3d TQFT

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#### Functorial quantum field theory
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#Contents#
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## Idea

[[TQFT]] of [[dimension]] 3.

## Constructions

From [[3-modules]] realized as [[sesquialgebra]], [[bialgebras]], [[Hopf algebra]], [[quantum groups]], [[fusion categories]]:

* [[Reshetikhin-Turaev model]]

* [[Turaev-Viro model]]

## Examples

* [[3d Chern-Simons theory]]

  * [[Dijkgraaf-Witten theory]]

  * [[analytically continued Chern-Simons theory]]

  * [[Courant sigma-model]]

* [[Levin-Wen model]]

## Related concepts

* [[TQFT]]

  * [[2d TQFT]]

    * [[2d Chern-Simons theory]]

  * [[3d TQFT]]

    * [[3d Chern-Simons theory]]

  * [[4d TQFT]]

    * [[4d Chern-Simons theory]]

## References
 {#References}

* [[Kevin Walker]], _On Witten's 3-manifold invariants_ (1991) [pdf](http://tqft.net/other-papers/KevinWalkerTQFTNotes.pdf)
  {#Walker1991}

* {#CarterKauffmanSaito96} [[Scott Carter]], [[Louis Kauffman]], Masahico Saito, _Diagrams, Singularities, and Their Algebraic Interpretations_, in "10th Brazilian Topology Meeting, S. Carlos, July 22 26, 1996 ([pdf](http://homepages.math.uic.edu/~kauffman/cksBrasil.pdf) (dead))

Discussion via the [[cobordism hypothesis]] with [[fusion categories]] as [[fully dualizable objects]] is in

* {#DSPS} [[Chris Douglas]], [[Chris Schommer-Pries]], [[Noah Snyder]], _The Structure of Fusion Categories via 3D TQFTs_ ([talk pdf](https://sites.google.com/site/chrisschommerpriesmath/Home/recent-and-upcoming-talks/UPennTalk.pdf?attredirects=0))

* {#DSPS13} [[Chris Douglas]], [[Chris Schommer-Pries]], [[Noah Snyder]], _Dualizable tensor categories_ ([arXiv:1312.7188](http://arxiv.org/abs/1312.7188))

* [[Bruce Bartlett]], [[Christopher Douglas]], [[Christopher Schommer-Pries]], [[Jamie Vicary]], _Extended 3-dimensional bordism as the theory of modular objects_ ([arXiv:1411.0945](http://arxiv.org/abs/1411.0945))

Constructions in the generality of non-[[semisimple category|semisimple]] [[tensor categories]]:

* Christian Blanchet, Marco De Renzi, _Modular Categories and TQFTs Beyond Semisimplicity_ ([arXiv:2011.12932](https://arxiv.org/abs/2011.12932))

[[!redirects 3d TQFTs]]

[[!redirects 3d topological field theory]]
[[!redirects 3d topological field theories]]


3d-3d correspondence

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###Context###
#### Duality in string theory
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#### String theory
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[[!include string theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A [[duality in string theory]]: relation between the
[[D=3 N=2 super Yang-Mills theory|D=3 N=2 SYM]]
[[worldvolume]] [[quantum field theory]] on [[M5-brane]] [[wrapped brane|wrapped]] on a ([[hyperbolic 3-manifold|hyperbolic]]) [[3-manifold]] and 3d [[Chern-Simons theory]]/[[analytically continued Chern-Simons theory]].

## Properties

### Relation to volume conjecture
 {#RelationToTheVolumeConjecture}

In [Gang-Kim-Lee 14b, 3.2](#GangKimLee14b), [Gang-Kim 18 (21)](#GangKim18) it is argued that the [[volume conjecture]] for [[Chern-Simons theory]] on [[hyperbolic 3-manifolds]] $\Sigma^3$ is the combined statement of two [[dualities in string theory]]

1. [[AdS/CFT duality]]

1. [[3d-3d correspondence]]

for the situation of [[M5-branes]] [[wrapped brane|wrapped on]] $\Sigma^3$ ([DGKV 10](#DGKV10)):

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

## Related concepts

[[wrapped brane|Wrapping]] the [[M5-brane]] on a [[torsion subgroup|torsion]] 3-cycle yields: [[fractional M2-brane]].

