
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#Contents#
* table of contents
{:toc}

## Idea

There is a sensible [[theory (physics)|theory]] of [[supergravity]] in a total of 12 [[spacetime]] [[dimensions]]. Even though this requires an exotic non-[[Lorentzian signature]] of $(10,2)$ (hence with a "2-dimensional [[time]]") it has been argued that this is a better starting point for obtaining low-dimensional [[supergravity]] theory by [[KK-compactification]], since it yields some lower-dimensional theories that are missed when starting with [[11-dimensional supergravity]], notably [[type IIB supergravity]] in 10 dimensions, hence relates to [[F-theory]] as [[11-dimensional supergravity]] relates to [[M-theory]] (e.g. [Nishino 97b](#Nishino97b), [Hewson 97](#Hewson97)). (A theory in $(9,3)$ signature has also been proposed in ([Kriz 05](#Kriz05)).)

It is an oft-repeated [[folklore]] that the highest number of [[spacetime]] [[dimensions]] for [[supergravity]] to make sense is 11, realized by [[11-dimensional supergravity]]. However, there are some assumptions that go into this conclusion. First of all, the argument goes that after [[KK-compactification]] to 4-dimensions there must not appear [[supermultiplets]] with [[mass]]-less fields of [[spin]] $\gt 2$, since another [[folklore]] argument states that [[quantum field theory]] in $3+1$ dimensions with fields of spin larger than 2 is inconsistent.

(This in turn needs further qualification: Consistent [[quantum field theory]] with an _infinite tower_ of higher spin fields _is_ consistent, this is called _[[higher spin gauge theory]]_ arising as the vanishing [[string tension]]-[[limit of a sequence|limit]] of [[string field theory]]. Ever since this discovery, the modified [[folklore]] is that field theories with a finite number of higher spin fields is inconsistent.)

Since acting with a [[supersymmetry]] generator on elements of a [[supermultiplet]] increases spin by 1/2, this argument requires that there are at most $(2 - (-2)) \times 2 = 8$ super charges in (3+1)d, hence corresponding to [[N=8 d=4 supergravity]].

This, in turn, requires, by the rules of [[KK-compactification]], that

1. there be only a single supercharge in dimension $10+1$, since the [[irreducible representation|irreducible]] [[real spin representation]] of $Spin(10,1)$ has real dimension 32, which [[branching rule|branches]] as $\mathbf{32} \mapsto 8 \cdot \mathbf{4}$ under $Spin(3,1) \hookrightarrow Spin(10,1)$;

1. there cannot be any supercharge in dimension $11+1$, since the [[irreducible representation|irreducible]] [[real spin representation]] of $Spin(11,1)$ has real dimension 64, which [[branching rule|branches]] as $\mathbf{64} \mapsto 16 \cdot \mathbf{4}$ under $Spin(3,1) \hookrightarrow Spin(11,1)$.

However, the second conclusion here is evaded by a change of [[spacetime signature]]: The [[irreducible representation|irreducible]] [[real spin representation]] of [[Spin(10,2)]] still happens to be of dimension 32 and still [[branching rule|branches]] as $\mathbf{32} \mapsto 8 \cdot \mathbf{4}$.

## Properties

### The $2+1$-brane in $10+2$ dimensions
 {#The21Brane}

There is supposed to be a consistent fundamental [[super p-brane]] on $10+2$-dimensional supergravity backgrounds, whose [[double dimensional reduction]] yields the [[M2-brane]] in [[11-dimensional supergravity]] and further the [[superstrings]] not just of [[type IIA supergravity]] but also (?) of [[type IIB supergravity]]. The [[worldvolume]] of this [[p-brane]] has 4 spacetime dimensions with signature $(2,2)$. Therefore some authors refer to this as a "2+2"-brane, even though this does not mesh well with the naming convention of $p$-branes in Lorentzian signature. Since Lorentzian $p$-branes have $(p+1)$-dimensional worldvolume, the systematic naming here would be "2+1"-brane.

See ([Blencowe-Duff 88, section 7](#BlencoweDuff88), [Hewson-Perry 96](#HewsonPerry96), [Nishino 97b](#Nishino97b))

## Related concepts

* [[D=11 supergravity]]

* [[F-theory]]

* [[D=14 supersymmetry]]

* [[bosonic M-theory]]

## References

### General

* [[Leonardo Castellani]], [[Pietro Fré]], F. Giani, K. Pilch, [[Peter van Nieuwenhuizen]], _Beyond $d=11$ Supergravity and Cartan Integrable Systems_, Phys.Rev. D26 (1982) 1481 ([spire:11999](http://inspirehep.net/record/11999))

* [[Itzhak Bars]], _Supersymmetry, p-brane duality and hidden space-time dimensions_, Phys. Rev. D54, 5203 (1996) ([arXiv: hep-th/9604139](https://arxiv.org/abs/hep-th/9604139)).

* [[Itzhak Bars]], _S-Theory_, Phys.Rev. D55 (1997) 2373-2381 ([arXiv:hep-th/9607112](https://arxiv.org/abs/hep-th/9607112))

* {#Nishino97} [[Hitoshi Nishino]], _Supergravity in 10 + 2 Dimensions as Consistent Background for Superstring_, ([arXiv:hep-th/9703214](https://arxiv.org/abs/hep-th/9703214))

* {#Nishino97b} [[Hitoshi Nishino]], _N=2 Chiral Supergravity in (10 + 2)-Dimensions As Consistent Background for Super (2 + 2)-Brane_, Phys. Lett. B437 (1998) 303-314 ([arXiv:hep-th/9706148](https://arxiv.org/abs/hep-th/9706148))

* {#Hewson97} Stephen Hewson, _An approach to F-theory_, Nucl. Phys. B534 (1998) 513-530 ([arXiv:hep-th/9712017](https://arxiv.org/abs/hep-th/9712017))

* {#Nishino98} [[Hitoshi Nishino]], _Supergravity Theories in $D \geq 12$ Coupled to Super p-Branes_, Nucl.Phys. B542 (1999) 217-261 ([arXiv:hep-th/9807199](https://arxiv.org/abs/hep-th/9807199))

* Stephen Hewson, _On supergravity in $(10,2)$_ ([arXiv:hep-th/9908209](https://arxiv.org/abs/hep-th/9908209))

* Tatsuya Ueno, _BPS States in 10+2 Dimensions_, JHEP 0012:006, 2000 ([arXiv:hep-th/9909007](https://arxiv.org/abs/hep-th/9909007))

* [[Leonardo Castellani]], _A locally supersymmetric SO(10,2) invariant action for D=12 supergravity_, ([arXiv:1705.00638](https://arxiv.org/abs/1705.00638))

### On the $2+1$-brane in $10+2$ dimensions

* {#BlencoweDuff88} [[Miles Blencowe]], [[Mike Duff]], _Supermembranes and the Signature of Space-time_, Nucl. Phys. B310 (1988) 387-404 ([spire:262142](inspirehep.net/record/262142), 10.1016/0550-3213(88)90155-1, [pdf](http://inspirehep.net/record/262142/files/cer-000099708.pdf))

* {#HewsonPerry96} S. F. Hewson, M. J. Perry, _The twelve dimensional super $(2+2)$-brane_, Nucl.Phys. B492 (1997) 249-277 ([arXiv:hep-th/9612008](https://arxiv.org/abs/hep-th/9612008))

* [Nishino 97b](#Nishino97b)

### On supergravity in $9 + 3$ dimensions

* {#Kriz05} [[Igor Kriz]], _Some remarks on fundamental physical F-theory_, ([arXiv:hep-th/0508046](https://arxiv.org/abs/hep-th/0508046))

### In bosonic M-theory

On organizing all these variants inside [[bosonic M-theory]] and [[KK-reduction]] on [[Cayley planes]] to actual [[M-theory]]:

* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], _The Geometry of Exceptional Super Yang-Mills Theories_, Phys. Rev. D **99** (2019) 046004  [[arXiv:1811.06101](https://arxiv.org/abs/1811.06101), [doi:10.1103/PhysRevD.99.046004](https://doi.org/10.1103/PhysRevD.99.046004)]

* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], *Exceptional Super Yang-Mills in $D=27+3$ and Worldvolume M-Theory*, Phys. Lett. B **808** (2020) 135674 [[arXiv:1906.10709](https://arxiv.org/abs/1906.10709), [doi:10.1016/j.physletb.2020.135674](https://doi.org/10.1016/j.physletb.2020.135674)]

reviewed in:

* [[Alessio Marrani]], *Exceptional super Yang-Mills in $27 + 3$ and worldvolume M-theory*, talk at *[M-Theory and mathematics 2023](https://ncatlab.org/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (Jan, 2023) [[web](/nlab/show/M-Theory+and+Mathematics#Marrani2023)]

[[!redirects 12d supergravity]]


D=14 supersymmetry

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## Idea

[[Supersymmetry]] in total [[spacetime]] [[dimension]] 14, of interest for [[spacetime signature]] $11+3$, hence super-extending [[Spin(11,3)]].