[[wrapped brane|Wrapping]] the [[M5-brane]] instead on a 2-manifold yields: [[AGT correspondence]].

* [[McKay correspondence]]

* [[AGT correspondence]]

* [[knots-quivers correspondence]]

## References

### General

Original articles include

* [[Yuji Terashima]], [[Masahito Yamazaki]], _$SL(2,\mathbb{R})$ Chern-Simons, Liouville, and Gauge Theory on Duality Walls_, JHEP 1108:135, 2011 ([arXiv:1103.5748](https://arxiv.org/abs/1103.5748))

* [[Yuji Terashima]], [[Masahito Yamazaki]], _Semiclassical Analysis of the 3d/3d Relation_, Phys.Rev.D88:026011, 2013 ([arXiv:1106.3066](https://arxiv.org/abs/1106.3066))

* [[Tudor Dimofte]], [[Davide Gaiotto]], [[Sergei Gukov]], _Gauge Theories Labelled by Three-Manifolds_ ([arXiv:1108.4389](https://arxiv.org/abs/1108.4389))

* [[Tudor Dimofte]], [[Davide Gaiotto]], [[Sergei Gukov]], _3-Manifolds and 3d Indices_ ([arXiv:1112.5179](https://arxiv.org/abs/1112.5179))

* [[Dongmin Gang]], Eunkyung Koh, Sangmin Lee, [[Jaemo Park]], _Superconformal Index and 3d-3d Correspondence for Mapping Cylinder/Torus_, JHEP01(2014)063 ([arXiv:1305.0937](https://arxiv.org/abs/1305.0937))

* [[Junya Yagi]], _3d TQFT from 6d SCFT_, JHEP08(2013)017 ([arXiv:1305.0291](https://arxiv.org/abs/1305.0291))

* Sungjay Lee, Masahito Yamazaki, _3d Chern-Simons Theory from M5-branes_, JHEP1312:035, 2013 ([arXiv:1305.2429](https://arxiv.org/abs/1305.2429))

* [[Clay Cordova]], [[Daniel Jafferis]], _Complex Chern-Simons from M5-branes on the Squashed Three-Sphere_ ([arXiv:1305.2891](https://arxiv.org/abs/1305.2891))

* [[Du Pei]], Ke Ye, _A 3d-3d appetizer_, JHEP 11 (2016) 008 ([arXiv:1503.04809](https://arxiv.org/abs/1503.04809))

* Sungbong Chun, [[Sergei Gukov]], Sunghyuk Park, Nikita Sopenko, _3d-3d correspondence for mapping tori_ ([arxiv:1911.08456](https://arxiv.org/abs/1911.08456))

* Hee-Joong Chung, _Index for a Model of 3d-3d Correspondence for Plumbed 3-Manifolds_ ([arXiv:1912.13486](https://arxiv.org/abs/1912.13486))

Specifically for [[Seifert 3-manifolds]] (such as [[lens spaces]]):

* [[Du Pei]], _3d-3d correspondence for Seifert manifolds_, 2016 ([spire:1469350](https://inspirehep.net/literature/1469350), [pdf](https://inspirehep.net/files/d509ff9e32448da3a5674f286b93224a))

Review is in

* {#Dimofte14} [[Tudor Dimofte]], _3d Superconformal Theories from Three-Manifolds_, In: [[Jörg  Teschner]] (ed.), *Exact Results on $\mathcal{N} = 2$ Supersymmetric Gauge Theories*, Springer 2015 ([arXiv:1412.7129](https://arxiv.org/abs/1412.7129), [doi:10.1007/978-3-319-18769-3_11](https://doi.org/10.1007/978-3-319-18769-3_11))

Relation of the [[AGT-correspondence]] to the [[D=6 N=(2,0) SCFT]] and the [[3d-3d correspondence]]:

* [[Clay Cordova]], [[Daniel Jafferis]], _Toda Theory From Six Dimensions_, J. High Energ. Phys. (2017) 2017: 106 ([arxiv:1605.03997](https://arxiv.org/abs/1605.03997))