## References

* Khaled Abdel-Khalek, _The Super P-Brane Scan and S Duality_ ([arXiv:hep-th/9607180](https://arxiv.org/abs/hep-th/9607180))

* [[Ergin Sezgin]], _Super Yang-Mills in $(11,3)$ Dimensions_, 	Phys.Lett. B403 (1997) 265-272 ([arXiv:hep-th/9703123](https://arxiv.org/abs/hep-th/9703123))

* [[Itzhak Bars]], _A case for 14 dimensions_, Phys. Lett. B403 (1997) 257-264 ([arXiv:hep-th/9704054](https://arxiv.org/abs/hep-th/9704054))

* [[Itzhak Bars]], _Duality Covariant Type IIB Supersymmetry and Nonperturbative Consequences_, Phys. Rev. D 56, 7954 (1997) ([arXiv:hep-th/9706185](https://arxiv.org/abs/hep-th/9706185))

* I. Rudychev, [[Ergin Sezgin]], P. Sundell, _Supersymmetry in Dimensions Beyond Eleven_, Nucl. Phys. Proc. Suppl. 68 (1998) 285-294 ([arXiv:hep-th/9711127](https://arxiv.org/abs/hep-th/9711127))

* [[Hitoshi Nishino]], _Supersymmetric Yang-Mills Theories in $D \geq 12$_, Nucl.Phys. B523 (1998) 450-464 ([arXiv:hep-th/9708064](https://arxiv.org/abs/hep-th/9708064))

In relation to [[bosonic M-theory]] on [[Cayley plane]]-[[fibers]] and [[12-dimensional supergravity]]:

* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], _The Geometry of Exceptional Super Yang-Mills Theories_, Phys. Rev. D **99** (2019) 046004  [[arXiv:1811.06101](https://arxiv.org/abs/1811.06101), [doi:10.1103/PhysRevD.99.046004](https://doi.org/10.1103/PhysRevD.99.046004)]

* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], *Exceptional Super Yang-Mills in $D=27+3$ and Worldvolume M-Theory*, Phys. Lett. B **808** (2020) 135674 [[arXiv:1906.10709](https://arxiv.org/abs/1906.10709), [doi:10.1016/j.physletb.2020.135674](https://doi.org/10.1016/j.physletb.2020.135674)]

reviewed in:

* [[Alessio Marrani]], *Exceptional super Yang-Mills in $27 + 3$ and worldvolume M-theory*, talk at *[M-Theory and mathematics 2023](https://ncatlab.org/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (Jan, 2023) [[web](/nlab/show/M-Theory+and+Mathematics#Marrani2023)]

## Related concepts

* [[D=11 supergravity]]

* [[D=12 supergravity]]

* [[Spin(11,3)]]

[[!redirects 14-dimensional supergravity]]


D=2 CFT as functorial field theory -- references

### $D=2$ CFT as functorial field theory
 {#2dCFTAsFunctorialQFTReferences}

Discussion of [[D=2 conformal field theory]] as a [[functorial field theory]], namely as a [[monoidal functor|monoidal]] [[functor]] from a 2d [[conformal cobordism category]] to [[Hilbert spaces]]:

* {#Segal88} [[Graeme Segal]], *The definition of conformal field theory*, in: K. Bleuler, M. Werner (eds.), *Differential geometrical methods in theoretical physics* (Proceedings of Research Workshop, Como 1987), NATO Adv. Sci. Inst., Ser. C: Math. Phys. Sci. **250** Kluwer Acad. Publ., Dordrecht (1988) 165-171 $[$[doi:10.1007/978-94-015-7809-7](https://doi.org/10.1007/978-94-015-7809-7)$]$

and including discussion of [[modular functors]]:

* {#Segal89} [[Graeme Segal]], *Two-dimensional conformal field theories and modular functors*, in: *Proceedings of the IXth International Congress on Mathematical Physics*, Swansea, 1988, Hilger, Bristol (1989) 22-37.

* {#Segal04} [[Graeme Segal]], *The definition of conformal field theory*, in: [[Ulrike Tillmann]] (ed.), *Topology, geometry and quantum field theory* , London Math. Soc. Lect. Note Ser. **308**, Cambridge University Press (2004) 421-577 $[$[doi:10.1017/CBO9780511526398.019](https://doi.org/10.1017/CBO9780511526398.019), [pdf](https://people.maths.ox.ac.uk/segalg/0521540496txt.pdf), [[SegalDefinitionCFT.pdf:file]]$]$

General construction for the case of [[rational 2d conformal field theory]] is given by the

* *[[FRS-theorem on rational 2d CFT]]*

See also:

* {#MooreSegal06} [[Greg Moore]], [[Graeme Segal]], _D-branes and K-theory in 2D topological field theory_ ([arXiv:hep-th/0609042](http://arxiv.org/abs/hep-th/0609042))

* [[Richard Blute]], [[Prakash Panangaden]], [[Dorette Pronk]], _Conformal field theory as a nuclear functor_, Electronic Notes in Theoretical Computer Science Volume 172, 1 April 2007, Pages 101-132 GDP Festschrift ([pdf](http://aix1.uottawa.ca/~rblute/conf.pdf), [doi:10.1016/j.entcs.2007.02.005](https://doi.org/10.1016/j.entcs.2007.02.005))

A different but closely analogous development for chiral 2d CFT ([[vertex operator algebras]], see [there](vertex+operator+algebra#AsOperadAlgebras) for more):

* {#Huang91} [[Yi-Zhi Huang]], _Geometric interpretation of vertex operator algebras_, Proc. Natl. Acad. Sci. USA __88__ (1991) pp. 9964-9968 ([doi:10.1073/pnas.88.22.9964](https://doi.org/10.1073/pnas.88.22.9964))

Discussion of the case of [[Liouville theory]]:

* [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], *Segal's axioms and bootstrap for Liouville Theory* [[arXiv:2112.14859](https://arxiv.org/abs/2112.14859)]

Early suggestions to refine this to an [[extended TQFT|extended]] [[2-functor|2-functorial]] construction:

* [[Stefan Stolz]], [[Peter Teichner]], _[[What is an elliptic object?]]_

A step towards generalization to [[2d super-conformal field theory]]:

* [[Stephan Stolz]], [[Peter Teichner]], _Supersymmetric field theories and generalized cohomology_, in: [[Hisham Sati|H. Sati]], [[Urs Schreiber|U. Schreiber]], _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics 83 (2011), 279--340 ([arXiv:1108.0189](https://arxiv.org/abs/1108.0189), [doi:10.1090/pspum/083/2742432](https://doi.org/10.1090/pspum/083))

Discussion of 2-functorial chiral 2d CFT:

* [[André Henriques]], *The complex cobordism 2-category*, 2021 ([video](http://andreghenriques.com/ComplexCob2CatandCentralExt.mp4))

[[!redirects D=2 CFT a functorial field theory -- references]]


D=2 gravity

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## Idea

One may consider [[gravity]] ([[general relativity]], [[Einstein's equations]]) on [[spacetimes]] of [[dimension]] 2.

## Examples

### Relativistic string worldsheets

If 2d gravity is coupled to [[scalar field|scalar]] "matter"-[[field (physics)|fields]] with values in some [[target space]], then this may be regarded as the [[worldsheet]]-[[sigma-model]] [[quantum field theory]] of a [[relativistic string]] propagating on that [[target spacetime]].

### 2d Dilaton gravity -- JT gravity

A form of 2d [[dilaton gravity]] is known as _[[Jackiw-Teitelboim gravity]]_, or _JT-gravity_, for short.

JT-gravity gives a good approximation to the [[anti de Sitter spacetime|AdS]]-factor in the [[near horizon geometry]] $AdS_2\times S^{d-2}$ of near-[[extremal black holes]] in $d$-dimensional spacetime
([NSST18](JT-gravity#NSST18), [MTV18](JT-gravity#MTV18)).

## Related concepts

* [[Wheeler superspace]]

* [[supersymmetric quantum mechanics]]

* [[relativistic string]] [[sigma-model]]

* [[Jackiw-Teitelboim gravity]]

* [[Liouville theory]]

## References

* [[Paul Ginsparg]], [[Gregory Moore]], _Lectures on 2D gravity and 2D string theory (TASI 1992)_ ([arXiv:hep-th/9304011](https://arxiv.org/abs/hep-th/9304011))

[[!redirects 2d gravity]]


D=2 N=(2,0) SCFT

[[!redirects 2d (2,0)-superconformal QFT]]

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## Idea

A [[2d superconformal field theory]] in which the holomorphic sector (the "right movers") has $N=2$ [[supersymmetry]], while the antiholomorphic sector (the "left movers") has no supersymmetry (or the other way around), is traditionally denoted as $N = (2,0)$ supersymmetry. The [[topological twist]] of this is also called the _[[half-twisted model]]_.

## Properties

### Relation to the heterotic string

In contrast, the [[heterotic string]] [[worldsheet]] [[2d SCFT]] a priori has only $N = (1,0)$ supersymmetry, while the [[type II superstring]] similarly has $N = (1,1)$. But for suitable [[target spaces]] and [[background gauge fields]] the [[supersymmetry]] of the heterotic string is enhanced.

In ([BDFF 88](#BDFF88)) it was shown that this enhancement of the $N=(1,0)$-superconformal worldsheet theory of the [[heterotic string]] to an $N=(2,0)$-superconformal theory happens precisely when the corresponding [[target space]] [[effective quantum field theory]] itself has $N=1$ [[supersymmetry]]. This in turn is the case notably when the target is a [[Calabi-Yau manifold]], for more on this see at _[[supersymmetry and Calabi-Yau manifolds]]_.

So to the extent that $N=1$ target space supersymmetry is regarded as important in [[string phenomenology]] (it used to be regarded as highly important but with the failure of the [[LHC]] [[experiment]] to detect evidence reflecting this assumption, this may be changing), then 2d $(2,0)$-superconfromal QFTs define the relevant [[heterotic string]] theory [[vacua]].

### Relation to topological strings and sheaves of quantum observables

The enhanced [[supersymmetry]] of the $2d$ $(2,0)$-models allows to consider the corresponding [[topological twist]] ([Kapustin 05](#Kapustin05), [Witten 05](#Kapustin05)). The resulting topologically twisted 2d (2,0)-model is on the one hand amenable to detailed mathematical analysis, since its [[quantum observables]] for the [[chiral de Rham complex]] [[sheaf of vertex operator algebras]] (see ([Malikov 06](#Malikov06)) for a mathematical treatment) and at the same time is argued to still detect important properties of the original geometric model, such as notably the [[Witten genus]].