* Sam van Leuven, Gerben Oling, _Generalized Toda Theory from Six Dimensions and the Conifold_, J. High Energ. Phys. (2017) 2017: 50 ([arxiv:1708.07840](https://arxiv.org/abs/1708.07840))

See also:

* Julius Eckhard, Heeyeon Kim, [[Sakura Schafer-Nameki]], Brian Willett, _Higher-Form Symmetries, Bethe Vacua, and the 3d-3d Correspondence_ ([arxiv:1910.14086](https://arxiv.org/abs/1910.14086))

* Yale Fan, *3D-3D Correspondence from Seifert Fibering Operators* ([arXiv:2008.13202](https://arxiv.org/abs/2008.13202))

### Black 5-branes and AdS/CFT

Discussion of [[D=11 N=1 supergravity]] solutions for the [[near horizon geometry]] of [[black brane|black]] [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-manifolds]] $\Sigma^3 = H^3/\Gamma$:

* {#DGKV10} Aristomenis Donos, [[Jerome Gauntlett]], [[Nakwoo Kim]], Oscar Varelam, _Wrapped M5-branes, consistent truncations and AdS/CMT_, JHEP 1012:003, 2010 ([arXiv:1009.3805](https://arxiv.org/abs/1009.3805))

* {#GangKimLee14a} [[Dongmin Gang]], [[Nakwoo Kim]], Sangmin Lee, _Holography of Wrapped M5-branes and Chern-Simons theory_, Physics Letters B, Volume 733, 2 June 2014, Pages 316-319 ([arXiv:1401.3595](https://arxiv.org/abs/1401.3595))

* {#GangKimLee14b} [[Dongmin Gang]], [[Nakwoo Kim]], Sangmin Lee, _Holography of 3d-3d correspondence at Large $N$_, JHEP04(2015) 091 ([arXiv:1409.6206](https://arxiv.org/abs/1409.6206))

* {#GangKim18} [[Dongmin Gang]], [[Nakwoo Kim]], _Large $N$ twisted partition functions in 3d-3d correspondence and Holography_, Phys. Rev. D 99, 021901 (2019) ([arXiv:1808.02797](https://arxiv.org/abs/1808.02797))

* {#GangKimPandoZayas19} [[Dongmin Gang]], [[Nakwoo Kim]], Leopoldo A. Pando Zayas, _Precision Microstate Counting for the Entropy of Wrapped M5-branes_ ([arXiv:1905.01559](https://arxiv.org/abs/1905.01559))

### Relation to perturbative CS-observables

Relation to [[perturbative quantization of 3d Chern-Simons theory]]:

specifically to [[Reidemeister torsion]]:

* [[Dongmin Gang]], Seonhwa Kim, Seokbeom Yoon,  _Adjoint Reidemeister torsions from wrapped M5-branes_ ([arXiv:1911.10718](https://arxiv.org/abs/1911.10718))

### Relation to volume conjecture

Discussion of the [[volume conjecture]] by combining the 3d/3d correspondence with [[AdS/CFT]] in these backgrounds:

* [Gang-Kim-Lee 14b, Section 3.2](#GangKimLee14b)

* [Gang-Kim 18, around (21)](#GangKim18)

Enhanced to a [[defect field theory]]:

* [[Dongmin Gang]], [[Nakwoo Kim]], Mauricio Romo, Masahito Yamazaki, _Aspects of Defects in 3d-3d Correspondence_, J. High Energ. Phys. (2016) ([arXiv:1510.05011](https://arxiv.org/abs/1510.05011))

More in:

* Jin-Beom Bae, [[Dongmin Gang]], Jaehoon Lee, _3d $\mathcal{N}=2$ minimal SCFTs from Wrapped M5-branes_, JHEP 08 (2017) 118 ([arXiv:1610.09259](https://arxiv.org/abs/1610.09259))

### Knot invariants

For discussion of [[knot invariants]]:

* [[Sergei Gukov]], Pavel Putrov, [[Cumrun Vafa]], _Fivebranes and 3-manifold homology_, J. High Energ. Phys. (2017) 2017: 71 ([arXiv:1602.05302](https://arxiv.org/abs/1602.05302))