## Related concepts

* [[D=6 N=(2,0) SCFT]]

* [[F/M-theory on elliptically fibered Calabi-Yau 5-folds]]

* [[gauged linear sigma model]]

## References

The relevance of 2d $(2,0)$-superconformal worldsheet theories for [[string phenonemology]] (see also at _[[supersymmetry and Calabi-Yau manifolds]]_) was first highligted in

* {#BDFF88} [[Tom Banks]], [[Lance Dixon]], [[Dan Friedan]], [[Emil Martinec]], _Phenomenology and Conformal Field Theory or Can String Theory Predict the Weak Mixing Angle?_, Nucl. Phys. B299 (1988) 613.  ([pdf](http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-4377.pdf))

Further discussion of 2d $(2,0)$-superconformal QFTs in the contest of [[heterotic string theory|heterotic]] [[string theory]] [[vacua]] includes

* [[Jacques Distler]], [[Brian Greene]], _Aspects Of $(2,0)$ String Compactifications_, Nucl. Phys. B304 (1988)

* [[Eva Silverstein]], [[Edward Witten]], _Criteria for Conformal Invariance of (0,2) Models_, Nucl.Phys.B444:161-190,1995 ([arXiv:hep-th/9503212](http://arxiv.org/abs/hep-th/9503212))

* [[Jock McOrist]], _The Revival of $(0,2)$ Linear Sigma Models_, Int.J.Mod.Phys.A26:1-41,2011 ([arXiv:1010.4667](http://arxiv.org/abs/1010.4667))

The topological twist leading to the [[half-twisted model]] is due to

* {#Witten91} [[Edward Witten]], _Mirror Manifolds And Topological Field Theory_ ([arXiv:hep-th/9112056](http://arxiv.org/abs/hep-th/9112056))

The original articles in the [[physics]] literature identifying the [[chiral de Rham complex]] in the topologically twisted 2d $(2,0)$ model are

* {#Kapustin05} [[Anton Kapustin]], _Chiral de Rham complex and the half-twisted sigma-model_ ([arXiv:hep-th/0504074](http://arxiv.org/abs/hep-th/0504074))

* {#Witten05} [[Edward Witten]], _Two-Dimensional Models With (0,2) Supersymmetry: Perturbative Aspects_ ([arXiv:hep-th/0504078](http://arxiv.org/abs/hep-th/0504078))

Approaches formalizing this mathematically in terms of [[sheaves of vertex operator algebras]] and the [[chiral de Rham complex]] is in

* {#Malikov06} [[Fyodor Malikov]], _Lagrangian approach to sheaves of vertex operator algebras_, ESI preprint 1828 (2006) ([pdf](http://www.mat.univie.ac.at/~esiprpr/esi1828.pdf))

and in terms of [[factorization algebras]] in

* {#Gwilliam} [[Owen Gwilliam]], section 6 of _Factorization algebras and free field theories_ PhD thesis ([pdf](http://math.berkeley.edu/~gwilliam/thesis.pdf))

A quick review with emphasis on the [[Stolz conjecture]] is in

* {#Yagi10} [[Junya Yagi]], _Vanishing Chiral Algebras and Höhn-Stolz Conjecture_ ([arXiv:1002.0028](http://arxiv.org/abs/1002.0028))

Relation to [[F/M-theory on elliptically fibered Calabi-Yau 5-folds]]:

* [[Sakura Schäfer-Nameki]], [[Timo Weigand]], _F-theory and 2d (0,2) Theories_, J. High Energ. Phys. 2016, 59 (2016) ([arXiv:1601.02015](https://arxiv.org/abs/1601.02015))

* [[Sakura Schäfer-Nameki]], _F-theory and $2d$ $(0,2)$ Theories_, 2016 ([[SchaeferNamekiFTheoryAnd2DQFT.pdf:file]])

[[!redirects 2d (2,0)-supersymmetric QFT]]

[[!redirects 2d (2,0)-CFT]]

[[!redirects 2d (2,0)-supersymmetric sigma-model]]
[[!redirects 2d (2,0)-supersymmetric sigma-models]]


D=2 QCD

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## Idea

[[quantum chromodynamics]] in 2 [[spacetime]] [[dimensions]]

First considered in ['t Hooft 74](#tHooft74), whence also called the _'t Hooft model_.

## References

### General

The original account is:

* {#tHooft74} [[Gerard ’t Hooft]], _A Two-Dimensional Model For Mesons_, Nucl. Phys. B75, 461 (1974) (doi:10.1016/0550-3213(74)90088-1)

See also:

* [[David Gross]], [[Washington Taylor]], _Two Dimensional QCD is a String Theory_, Nucl. Phys. B400:181-210, 1993 ([arxiv:hep-th/9301068](https://arxiv.org/abs/hep-th/9301068))

* [[Michael Douglas]], Keke Li, Matthias Staudacher, _Generalized Two-Dimensional QCD_ ([arXiv:hep-th/9401062](https://arxiv.org/abs/hep-th/9401062))

* [[Emanuel Katz]], Gustavo Marques Tavares, Yiming Xu, _A solution of 2D QCD at Finite $N$ using a conformal basis_ ([arxiv:1405.6727](https://arxiv.org/abs/1405.6727))

### In Ads/CFT

In view of [[AdS/CFT duality]] and [[holographic entanglement entropy]]:

* William Donnelly, Sydney Timmerman, Nicolás Valdés-Meller,
_Entanglement entropy and the large $N$ expansion of two-dimensional Yang-Mills theory_ ([arxiv:1911.09302](https://arxiv.org/abs/1911.09302))

### On intersecting D-branes

Specifically [[geometric engineering of QFT|geometric engineering]] of 2d QCD in [[intersecting D-brane models]] ([[AdS/QCD]]):

#### Top-down

[[bottom-up and top-down model building|top-down model building]]
 construction of 2d QCD analogous to the [[Witten-Sakai-Sugimoto model]] for 4d [[QCD]]

* Yi-hong Gao, Weishui Xu, Ding-fang Zeng, _NGN, $QCD_2$ and chiral phase transition from string theory_, Nucl.Phys. B400:181-210, 1993 ([arXiv:hep-th/0605138](https://arxiv.org/abs/hep-th/0605138))

Specifically concerning the 3d [[Chern-Simons theory]] on the [[D8-branes]]:

* Ho-Ung Yee, Ismail Zahed, _Holographic two dimensional QCD and Chern-Simons term_, JHEP 1107:033, 2011 ([arXiv:1103.6286](https://arxiv.org/abs/1103.6286))

and its relation to [[baryons]]:

* Hideo Suganuma, Yuya Nakagawa, Kohei Matsumoto, _1+1 Large $N_c$ QCD and its Holographic Dual $\sim$ Soliton Picture of Baryons in Single-Flavor World_, JPS Conf. Proc. 13, 020013 (2017) ([arXiv:1610.02074](https://arxiv.org/abs/1610.02074))

#### Bottom-up

[[bottom-up and top-down model building|bottom up model building]]

* [[Emanuel Katz]], Takemichi Okui, _The 't Hooft Model As A Hologram_, JHEP 0901:013, 2009 ([arXiv:0710.3402](https://arxiv.org/abs/0710.3402))

in terms of [[worldline formalism]]:

* Adi Armoni, Oded Mintakevich, _Comments on Mesonic Correlators in the Worldline Formalism_,
 Nuclear Physics B, Volume 852, Issue 1, 1 November 2011, Pages 61-70 ([arxiv:1102.5318](https://arxiv.org/abs/1102.5318))

### As a string theory

On [[D=2 Yang-Mills theory]] as the [[effective field theory]] of a [[string theory]]:

* [[Ofer Aharony]], Suman Kundu, Tal Sheaffer, *A String Theory for Two Dimensional Yang-Mills Theory I* [[arXiv:2312.12266](https://arxiv.org/abs/2312.12266)]

[[!redirects 2d QCD]]

[[!redirects D=2 quantum chromodynamics]]
[[!redirects 2d quantum chromodynamics]]

[[!redirects 't Hooft model]]
[[!redirects 't Hooft models]]

[[!redirects D=2 Yang-Mills theory]]


D=2 supergravity

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## Idea

[[supergravity]] in [[dimension]] 2.

## Properties

### U-duality

The [[U-duality]] group (see there) of maximal 2d SuGra is [[E9]] (its [[split real form]] or, quantumly, a discrete subgroup of that).

## Related entries

* [[superstring]]

* [[2d SCFT]]

## References

* T. Fujiwara, Y. Igarashi, R. Kuriki, T. Tabei, _BFV-BRST Quantization of 2D Supergravity_ ([arXiv:hep-th/9502127](http://arxiv.org/abs/hep-th/9502127))

* S. Bellucci, E. Ivanov, _$N=(4,4)$, 2D supergravity in $SU(2)\times SU(2)$ harmonic superspace_ ([arXiv:hep-th/0003154](http://arxiv.org/abs/hep-th/0003154))

On maximal [[gauged supergravity|gauged]] $D=2$ supergravity via [[KK-reduction]] of [[D=11 supergravity]]:

* [[Guillaume Bossard]], [[Franz Ciceri]], [[Gianluca Inverso]], [[Axel Kleinschmidt]], *Maximal $D=2$ supergravities from higher dimensions* [[arXiv:2309.07232](https://arxiv.org/abs/2309.07232)]

* [[Guillaume Bossard]], [[Franz Ciceri]], [[Gianluca Inverso]], [[Axel Kleinschmidt]], *Consistent truncation of eleven-dimensional supergravity on $S^8 \times S^1$* [[arXiv:2309.07233](https://arxiv.org/abs/2309.07233)]

[[!redirects 2d supergravity]]
[[!redirects 2-dimensional supergravity]]


D=3 N=1 super Yang-Mills theory

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## Idea

The special case of [[super Yang-Mills theory]] over a [[spacetime]] of [[dimension]] 3 and with $\mathcal{N}=1$ [[number of supersymmetries]].

## References

### General

(...)

### Witten index

* [[Edward Witten]], _Supersymmetric Index Of Three-Dimensional Gauge Theory_, in: _[[The many faces of the superworld]]_ ([arXiv:hep-th/9903005](https://arxiv.org/abs/hep-th/9903005))


D=3 N=2 super Yang-Mills theory

[[!redirects D=3 N=2 super Yanf-Mills theory]]

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## Idea

The special case of [[super Yang-Mills theory]] in [[dimension]] 3 and [[number of supersymmetries]] 2.

## Properties

* [[3d-3d correspondence]]

## Related concept

* [[3d superconformal gauge field theory]]

  * [[D=3 N=4 super Yang-Mills theory]]

## References

* [[Jan de Boer]], [[Kentaro Hori]], [[Yaron Oz]], _Dynamics of N=2 Supersymmetric Gauge Theories in Three Dimensions_, Nucl. Phys. B500:163-191, 1997 ([arXiv:hep-th/9703100](https://arxiv.org/abs/hep-th/9703100))

* [[Ofer Aharony]], [[Amihay Hanany]], [[Kenneth Intriligator]], [[Nathan Seiberg]], [[Matthew Strassler]],
_Aspects of $\mathcal{N}=2$ Supersymmetric Gauge Theories in Three Dimensions_, Nucl. Phys. B499:67-99, 1997 ([arXiv:hep-th/9703110](https://arxiv.org/abs/hep-th/9703110))

On the [[Witten index]] for [[D=3 N=2 super Yang-Mills theory]]:

* [[Mathew Bullimore]], Andrea E. V. Ferrari, Heeyeon Kim, _The 3d Twisted Index and Wall-Crossing_ ([arXiv:1912.09591](https://arxiv.org/abs/1912.09591))

Relation to [[quantum K-theory rings]]:

Relation to [[D=3 N=2 super Yang-Mills theory]]:

* [[Hans Jockers]], [[Peter Mayr]], *A 3d Gauge Theory/Quantum K-Theory Correspondence*, Advances in Theoretical and Mathematical Physics **24** 2 (2020) [[arXiv:1808.02040](https://arxiv.org/abs/1808.02040), [doi:10.4310/ATMP.2020.v24.n2.a4](https://dx.doi.org/10.4310/ATMP.2020.v24.n2.a4)]

Construction in [[M-theory]]:

* Marwan Najjar, Jiahua Tian, Yi-Nan Wang, *3d $\mathcal{N}=2$ theories from M-theory on CY4 and IIB brane box* [[arXiv:2312.17082](https://arxiv.org/abs/2312.17082)]

[[!redirects D=3 N=2 SYM]]


D=3 N=4 super Yang-Mills theory

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#Contents#
* table of contents
{:toc}

## Idea

The special case of [[super Yang-Mills theory]] over a [[spacetime]] of [[dimension]] 3 and with $\mathcal{N}=4$ [[number of supersymmetries]].

## Properties

### Coulomb- and Higgs-branches

Both the [[Coulomb branches]] and the [[Higgs branch]] of [[D=3 N=4 super Yang-Mills theory]] are [[hyperkähler manifolds]]. In special cases they are [[compact hyperkähler manifolds]] (e.g. [dBHOO 96](#dBHOO96)).

### Reduction from $N = 2$, $D = 4$

The $N = 4$, $D = 3$ SYM theory can be obtained by [[Kaluza-Klein mechanism|dimensional reduction]] from [[N=2 D=4 super Yang-Mills theory]] ([Seiberg-Witten 96](#SeibergWitten96))

### Mirror symmetry

A version of [[mirror symmetry]] acts on the $N = 4$, $D = 3$ SYM [[moduli space]] of vacua and exchanges the [[Coulomb branch]] with the [[Higgs branch]]. ([Intriligator-Seiberg 96](#IntriligatorSeiberg96))

See also the discussion at _[[symplectic duality]]_.

### Topological twist and Rozansky-Witten theory

A [[topological twist]] of [[D=3 N=4 super Yang-Mills theory]] is [[Rozansky-Witten theory]].

## Related concepts

* [[3d superconformal gauge field theory]]

  * [[D=3 N=4 super Yang-Mills theory]]

* [[3d-3d correspondence]]

* [[N=2 D=4 super Yang-Mills theory]]

* [[N=4 D=4 super Yang-Mills theory]]

## References

### General

The construction of [[D=3 N=4 super Yang-Mills theory]] by [[Kaluza-Klein mechanism|dimensional reduction]] from [[N=2 D=4 super Yang-Mills theory]] was first considered in

* {#SeibergWitten96} [[Nathan Seiberg]], [[Edward Witten]], _Gauge Dynamics And Compactification To Three Dimensions_, In: J.M. Drouffe, J.B. Zuber (eds.) _The mathematical beauty of physics: A memorial volume for Claude Itzykson_ Proceedings, Conference, Saclay, France, June 5-7, 1996 ([arXiv:hep-th/9607163](http://arxiv.org/abs/hep-th/9607163), [spire:420925](http://inspirehep.net/record/420925))

Discussion as the [[worldvolume]]-theory of [[D3-D5 brane intersections]]:

* [[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))

Review of the [[moduli space]] of [[vacua]]:

* Federici Carta, _Moduli Spaces of $\mathcal{N} = 4$, $d = 3$ Quiver Gauge Theories and Mirror Symmetry_, ([tesi.cab.unipd.it/46485/](http://tesi.cab.unipd.it/46485/))

Via [[KK-compactification]] from [[little string theory]]:

* Antonio Amariti, Gianmarco Formigoni, _A note on $4d$ $\mathcal{N} = 3$ from little string theory_ ([arXiv:2003.05983](https://arxiv.org/abs/2003.05983))

and from [[heterotic string theory]] on [[ADE-singularities]]:

* [[Michele Del Zotto]], [[Marco Fazzi]], Suvendu Giri, *A new vista on the Heterotic Moduli Space from Six and Three Dimensions* [[arXiv:2307.10356](https://arxiv.org/abs/2307.10356)]

See also:

* Mikhail Evtikhiev, _$\mathcal{N} = 3$ SCFTs in 4 dimensions and non-simply laced groups_ ([arXiv:2004.03919](https://arxiv.org/abs/2004.03919))

### Mirror symmetry for $D=3$ $\mathcal{N}=4$ SYM

On [[mirror symmetry]] for [[D=3 N=4 super Yang-Mills theory]]

The mirror symmetry operation was discussed in

* {#IntriligatorSeiberg96} [[Ken Intriligator]], [[Nathan Seiberg]], _Mirror Symmetry in Three Dimensional Gauge Theories_, Phys. Lett.B 387 : 513-519,1996 ([arXiv:hep-th/9607207](http://arxiv.org/abs/hep-th/9607207))

Discussion with emphasis of [[Higgs branches]]/[[Coulomb branches]] as [[Hilbert schemes of points]]

* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], _Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes_, Nucl. Phys. B493:101-147, 1997 ([arXiv:hep-th/9611063](https://arxiv.org/abs/hep-th/9611063))

* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], Zheng Yin, _Mirror Symmetry in Three-Dimensional Gauge Theories, $SL(2,\mathbb{Z})$ and D-Brane Moduli Spaces_, Nucl. Phys. B493:148-176, 1997 ([arXiv:hep-th/9612131](https://arxiv.org/abs/hep-th/9612131))

* Mathew Bullimore, Andrea Ferrari, Heeyeon Kim, *Supersymmetric Ground States of 3d $\mathcal{N}=4$ Gauge Theories on a Riemann Surface* ([arXiv:2105.08783](https://arxiv.org/abs/2105.08783))

### Lift to M-theory

Lift to [[M-theory]]:

* M. Porrati, [[Alberto Zaffaroni]], _M-Theory Origin of Mirror Symmetry in Three Dimensional Gauge Theories_, Nucl. Phys. B490 (1997) 107-120 ([arXiv:hep-th/9611201](https://arxiv.org/abs/hep-th/9611201))

### Coulomb branch and monopole moduli

Review of [[Coulomb branches]] of [[D=3 N=4 super Yang-Mills theory]]:

* [[Marcus Sperling]], chapter III of: _Two aspects of gauge theories : higher-dimensional instantons on cones over Sasaki-Einstein spaces and Coulomb branches for 3-dimensional $\mathcal{N}=4$ gauge theories_ ([spire:1495766/](http://inspirehep.net/record/1495766),  [pdf](https://www.itp.uni-hannover.de/fileadmin/arbeitsgruppen/ag_lechtenf/Theses/sperling.pdf), [[SperlingGaugeTheory.pdf:file]])

Identification of the [[Coulomb branch]] of [[D=3 N=4 super Yang-Mills theory]] with the [[moduli space of monopoles]] in [[Yang-Mills theory]]:

* [Seiberg-Witten 96](#SeibergWitten96)

* N. Dorey, V. V. Khoze, M. P. Mattis, [[David Tong]], S. Vandoren, _Instantons, Three-Dimensional Gauge Theory, and the Atiyah-Hitchin Manifold_, Nucl. Phys. B502 (1997) 59-93 ([arXiv:hep-th/9703228](https://arxiv.org/abs/hep-th/9703228))

* [[David Tong]], _Three-Dimensional Gauge Theories and ADE Monopoles_, Phys. Lett. B448 (1999) 33-36 ([arXiv:hep-th/9803148](https://arxiv.org/abs/hep-th/9803148))

* [[Mathew Bullimore]], [[Tudor Dimofte]], [[Davide Gaiotto]], _The Coulomb Branch of 3d $\mathcal{N}=4$ Theories_, Commun. Math. Phys. (2017) 354: 671 ([arXiv:1503.04817](https://arxiv.org/abs/1503.04817))

On [[D=3 N=4 super Yang-Mills theories]] with [[compact hyperkähler manifold]] [[Coulomb branches]] obtained by [[KK-compactification]] of [[little string theories]]:

* {#Intriligator99} [[Kenneth Intriligator]], _Compactified Little String Theories and Compact Moduli Spaces of Vacua_, Phys. Rev. D61:106005, 2000 ([arXiv:hep-th/9909219](https://arxiv.org/abs/hep-th/9909219))

The [[Rozansky-Witten invariants]] of these moduli spaces:

* {#RozanskyWitten96} [[Lev Rozansky]], [[Edward Witten]], p. 36 of: _Hyper-Kähler geometry and invariants of 3-manifolds_, Selecta Math., New Ser. __3__ (1997), 401–458 ([arXiv:hep-th/9612216](https://arxiv.org/abs/hep-th/9612216), [doi:10.1007/s000290050016](https://doi.org/10.1007/s000290050016), [MR98m:57041](http://www.ams.org/mathscinet-getitem?mr=98m:57041))

On a mathematical definition of quantum [[Coulomb branches]] of [[D=3 N=4 super Yang-Mills theory]]:

* [[Hiraku Nakajima]], _Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N} = 4$ gauge theories_ ([arXiv:1706.05154](http://arxiv.org/abs/1706.05154))

* [[Hiraku Nakajima]], _Towards a mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N} = 4$ gauge theories, I_ ([arXiv:1503.03676](http://arxiv.org/abs/1503.03676))

* [[Alexander Braverman]], [[Michael Finkelberg]], [[Hiraku Nakajima]], _Towards a mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N} = 4$ gauge theories, II_, Adv. Theor. Math. Phys. 22 (2018) 1071-1147 ([arXiv:1601.03586](http://arxiv.org/abs/1601.03586))

* [[Alexander Braverman]], [[Michael Finkelberg]], [[Hiraku Nakajima]], _Line bundles over Coulomb branches_ ([arXiv:1805.11826](https://arxiv.org/abs/1805.11826))

  (relation to [[Hilbert schemes]])

### Hilbert schemes and Higgs/Coulomb branches

Identification of [[Higgs branches]]/[[Coulomb branches]] in [[D=3 N=4 super Yang-Mills theory]] with [[Hilbert schemes of points]] of [[complex curves]]:

* {#dBHOO96} [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], _Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes_, Nucl. Phys. B493:101-147, 1997 ([arXiv:hep-th/9611063](https://arxiv.org/abs/hep-th/9611063))

* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], Zheng Yin, _Mirror Symmetry in Three-Dimensional Gauge Theories, $SL(2,\mathbb{Z})$ and D-Brane Moduli Spaces_, Nucl. Phys. B493:148-176, 1997 ([arXiv:hep-th/9612131](https://arxiv.org/abs/hep-th/9612131))

* [[Stefano Cremonesi]], [[Amihay Hanany]], [[Alberto Zaffaroni]], around (4.4) of: _Monopole operators and Hilbert series of Coulomb branches of 3d $\mathcal{N} = 4$ gauge theories_, JHEP 01 (2014) 005 ([arXiv:1309.2657](https://arxiv.org/abs/1309.2657))

* [[Alexander Braverman]], [[Michael Finkelberg]], [[Hiraku Nakajima]], _Line bundles over Coulomb branches_ ([arXiv:1805.11826](https://arxiv.org/abs/1805.11826))

* Mykola Dedushenko, Yale Fan, Silviu Pufu, Ran Yacoby, Section E.2 of: _Coulomb Branch Quantization and Abelianized Monopole Bubbling_, JHEP 10 (2019) 179 ([arXiv:1812.08788](https://arxiv.org/abs/1812.08788))

### Witten index

Discussion of the [[Witten index]] of [[D=3 N=4 super Yang-Mills theory]]:

* [[Mathew Bullimore]], Andrea E.V. Ferrari, Heeyeon Kim, _Twisted Indices of 3d N=4 Gauge Theories and Enumerative Geometry of Quasi-Maps_ ([arXiv:1812.05567](https://arxiv.org/abs/1812.05567))

* [[Davide Gaiotto]], Tadashi Okazaki, _Sphere correlation functions and Verma modules_ ([arXiv:1911.11126](https://arxiv.org/abs/1911.11126))

* [[Mathew Bullimore]], Samuel Crew, Daniel Zhang, _Boundaries, Vermas, and Factorisation_ ([arXiv:2010.09741](https://arxiv.org/abs/2010.09741))

using discussion in

* [[Andrei Okounkov]], Section 3.4 of: _Lectures on K-theoretic computations in enumerative geometry_ ([arXiv:1512.07363](https://arxiv.org/abs/1512.07363))

See also on the [[Witten index]] for [[D=3 N=2 super Yang-Mills theory]]:

* [[Mathew Bullimore]], Andrea E. V. Ferrari, Heeyeon Kim, _The 3d Twisted Index and Wall-Crossing_ ([arXiv:1912.09591](https://arxiv.org/abs/1912.09591))

### Wilson loop operators

On [[Wilson loop operators]] in [[D=3 N=4 super Yang-Mills theory]]:

* [[Tudor Dimofte]], Niklas Garner, Michael Geracie, Justin Hilburn, _Mirror symmetry and line operators_ ([arXiv:1908.00013](https://arxiv.org/abs/1908.00013))

[[!redirects D=3 N=4 super Yang-Mills theories]]

[[!redirects N=4 D=3 super Yang-Mills theory]]
[[!redirects N=4 D=3 super Yang-Mills theories]]

[[!redirects N=4 D=3 sYM]]
[[!redirects N=4 D=3 SYM]]

[[!redirects D=3 N=4 SYM]]


D=3 supergravity

[[!redirects 3-dimensional supergravity]]

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#Contents#
* table of contents
{:toc}

## Idea

([[gauged supergravity|gauged]]) [[supergravity]] in [[dimension]] 3.

## Properties

### Possible gaugings
 {#PossibleGaugings}

The ([[U-duality]]-)[[global gauge group]] of maximally [[supersymmetry|supersymmetric]] 3d supergravity is [[E8]] (its [[split real form]] $E_{8(8)}$). Various [[subgroups]] of this may be promoted to [[local gauge groups]] (with [[gauge fields]] in [[gauged supergravity]]), which may be obtained via ([[flux compactification|fluxed]]) [[KK-compactification]] of [[11-dimensional supergravity]]. However, 3d supergravity also admits a maximal gauging where all of $E_{8(8)}$ is promoted to a local gauge group
([Nicolai-Samtleben 00](#NicolaiSamtleben00), [Nicolai-Samtleben 01, table 1](#NicolaiSamtleben01))). This maximal gauging in 3d supergravity is not obtained by reduction from standard [[11-dimensional supergravity]], see the remarks in ([Nicolai-Samtleben 01, section 7](#NicolaiSamtleben01)) and see the followup ([Hohm-Samtleben 13](#HohmSamtleben13)). In ([de Wit-Nicolai 13, section 13](#deWitNicolai13)) it is suggested that the seemingly missing degrees of freedom necessary to accomplish for [[U-duality]]-gauge enhancement after reduction may be sitting in a [[non-perturbative effect|non-perturvative]] [[electric-magnetic duality|dual]] of the [[graviton]] ([[dual graviton]]).

## Related concepts

* [[11-dimensional supergravity]]

* 10-dimensional [[type II supergravity]], [[heterotic supergravity]]

* [[7-dimensional supergravity]]

* [[5-dimensional supergravity]]

* [[4-dimensional supergravity]]

* **3-dimensional supergravity**

  * [[M-theory on 8-manifolds]]

  * [[3d quantum gravity]]

* [[ABJM model]]

## References

### Generally

Lecture notes include

* F. Ruiz Ruiz, [[Peter van Nieuwenhuizen]], _Lectures on Supersymmetry and Supergravity in $2+1$ Dimensions and regularization of supersymmetric gauge theories_ ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.17.3667&rep=rep1&type=pdf))

A complete list of un-gauged supergravities in 3 dimensions was given in

* [[Bernard de Wit]], A.K. Tollsten, [[Hermann Nicolai]], _Locally supersymmetric D=3 non-linear sigma models_, Nucl. Phys. B392 (1993) 3 ([arXiv:hep-th/9208074](http://xxx.lanl.gov/abs/hep-th/9208074))

See also

* Gabriele Tartaglino-Mazzucchelli, _Topics in $3D$ $N=2$ AdS supergravity in superspace_, ([arXiv:1202.0109](http://arxiv.org/abs/1202.0109))

* Martin Poláček, [[Warren Siegel]], _T-duality off shell in 3D Type II superspace_ ([arXiv:1403.6904](http://arxiv.org/abs/1403.6904))

### Branes

Discussion of [[D-branes]] in 3d supergravity includes

* Björn Brinne, Svend E. Hjelmeland, [[Ulf Lindström]], _World-Volume Locally Supersymmetric Born-Infeld Actions_, Phys.Lett. B459 (1999) 507-514 ([arXiv:hep-th/9904175](http://arxiv.org/abs/hep-th/9904175))

* Björn Brinne, _3D supergravity and a spinning D2-brane _

### Gauged

Topological gauged supergravity in dimension three was first considered in

* A. Achúcarro and [[Paul Townsend]], _A Chern-Simons action for
three-dimensional anti-de Sitter supergravity theories_, Phys. Lett. B180
(1986) 89

Gauged supergravity via [[KK-compactification]] of [[11-dimensional supergravity]] on an 8-torus and with global [[E8]] [[U-duality]] and local $SO(16)$ [[gauge field]] was discussed in

* [[Bernard Julia]], _Application of supergravity to gravitation theories_, in _Unified field theories in more than 4 dimensions_ (V. D. Sabbata and E. Schmutzer, eds.), (Singapore), pp. 215–236, World Scientific, 1983.

* N. Marcus, [[John Schwarz]], _Three-dimensional supergravity theories_,
Nucl. Phys. B228 (1983) 145.