### Entropy computation

Applied to computation of [[Bekenstein-Hawking entropy]] for [[black holes in string theory]]:

* [Gang-Kim-Pando-Zayas 19](#GangKimPandoZayas19)

* Francesco Benini, Dongmin Gang, Leopoldo A. Pando Zayas, _Rotating Black Hole Entropy from M5 Branes_ ([arXiv:1909.11612](https://arxiv.org/abs/1909.11612))

[[!redirects 3d-3d correspondences]]

[[!redirects 3d-3d relation]]
[[!redirects 3d-3d relations]]

[[!redirects 3d/3d correspondence]]
[[!redirects 3d/3d correspondences]]

[[!redirects 3d/3d relation]]
[[!redirects 3d/3d relations]]


3x3 lemma

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### Context
#### Diagram chasing lemmas
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#### Homological algebra
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## Idea

The _$3 \times 3$-lemma_ or _nine lemma_ is one of the basic [[diagram chasing lemmas]] in [[homological algebra]].

## Statement

+-- {: .num_lemma}
###### Lemma

Let

$$
  0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0
$$

be a [[short exact sequence]] of [[chain complexes]]. Then
if two of the three complexes $A_\bullet, B_\bullet, C_\bullet$
are [[long exact sequence|exact]], so is the remaining third.

=--

+-- {: .num_lemma}
###### Lemma

Let

$$
  \array{
     && 0 && 0 && 0 &&
     \\
     && \downarrow && \downarrow && \downarrow &&
     \\
     0 &\to& A' &\to& B' &\to& C' &\to& 0
     \\
     && \downarrow && \downarrow && \downarrow &&
     \\
     0 &\to& A &\to& B &\to& C &\to& 0
     \\
     && \downarrow && \downarrow && \downarrow &&
     \\
     0 &\to& A'' &\to& B'' &\to& C'' &\to& 0
     \\
     && \downarrow && \downarrow && \downarrow &&
     \\
     && 0 && 0 && 0 &&
  }
$$

be a [[commuting diagram]] in some [[abelian category]] such that each of the three columns is an [[exact sequence]].  Then

1. If the two bottom rows are exact, then so is the top.

1. If the top two rows are exact, then so is the bottom.

1. If the top and bottom rows are exact _and_ $A \to C$ is the [[zero morphism]], then also the middle row is exact.

=--

A proof by way of the [[salamander lemma]] is spelled out in detail at _[Salamander lemma - Implications - 3x3 lemma](http://ncatlab.org/nlab/show/salamander+lemma#3x3Lemmas)_.

## Related concepts

* [[salamander lemma]]

* [[snake lemma]], [[5-lemma]]

* [[horseshoe lemma]]

## References

### In abelian categories

An early appearance of the $3 \times 3$-lemma is as lemma (5.5) in

* D. A. Buchsbaum, _Exact categories and duality_, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 ([JSTOR](http://www.jstor.org/stable/1993003))

In

* [[Charles Weibel]], _[[An introduction to homological algebra]]_

it appears as exercise 1.3.2.

The sharp $3 \times 3$-lemma appears as lemma 2 in

* Temple Fay, [[Keith Hardie]], [[Peter Hilton]], _The two-square lemma_, Publicacions Matemàtiques, Vol 33 (1989) ([pdf](http://dmle.cindoc.csic.es/pdf/PUBLICACIONSMATEMATIQUES_1989_33_01_10.pdf))

Also lemma 3.2-3.4 of

* [[Saunders MacLane]], _Homology_, Grundlehren der math. Wissenshaften vol 114, Springer (1995)

### In non-abelian categories

Discussion of generalization to non-abelian categories is in

* Marino Gran, Diana Rodelo, _Goursat categories and the $3 \times 3$-lemma_, Applied Categorical Structures, Vol. 20, No 3, 2012, 229-238. ([journal](http://www.google.com/url?q=http%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%252Fs10485-010-9236-x&sa=D&sntz=1&usg=AFQjCNG1kEggBxeeDfQlkWOa4ShUo8vwLA), [pdf slides](http://ct2010.disi.unige.it/slides/Rodelo_CT2010.pdf))