The maximally supersymmetric gauged 3d supergravitites (and their [exceptional gaugings](#PossibleGaugings)) are listed in

* {#NicolaiSamtleben00} [[Hermann Nicolai]], [[Henning Samtleben]], _Maximal gauged supergravity in three dimensions_, Phys.Rev.Lett. 86 (2001) 1686-1689 ([arXiv:hep-th/0010076](http://xxx.lanl.gov/abs/hep-th/0010076))

with details in

* {#NicolaiSamtleben01} [[Hermann Nicolai]], [[Henning Samtleben]], _Compact and Noncompact Gauged Maximal Supergravities in Three Dimensions_ ([arXiv:hep-th/0103032](http://arxiv.org/abs/hep-th/0103032))

* {#HohmSamtleben13} [[Olaf Hohm]], [[Henning Samtleben]], _Exceptional Form of $D=11$ Supergravity_, Phys. Rev. Lett. 111, 231601 (2013) ([arXiv:1308.1673](http://arxiv.org/abs/1308.1673))

* {#deWitNicolai13} [[Bernard de Wit]], [[Hermann Nicolai]], _Deformations of gauged SO(8) supergravity and supergravity in eleven dimensions_ ([arXiv:1302.6219](http://arxiv.org/abs/1302.6219))

(see also at _[[exceptional generalized geometry]]_).

See also

* Hitoshi Nishino, Subhash Rajpoot, _Topologican Gauging of N=16 Supergravity in Three-Dimensions_, Phys.Rev. D67 (2003) 025009 ([arXiv:hep-th/0209106](http://arxiv.org/abs/hep-th/0209106))

* Eoin Ó Colgáin, [[Henning Samtleben]], _3D gauged supergravity from wrapped M5-branes with AdS/CMT applications_, JHEP 1102:031,2011 ([arXiv:1012.2145](http://arxiv.org/abs/1012.2145))

* Edi Gava, Parinya Karndumri, K. S. Narain, _3D gauged supergravity from SU(2) reduction of N=1 6D supergravity_, JHEP 09 (2010) 028 ([arXiv:1006.4997](http://arxiv.org/abs/1006.4997))

[[!redirects 3-dimensional supergravities]]

[[!redirects 3d supergravity]]
[[!redirects 3d supergravities]]


D=4 N=1 super Yang-Mills theory

[[!redirects N=1 D=4 super Yang-Mills theory]]
[[!redirects N=1 D=4 super Yang-Mills theory]]

#Contents#
* table of contents
{:toc}

## Idea

[[super Yang-Mills theory]] on a 4-[[dimension|dimensional]] [[spacetime]] with $N = 1$ [[supersymmetry]].

[[!include superconformal symmetry -- table]]

## Related concepts

* [[super Yang-Mills theory]]

  * [[N=2 D=4 super Yang-Mills theory]]

  * [[N=4 D=3 super Yang-Mills theory]]

  * [[N=4 D=4 super Yang-Mills theory]]

  * [[D=5 super Yang-Mills theory]]

  * [[topologically twisted D=4 super Yang-Mills theory]]

* [[geometric engineering of quantum field theory]]

[[!include superconformal symmetry -- table]]

## References

### General

Original articles:

* [[Hugh Osborn]], _$\mathcal{N}=1$ Superconformal Symmetry in Four Dimensional Quantum Field Theory_, Annals Phys. 272:243-294, 1999 ([arXiv:hep-th/9808041](https://arxiv.org/abs/hep-th/9808041))

Review:

* Yuji Tachikawa, _Lectures on $4d$ $N=1$ dynamics and related topics_ ([arXiv:1812.08946](https://arxiv.org/abs/1812.08946))

See also at _[[N=2 D=4 super Yang-Mills theory]]_.

The [[KK-compactification]] of the [[D=6 N=(1,0) SCFT]] (on [[M5-branes]]) to [[D=4 N=1 super Yang-Mills]]:

* Ibrahima Bah, Christopher Beem, Nikolay Bobev, Brian Wecht, _Four-Dimensional SCFTs from M5-Branes_ ([arXiv:1203.0303](http://arxiv.org/abs/1203.0303))

* Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _$4d$ $\mathcal{N} = 1$ from $6d (1,0)$_, J. High Energ. Phys. (2017) 2017: 64 ([arXiv:1610.09178](https://arxiv.org/abs/1610.09178))

* Ibrahima Bah, [[Amihay Hanany]], Kazunobu Maruyoshi, Shlomo S. Razamat, Yuji Tachikawa, Gabi Zafrir, _$4d$ $\mathcal{N}=1$ from $6d$ $\mathcal{N}=(1,0)$ on a torus with fluxes_ ([arXiv:1702.04740](https://arxiv.org/abs/1702.04740))

* Hee-Cheol Kim, Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _E-String Theory on Riemann Surfaces_, Fortsch. Phys. ([arXiv:1709.02496](https://arxiv.org/abs/1709.02496))

* Hee-Cheol Kim, Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _D-type Conformal Matter and SU/USp Quivers_, JHEP06(2018)058 ([arXiv:1802.00620](https://arxiv.org/abs/1802.00620))

* Hee-Cheol Kim, Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _Compactifications of ADE conformal matter on a torus_, JHEP09(2018)110 ([arXiv:1806.07620](https://arxiv.org/abs/1806.07620))

* Shlomo S. Razamat, Gabi Zafrir, _Compactification of 6d minimal SCFTs on Riemann surfaces_, Phys. Rev. D 98, 066006 (2018) ([arXiv:1806.09196](https://arxiv.org/abs/1806.09196))

* Jin Chen, Babak Haghighat, Shuwei Liu, Marcus Sperling, _4d N=1 from 6d D-type N=(1,0)_ ([arXiv:1907.00536](https://arxiv.org/abs/1907.00536))

Via [[D3 branes]] at resolved [[orbifold singularities]] and invoking a generalized [[McKay correspondence]]:

* [[Pietro G. Fré]], *Lectures on resolutions à la Kronheimer of orbifold singularities, McKay quivers for Gauge Theories on D3 branes, and the issue of Ricci flat metrics on the resolved three-folds* [[arXiv:2308.14022](https://arxiv.org/abs/2308.14022)]

The [[Witten index]]:

* [[Leonardo Rastelli]], Shlomo S. Razamat, _The supersymmetric index in four dimensions_, Journal of Physics A: Mathematical and Theoretical, Volume 50, Number 44 ([arXiv:1608.02965](https://arxiv.org/abs/1608.02965))

[[!redirects N=1 D=4 sYM]]

[[!redirects D=4 N=1 super Yang-Mills]]

[[!redirects N=1 D=4 SYM]]
[[!redirects D=4 N=1 SYM]]


D=4 N=2 super Yang-Mills theory

[[!redirects N=2 D=4 super Yang-Mills theory]]
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#Contents#
* table of contents
{:toc}

## Idea

The special case of [[super Yang-Mills theory]] over a [[spacetime]] of [[dimension]] 4 and with $N = 2$ [[supersymmetry]].

[[!include superconformal symmetry -- table]]

## Properties

### Moduli space of vacua
 {#ModuliSpacesOfVacua}

A speciality of $N=2$, $D = 4$ SYM is that its [[moduli space]] of [[vacua]] has two "branches" called the _[[Coulomb branch]]_ and the _[[Higgs branch]]_. This is the content of what is now called _[[Seiberg-Witten theory]]_ ([Seiberg-Witten 94](#SeibergWitten94)). Review includes ([Albertsson 03, section 2.3.4](#Albertsson03)).

### Confinement

While [[confinement]] in plain [[Yang-Mills theory]] is still waiting for mathematical formalization and [[proof]] (see [Jaffe-Witten](confinement#JaffeWitten)), N=2 D=4 super Yang-Mills theory is has been obsrved in ([Seiberg-Witten 94](#SeibergWitten94)).

### Reduction to $D = 3$ super Yang-Mills

By [[Kaluza-Klein mechanism|dimensional reduction]] on $\mathbb{R}^3 \times S^1$ families of $N = 2, D = 4$ SYM theories interpolate to [[N=4 D=3 super Yang-Mills theory]]. ([Seiberg-Witten 96](#SeibergWitten96)).

### Construction by compactification of 5-branes
 {#ConstructionByCompactificationOf5Branes}

$N=2$ super Yang-Mills theory can be realized as the [[worldvolume]] theory of [[M5-branes]] [[Kaluza-Klein compactification|compactified]] on a [[Riemann surface]] ([Klemm-Lerche-Mayr-Vafa-Warner 96](#KlemmLercheMayrVafaWarner96), [Witten 97](#Witten97), [Gaiotto 09](#Gaiotto09)), hence as a compactifiction of the [[6d (2,0)-superconformal QFT]] on the M5. This in particular gives a geometric interpretation of [[Seiberg-Witten duality]] in 4d in terms of the 6d 5-brane geometry.

Specifically the _[[AGT correspondence]]_ expresses this relation in terms of the [[partition function]] of the theory and a [[2d CFT]] on the [[Riemann surface]] on which the 5-brane is compactified. See at _[[AGT correspondence]]_ for more on this.

[[!include gauge theory from AdS-CFT -- table]]

## Related concepts

* [[quiver gauge theory]], [[fractional D-brane]]

* [[super Yang-Mills theory]]

  * [[N=4 D=4 super Yang-Mills theory]]

  * [[N=4 D=3 super Yang-Mills theory]]

  * [[topologically twisted D=4 super Yang-Mills theory]]

    * [[chiral ring]], [[quantum cohomology]]

  * [[Seiberg-Witten theory]], [[electric-magnetic duality]], [[symplectic duality]]

  * [[AGT correspondence]]

* [[spectral network]]

* [[geometric engineering of quantum field theory]]

[[!include superconformal symmetry -- table]]

## References

### Introductions and surveys
 {#Surveys}

* [[Davide Gaiotto]], _Recent progress in $N=2$ $4d$ field theory_ (2009) ([pdf](http://www.physics.rutgers.edu/het/video/gaiotto09a.pdf))

* {#Moore12} [[Gregory Moore]], _Four-dimensional $N=2$ Field Theory and Physical Mathematics_ ([arXiv:1211.2331](http://arxiv.org/abs/1211.2331))

* [[Greg Moore]], _Surface Defects and the BPS Spectrum of $4d$ $N=2$ Theories_ ([pdf](http://www.physics.rutgers.edu/~gmoore/SolvayMay19Final.pdf))

* [[Yuji Tachikawa]], *$\mathcal{N}=2$ supersymmetric dynamics for pedestrians*, Lecture Notes in Physics, vol. 890, 2014 ([arXiv:1312.2684](https://arxiv.org/abs/1312.2684), [doi:10.1007/978-3-319-08822-8](https://link.springer.com/book/10.1007%2F978-3-319-08822-8), [web version](http://www-hep.phys.s.u-tokyo.ac.jp/~yujitach/tmp/dummies/dummies.html#dummiesse7.html))

* Mohammad Akhond, Guillermo Arias-Tamargo, Alessandro Mininno, Hao-Yu Sun, Zhengdi Sun, Yifan Wang, Fengjun Xu, *The Hitchhiker's Guide to 4d $\mathcal{N}=2$ Superconformal Field Theories* [[arXiv:2112.14764](https://arxiv.org/abs/2112.14764)]

### General

The terminology "Coulomb branch" and "Higgs branch" first appears in

* {#SeibergWitten94} [[Nathan Seiberg]], [[Edward Witten]], _Monopoles, Duality and Chiral Symmetry Breaking in $N=2$ Supersymmetric QCD_ ([arXiv:hep-th/9408099](http://arxiv.org/abs/hep-th/9408099))

See also at *[[Seiberg-Witten theory]]:

The dimensional reduction to $D = 3$ was first considered in

* {#SeibergWitten96} [[Nathan Seiberg]], [[Edward Witten]], _Gauge Dynamics And Compactification To Three Dimensions_ ([arXiv:hep-th/9607163](http://arxiv.org/abs/hep-th/9607163))

The [[confinement]]-phenomenon was observed in

* {#SeibergWitten94} [[Nathan Seiberg]], [[Edward Witten]], _Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory_,  	Nucl.Phys.B426:19-52,1994; Erratum-ibid.B430:485-486,1994 ([arXiv:hep-th/9407087](http://arxiv.org/abs/hep-th/9407087))

Reviews of this confinement mechanism:

* Alexei Yung, _What Do We Learn about Confinement from the Seiberg-Witten Theory_ ([arXiv:hep-th/0005088](http://arxiv.org/abs/hep-th/0005088))

* {#Albertsson03} Cecilia Albertsson, _Superconformal D-branes and moduli spaces_ ([arXiv:hep-th/0305188](http://arxiv.org/abs/hep-th/0305188))

* [[Yang-Hui He]], _Lectures on D-branes, Gauge Theories and Calabi-Yau Singularities_, International Summer School in Mathematical Physics ([arXiv:hep-th/0408142](https://arxiv.org/abs/hep-th/0408142))

* [[Emily Nardoni]], *From $\mathcal{N} = 2$ Supersymmetry to adjoint QCD*, talk at *[[Strings 2022]]* [[indico:4940894](https://indico.cern.ch/event/1085701/contributions/4940894), [pdf](https://indico.cern.ch/event/1085701/contributions/4940894/attachments/2484206/4265102/Slides_Nardoni.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=035dfeb0-ef40-4530-8f20-01e5b890f43d)]

For references on [[wall crossing]] of [[BPS states]] see the references given there.

### Construction from 5-branes
 {#ReferencesConstructionFrom5Branes}

$N=2$ $D=4$ SYM including its [[Seiberg-Witten theory]] ([Seiberg-Witten 94](SeibergWitten94)) may be understood as being the [[Kaluza-Klein mechanism|compactification]] of the [[6d (2,0)-superconformal QFT]] on the [[worldvolume]] of [[M5-branes]] on a [[Riemann surface]]: the Riemann surface is identified with the _[[Seiberg-Witten curve]]_ of complexified coupling constants. This observation goes back to

* {#KlemmLercheMayrVafaWarner96} [[Albrecht Klemm]], [[Wolfgang Lerche]], [[Peter Mayr]], [[Cumrun Vafa]], [[Nicholas Warner]], _Self-Dual Strings and $\mathcal{N}=2$ Supersymmetric Field Theory_, Nucl. Phys. B **477** (1996) 746-766 [[arXiv:hep-th/9604034](http://arxiv.org/abs/hep-th/9604034), doi:10.1016/0550-3213(96)00353-7]

* {#Witten97} [[Edward Witten]], *Solutions Of Four-Dimensional Field Theories Via M Theory*, Nucl. Phys. B **500** (1997) 3-42 [[arXiv:hep-th/9703166](http://arxiv.org/abs/hep-th/9703166), doi:10.1016/S0550-3213(97)00416-1]

review in:

* Csaba Csaki, Joshua Erlich, John Terning, pp. 4 of; *Seiberg-Witten Description of the Deconstructed 6D $(0,2)$ Theory*, Phys. Rev. D **67** 025019 (2003) [[arXiv:hep-th/0208095](https://arxiv.org/abs/hep-th/0208095), [doi:10.1103/PhysRevD.67.025019](https://doi.org/10.1103/PhysRevD.67.025019)]

* Ling Bao, Elli Pomoni, Masato Taki, Futoshi Yagi, p. 5 of: *M5-branes, toric diagrams and gauge theory duality*, J. High Energ. Phys. **2012** 105 (2012) [[arXiv:1112.5228](https://arxiv.org/abs/1112.5228), doi:10.1007/JHEP04(2012)105]

* [[Taro Kimura]], §4.5.2 in: *Seiberg–Witten Geometry*, Chapter 4 in: *Instanton Counting, Quantum Geometry and Algebra*, Mathematical Physics Studies, Springer (2021) [[doi:10.1007/978-3-030-76190-5_4](https://doi.org/10.1007/978-3-030-76190-5_4)]

The further observation that therefore the sewing of [[Riemann surfaces]] on which one compactifies the M5-brane yields a gluing operation on N=2 SYM theories is due to

* {#Gaiotto09} [[Davide Gaiotto]], _$N=2$ dualities_, JHEP 08 (2012) 034 [[arXiv:0904.2715](https://arxiv.org/abs/0904.2715)]

cf. at *[[AGT correspondence]]*.

The [[topologically twisted D=4 super Yang-Mills theory|topological twisting]] of the compactification which is used around (2.27) there was previously introduced in section 3.1.2 of:

* [[Davide Gaiotto]], [[Gregory Moore]] [[Andrew Neitzke]], _Wall-crossing, Hitchin Systems, and the WKB Approximation_, ([arXiv:0907.3987](http://arxiv.org/abs/0907.3987))

and is discussed also for instance in section 5.1 of

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

(This is possibly also the mechanism behind the [[AGT correspondence]], though the details behind that statement seem to be unclear.)

A brief review of these matters is in ([Moore 12, section 7](#Moore12)). A formalization of the [[topological twist]] in  [[perturbation theory]] formalized by [[factorization algebras]] with values in [[BV complexes]] is in section 16 of

* {#Costello11} [[Kevin Costello]], *Notes on supersymmetric and holomorphic field theories in dimension 2 and 4*, talk at *[Geometric and Algebraic Structures in Mathematics](https://www.math.stonybrook.edu/dennisfest/)*, Stony Brook (2011) [[arXiv:1111.4234](https://arxiv.org/abs/1111.4234)]

For more on this see at _[[topologically twisted D=4 super Yang-Mills theory]]_.

An amplification of the relevance of this to the understanding of [[S-duality]]/[[electric-magnetic duality]] is in

* [[Edward Witten]], _[[Conformal field theory in four and six dimensions]]_, in [[Ulrike Tillmann]] (ed.) _Topology, geometry and quantum field theory_ LMS Lecture Note Series (2004) ([arXiv:0712.0157](http://arxiv.org/abs/0712.0157))
 {#Witten07}

and the resulting relation to the [[geometric Langlands correspondence]] is discussed in

* [[Edward Witten]], _Geometric Langlands From Six Dimensions_ ([arXiv:0905.2720](http://arxiv.org/abs/0905.2720))
 {#Witten09}.

The corresponding dual theory under [[AdS-CFT duality]] is discussed in

* [[Davide Gaiotto]], [[Juan Maldacena]], _The gravity duals of N=2 superconformal field theories_ ([arXiv:0904.4466](http://arxiv.org/abs/0904.4466))

Discussion of construction of just [[N=1 D=4 super Yang-Mills theory]] this way is in

* [[Ibrahima Bah]], Christopher Beem, Nikolay Bobev, Brian Wecht, _Four-Dimensional SCFTs from M5-Branes_ ([arXiv:1203.0303](http://arxiv.org/abs/1203.0303))

### Relation to 3-branes in M5-branes (M5 self-intersections)

Relation to the [[3-brane in 6d]]:

* [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _Classical M-Fivebrane Dynamics and Quantum $N=2$ Yang-Mills_, Phys. Lett. B418 (1998) 85-90 ([arXiv:hep-th/9710034](https://arxiv.org/abs/hep-th/9710034))

* [[Neil Lambert]], [[Peter West]], _Gauge Fields and M-Fivebrane Dynamics_, Nucl. Phys. B524 (1998) 141-158 ([arXiv:hep-th/9712040](https://arxiv.org/abs/hep-th/9712040))

* {#LambertWest98} [[Neil Lambert]], [[Peter West]], _$N=2$ Superfields and the M-Fivebrane_, Phys. Lett. B424 (1998) 281-287 ([arXiv:hep-th/9801104](https://arxiv.org/abs/hep-th/9801104))

* [[Neil Lambert]], [[Peter West]], _Monopole Dynamics from the M-Fivebrane_, Nucl. Phys. B556 (1999) 177-196 ([arXiv:hep-th/9811025](https://arxiv.org/abs/hep-th/9811025))

and via [[F-theory]] in

* Robert de Mello Koch, Alastair Paulin-Campbell, Joao P. Rodrigues, _Monopole Dynamics in $\mathcal{N}=2$ super Yang-Mills Theory From a Threebrane Probe_, Nucl. Phys. B559 (1999) 143-164 ([arXiv:hep-th/9903207](https://arxiv.org/abs/hep-th/9903207))

[[!redirects N=2 D=4 sYM]]

[[!redirects D=4 N=2 SYM]]

[[!redirects N=2 super Yang-Mills theory]]
[[!redirects N=2 super Yang-Mills theories]]


D=4 N=4 super Yang-Mills theory

[[!redirects N=4 D=4 super Yang-Mills theory]]
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#Contents#
* table of contents
{:toc}

## Idea

The [[super Yang-Mills theory]] in [[dimension]] 4 with the
maximum number $N = 4$ of [[supersymmetries]].