* Marino Gran, Zurab Janelidze and Diana Rodelo, _$3 \times 3$ lemma for star-exact sequences_, Homology, Homotopy and Applications, Vol. 14 (2012), No. 2, pp.1-22. ([journal](http://www.intlpress.com/HHA/v14/n2/a1/))

* [[Dominique Bourn]], _$3 \times 3$-lemma and protomodularity_, Journal of Algebra, Volume 236, Number 2, 15 February 2001 , pp. 778-795(18)

* [[Dominique Bourn]], _The denormalized $3 \times 3$ lemma_, Journal of Pure and Applied Algebra, Volume 177, Issue 2, 24 January 2003, Pages 113-129, doi:[10.1016/S0022-4049(02)00143-3](https://doi.org/10.1016/S0022-4049%2802%2900143-3)

[[!redirects nine lemma]]
[[!redirects 9 lemma]]
[[!redirects 3x3-lemma]]

[[!redirects sharp 3x3 lemma]]


4-3-2 8-7-6

The talk

* [[Daniel Freed]],

  **4-3-2 8-7-6**

  talk at _[ASPECTS of Topology](https://people.maths.ox.ac.uk/tillmann/ASPECTS.html)_ Dec 2012

  [pdf](https://people.maths.ox.ac.uk/tillmann/ASPECTSfreed.pdf), [[Freed432876.pdf:file]]

discusses aspects and open problems of understanding

* [[3d Chern-Simons theory]] $\leftarrow$[[holographic principle]]$\to$ [[2d WZW model]]

* [[7d Chern-Simons theory]] $\leftarrow$[[holographic principle]]$\to$ [[6d (2,0)-supersymmetric QFT]]

in the context of [[extended TQFT]].

See also

* [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], _[[Topological Quantum Field Theories from Compact Lie Groups]]_

(Other talks at this conference include [[David Ben-Zvi]]: _[[Loop Groups, Characters and Elliptic Curves]]_.)

One explicit version of

* [[3d Chern-Simons theory]] $\leftarrow$[[holographic principle]]$\to$ [[2d WZW model]]

is the relation [[Turaev-Viro model]] $\leftrightarrow$ [[Crane-Yetter model]] due to theorem 2 in

* [[John Barrett]], J. Garcia-Islas, [[João Faria Martins]], _Observables in the Turaev-Viro and Crane-Yetter models_, J. Math. Phys. 48:093508, 2007 ([arXiv:math/0411281](http://arxiv.org/abs/math/0411281))
  {#BarrettGarciIslasMartins04}

category: reference


4-manifold

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#### Geometry
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## Idea

a [[manifold]] of [[dimension]] 4.

## Examples

* [[4-sphere]]

* [[spacetime]]

## Properties

#### Cohomotopy

Let $X$ be a [[4-manifold]] which is [[connected topological space|connected]] and [[orientation|oriented]].

The [[Pontryagin-Thom construction]] as [above](cohomotopy#RelationToCobordismGroup) gives for $n \in \mathbb{Z}$ the [[commuting diagram]] of sets

$$
  \array{
     \pi^n(X)
       &\overset{\simeq}{\longrightarrow}&
     \mathbb{F}_{4-n}(X)
     \\
     {}^{ \mathllap{h^n} }
     \downarrow
       &&
     \downarrow^{ h_{4-n} }
     \\
     H^n(X,\mathbb{Z})
       &\underset{\simeq}{\longrightarrow}&
     H_{4-n}(X,\mathbb{Z})
     \,,
  }
$$

where $\pi^\bullet$ denotes [[cohomotopy]] sets, $H^\bullet$ denotes [[ordinary cohomology]], $H_\bullet$ denotes [[ordinary homology]] and $\mathbb{F}_\bullet$ is [[normal bundle|normally]] [[framing|framed]] [[cobordism classes]] of [[normal bundle|normally]] [[framing|framed]] [[submanifolds]]. Finally $h^n$ is the operation of pullback of the generating integral cohomology class on $S^n$ (by the nature of [[Eilenberg-MacLane spaces]]):

$$
  h^n(\alpha) \;\colon\; X \overset{\alpha}{\longrightarrow} S^n \overset{generator}{\longrightarrow} B^n \mathbb{Z}
  \,.
$$