[[!include superconformal symmetry -- table]]

## Properties of 4d SYM

### Conformal invariance

$N=4$ $D=4$ SYM is an [[SCFT]].

### Closed expressions for physical observables
 {#ClosedExpressionsForPhysicalObservables}

Among all [[gauge theory]] [[Lagrangians]] that of $N=4$, $D = 4$ SYM is special in several ways, in particular of course in that it is conformally invariant and in that it has maximal [[supersymmetry]]; and ultimately by the fact that it is the [[Kaluza-Klein mechanism|KK-reduction]] of the very special [[6d (2,0)-superconformal QFT]] and related by [[AdS-CFT|AdS7-CFT6 duality]] to the very special theory of [[11-dimensional supergravity]]/[[M-theory]].

Accordingly, it is to be expected that the [[quantum observables]] of $N=4$, $D = 4$ SYM satisfy special relations that make them more tractable than the observables of a generic [[gauge theory]], in particular by having closed-form expressions.
Indeed, such relations have been and are being uncovered in the last years, in particular in what is called the _[[planar limit]]_ of the theory, where [[scattering amplitudes]] are dominated by [[Feynman diagrams]] that can be given the structure of planar [[graphs]].

This includes notably the following phenomena:

1. The [[operator spectrum]] of the dilatation operator (the part of the [[stress-energy tensor]] which induces conformal transformations) can be expressed in closed form, indeed when regarded as a [[Hamiltonian]] it defines an [[integrable system]] equivalent to [[spin chain]] models. This has been used in particular to explicitly check aspects of the conjectured [[AdS-CFT duality]] of $N=4$, $D= 4$ SYM with [[type II string theory]] on [[anti de Sitter spacetimes]]. See the review ([Beisert et al](#Beisert)).

1. Certain [[scattering amplitudes]] called _maximally helicity violating amplitudes_ ("[[MHV amplitudes]]") simplify drastically as compared to the generic situation and in fact are controled by a certain [[twistor string theory]] whose [[target space]] is a [[twistor space]]. See ([Monteiro](#Monteiro)) for a review.

1. Generally, the [[scattering amplitudes]] of the theory in the planar limit have certain closed-form combinatorial expressions. See ([Arkani-Hamed et al](#Arkani-Hamed)).

Such "exact solutions" of the theory are of interest in that even though N=4, D=4 SYM is very different from [[phenomenology|phenomenologically viable]] models such that [[QCD]] in the [[standard model of particle physics]] in that it is highly (super-)symmetric and conformal, it is still similar enough (being a nonabelian [[gauge theory]] [[minimal coupling|minimally coupled]] to [[fermions]]) that one can or can hope to deduce from these exact results approximate information about these less symmetric theories.

In other words, because understanding [[observables]] in [[QCD]]/[[Yang-Mills theory]] in general is difficult, going to special points in the space of all such theories -- such as the point of N=4, D=4 SYM -- may be hoped to yield a tractable approximation. For more on this way of studying [[QCD]] and other realistic theories by studying instead their highly symmetric but phenomenologically unrealistic siblings, see also at _[[string theory results applied elsewhere]]_.

### Holography, AdS-CFT

$N=4$ $d=4$ SYM is supposed to be related under the [[AdS/CFT correspondence]] to [[type II superstring theory]] [[Kaluza-Klein mechanism|compactified]] on a 5-[[sphere]] to an asymptotically [[anti de Sitter spacetime]].

[[!include gauge theory from AdS-CFT -- table]]

### Twistor space formulation

There is a natural reformulation of the theory using [[twistor]] [[field (physics)|fields]]. See the references [below](#ReferencesTwistorSpace). And see at _[[twistor string theory]]_.

### Topological twists

There is a topological twist of 4d SYM to a [[TQFT]] -- the [[Kapustin-Witten TQFT]]. Its [[S-duality]] is supposed to contain [[geometric Langlands duality]] as a special case.

See at _[[topologically twisted D=4 super Yang-Mills theory]]_.

## Related concepts

* [[super Yang-Mills theory]]
* [[N=1 D=4 super Yang-Mills theory]]

* [[N=2 D=4 super Yang-Mills theory]]

* [[N=4 D=3 super Yang-Mills theory]]

* [[D=5 super Yang-Mills theory]]

* [[topologically twisted D=4 super Yang-Mills theory]]

[[!include table of branes]]

## References

### General

Discussion of [[D=4 N=4 SYM]] with emphasis on [[multi-trace operators]]:

* P. J. Heslop, [[Paul Howe]], _Aspects of $\mathcal{N}$=4 SYM_, JHEP 0401 (2004) 058 ([arXiv:hep-th/0307210](https://arxiv.org/abs/hep-th/0307210))

[[superconformal group|Superconformal]] invariance of $\mathcal{N}=4$, $D=4$ SYM can be shown with the result of

* Robert Leigh, [[Matthew Strassler]], _Exactly Marginal Operators and Duality in Four Dimensional $\mathcal{N}=1$ Supersymmetric Gauge Theory_ ([arXiv:hep-th/9503121](http://arxiv.org/abs/hep-th/9503121))

(after regarding it as $\mathcal{N}=1$ SYM with three adjoint chiral [[superfield]]s).

Introductions in view of [[single trace operators]] as [[spin chain]] [[integrable systems]] in the [[AdS/CFT correspondence]]:

* Joseph A. Minahan _Review of AdS/CFT Integrability, Chapter I.1: Spin Chains in $\mathcal{N}=4$ Super Yang-Mills_ ([arXiv:1012.3983](http://arxiv.org/abs/1012.3983))

More recent results are in

* Simon Caron-Huot, _Superconformal symmetry and two-loop amplitudes in planar N=4 super Yang-Mills_ ([arXiv:1105.5606](http://arxiv.org/abs/1105.5606))

### Matrix model description

On the [[BMN matrix model]] as a [[KK-compactification]] of [[D=4 N=4 super Yang-Mills theory]] on the [[3-sphere]]:

* [[Nakwoo Kim]], [[Thomas Klose]], [[Jan Plefka]], _Plane-wave Matrix Theory from N=4 Super Yang-Mills on $\mathbb{R} \times S^3$_, Nucl. Phys. B671:359-382, 2003 ([arXiv:hep-th/0306054](https://arxiv.org/abs/hep-th/0306054))

Using the [[KK-compactification]] of [[D=4 N=4 super Yang-Mills theory]] to the [[BMN matrix model]] for [[lattice gauge theory]]-computations in [[D=4 N=4 SYM]] and for numerical checks of the [[AdS-CFT correspondence]]:

* Masazumi Honda, [[Goro Ishiki]], Sang-Woo Kim, Jun Nishimura, Asato Tsuchiya, _Direct test of the AdS/CFT correspondence by Monte Carlo studies of N=4 super Yang-Mills theory_, JHEP 1311 (2013) 200 [[arXiv:1308.3525](https://arxiv.org/abs/1308.3525)]

### Planar sector, integrability, MHV amplitudes

A comprehensive discussion of the [[integrability]] related to anomalous dimension in the planar sector is in

* [[Niklas Beisert]] et al., _Review of AdS/CFT Integrability, An Overview_
Lett. Math. Phys. vv, pp (2011), ([arXiv:1012.3982](http://arxiv.org/abs/1012.3982)).
 {#Beisert}

A review of [[MHV amplitudes]] is in

* Gustavo Machado Monteiro, _MHV Tree Amplitudes in Super-Yang-Mills and in Superstring Theory_ (2010) ([pdf](http://www.ift.unesp.br/posgrad/gustavo.pdf))
  {#Monteiro}

Discussion of special properties of [[scattering amplitudes]] in the planar sector is in

* [[Nima Arkani-Hamed]], Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, Jaroslav Trnka, _Scattering Amplitudes and the Positive Grassmannian_ ([arXiv:1212.5605](http://arxiv.org/abs/1212.5605))
 {#Arkani-Hamed}

For mathematical background see

* Alexander Postnikov, _Total positivity, Grassmannians, and networks_ ([arXiv:math/0609764](http://arxiv.org/abs/math/0609764))

### Twistor space formulation
 {#ReferencesTwistorSpace}

The [[twistor]] space formulation of $N=4$ $D = 4$ SYM was originally found from the [[B-model]] [[string theory]] in

* [[Edward Witten]], _Perturbative Gauge Theory As A String Theory In Twistor Space_, Commun. Math. Phys. 252:189-258,2004 ([arXiv:hep-th/0312171/](http://arxiv.org/abs/hep-th/0312171/))

A comprehensive discussion is in

* Rutger Boels, Lionel Mason, David Skinner, _Supersymmetric Gauge Theories in Twistor Space_, JHEP 0702:014,2007 ([arXiv:hep-th/0604040](http://arxiv.org/abs/hep-th/0604040))

See also ([Monteiro](#Monteiro)) below.

### Scattering amplitudes

For the moment see at

* _[[scattering amplitude]]_

and also at

* _[[motivic multiple zeta values]]_.

### Relation to number theory

Relating the [[abc conjecture]] to [[D=4 N=4 super Yang-Mills theory]]:

* [[Yang-Hui He]], Zhi Hu, Malte Probst, James Read, _Yang-Mills Theory and the ABC Conjecture_,  International Journal of Modern Physics A Vol. 33, No. 13, 1850053 (2018)  ([arXiv:1602.01780](https://arxiv.org/abs/1602.01780), [doi:10.1142/S0217751X18500537](https://doi.org/10.1142/S0217751X18500537))

[[!redirects N=4 D=4 SYM]]

[[!redirects N=4 D=4 sYM]]

[[!redirects D=4 N=4 SYM]]
[[!redirects D=4 N=4 super Yang-Mills]]