Now

* $h^0$, $h^1$, $h^4$ are [[isomorphisms]]

* $h^3$ is an isomorphism if $X$ is "odd" in that it contains at least one closed oriented [[surface]] of odd self-intersection, otherwise $h^3$ becomes an isomorphism on a $\mathbb{Z}/2$-[[quotient group]] of $\pi^3(X)$ (which is a group via the [[group]]-[[structure]] of the [[3-sphere]] ([[special unitary group|SU(2)]]))

([Kirby-Melvin-Teichner 12](#KirbyMelvinTeichner12))

## Related concepts

* [[Yang-Mills instanton]]

* [[Donaldson-Thomas invariants]]

* [[2-manifold]], [[3-manifold]], [[8-manifold]]

## References

### General

All [[PL manifold|PL]] [[4-manifolds]] are _simple_ [[branched covers]] of the  [[4-sphere]]:

* {#Piergallini95} [[Riccardo Piergallini]], _Four-manifolds as 4-fold branched covers of $S^4$_, Topology Volume 34, Issue 3, July 1995 (doi:10.1016/0040-9383(94)00034-I, [pdf](https://core.ac.uk/download/pdf/82379618.pdf))

* {#IoriPiergallini02} Massimiliano Iori, [[Riccardo Piergallini]], _4-manifolds as covers of the 4-sphere branched over non-singular surfaces_, Geom. Topol. 6 (2002) 393-401 ([arXiv:math/0203087](https://arxiv.org/abs/math/0203087))

On [[cohomotopy]] of 4-manifolds:

* [[Daniel Freed]], [[Karen Uhlenbeck]], Appendix B of: _Instantons and Four-Manifolds_, Mathematical Sciences Research Institute Publications, Springer 1991 ([doi:10.1007/978-1-4613-9703-8](https://link.springer.com/book/10.1007/978-1-4613-9703-8))

* {#KirbyMelvinTeichner12} [[Robion Kirby]], [[Paul Melvin]], [[Peter Teichner]], _Cohomotopy sets of 4-manifolds_, GTM 18 (2012) 161-190 ([arXiv:1203.1608](https://arxiv.org/abs/1203.1608))

### Relation to 2d CFTs via 6d CFT

On [[KK-compactification]] of [[D=6 N=(2,0) SCFT]] on [[4-manifolds]] to [[2d CFTs]]:

* [[Abhijit Gadde]], [[Sergei Gukov]], [[Pavel Putrov]], *Fivebranes and 4-manifolds*, in: *Arbeitstagung Bonn 2013*, Progress in Mathematics **319**, Birkhäuser (2016) [[arXiv:1306.4320](https://arxiv.org/abs/1306.4320), [doi:10.1007/978-3-319-43648-7_7](https://doi.org/10.1007/978-3-319-43648-7_7)]

* [[Mykola Dedushenko]], [[Sergei Gukov]], [[Pavel Putrov]], *Vertex algebras and 4-manifold invariants*, Chapter 11 in: *Geometry and Physics: Volume I*  (2018) 249–318 [[arXiv:1705.01645](https://arxiv.org/abs/1705.01645), [doi:10.1093/oso/9780198802013.003.0011](https://doi.org/10.1093/oso/9780198802013.003.0011)]

* [[Boris Feigin]], [[Sergei Gukov]], $VOA[M_4]$, J. Math. Phys. **61** 012302 (2020) [[arXiv:1806.02470](https://arxiv.org/abs/1806.02470), [doi:10.1063/1.5100059](https://doi.org/10.1063/1.5100059)]

In relation to [[M5-brane elliptic genus]]:

* {#GukovPeiPutrovVafa18} [[Sergei Gukov]], [[Du Pei]], [[Pavel Putrov]], [[Cumrun Vafa]], *4-manifolds and topological modular forms*,  J. High Energ. Phys. **2021** 84 (2021) [[arXiv:1811.07884](https://arxiv.org/abs/1811.07884), doi:10.1007/JHEP05(2021)084, [spire:1704312](https://inspirehep.net/literature/1704312)]

and in relation to [[QFT with defects|defects]]:

* [[Jin Chen]], [[Wei Cui]], [[Babak Haghighat]], [[Yi-Nan Wang]], *SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds*, JHEP **2023** 208 (2023) [[arXiv:2305.09734](https://arxiv.org/abs/2305.09734), doi:10.1007/JHEP11(2023)208]

[[!redirects 4-manifolds]]


4-sphere

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* table of contents
{:toc}

## Idea

The [[sphere]] of [[dimension]] 4.

## Properties

### Basic differential geometry
  {#Differential geometry}

An embedded radius-$R$ 4-sphere inherits a [[volume form]] (a degree-4 [[differential form]] from the ambient $5$-dimensional [[Euclidean space]], namely
$$
  \omega = \frac{1}{R} \sum_{j=1}^{5} (-1)^{j-1} x_j d x_1 \wedge \ldots \wedge \widehat{d x_j} \wedge \ldots \wedge d x_{5}
$$
where the hat means omit that factor. This is equal to $\ast d r$, where $\ast$ is the [[Hodge star operator]] in $\mathbb{R}^5$ for the Euclidean [[Riemannian metric|metric]], and $d r$ is the [[exterior derivative]] of the radius function.

The volume of the manifold $S^4$ with this volume form is then given by $8\pi^2R^4/3$.

### Coset space structure
 {#CosetSpaceStructure}

As any [[sphere]], the [[4-sphere]] has the [[coset space]] [[structure]]

$$
  S^4 \simeq O(5)/O(4) \simeq SO(5)/SO(4) \simeq Spin(5)/Spin(4)\simeq Pin(5)/Pin(4).
$$

There is also this:

+-- {: .num_example #Sp2Sp1BySp1Sp1Sp1IsS4}
###### Example

The [[coset space]] of [[Sp(2).Sp(1)]] ([this Def.](SpnSp1#SpnSp1)) by [[Sp(1)Sp(1)Sp(1)]] ([this Def.](SpnSp1#Spin4Spin3)) is the [[4-sphere]]:

$$
  \frac{
    Sp(2)\cdot Sp(1)
  }
  {
    Sp(1)Sp(1)Sp(1)
  }
  \;\simeq\;
  S^4
  \,.
$$

This follows essentially from the [[quaternionic Hopf fibration]] and its $Sp(2)$-[[equivariant function|equivariance]]...

=--

(e.g. [Bettiol-Mendes 15, (3.1), (3.2), (3.3)](#BettiolMendes15))

### Homotopy groups
 {#HomotopyGroups}

The [[homotopy groups]] of the 4-sphere in low degree are:

| $k$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11  | 12 |
|-----|---|---|---|---|---|---|---|---|---|---|----|-----|----|
| $\pi_k(S^4)$ | $\ast$ | 0 | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z} \times \mathbb{Z}_{12}$ | $\mathbb{Z}_2^2$ | $\mathbb{Z}_2^2$ | $\mathbb{Z}_{24} \times \mathbb{Z}_3$ | $\mathbb{Z}_{15}$ | $\mathbb{Z}_2$ |

For more see at *[[Serre finiteness theorem]]* and at *[[homotopy groups of spheres]]*.

### Bundles over the 4-sphere

#### The quaternionic Hopf fibration

The 4-sphere participates in the [[quaternionic Hopf fibration]], the analog of the complex [[Hopf fibration]] with the field of [[complex numbers]] replaced by the division ring of [[quaternions]] or Hamiltonian numbers $\mathbb{H}$.

$$
  \array{
    S^3 &\hookrightarrow& S^7
    \\
    && \downarrow^\mathrlap{p}
    \\
    && S^4 &\stackrel{}{\longrightarrow}& \mathbf{B} SU(2)
  }
$$

Here the idea is that $S^7$ may be construed as

$$
  \array{
  S^7
    &\simeq
  S(\mathbb{H}^4)
  \\
  & \simeq
  \{(x, y) \in \mathbb{H}^2: {|x|}^2 + {|y|}^2 = 1\},
 }
$$

with $p$ mapping $(x, y)$
to $x/y$ as an element in the [[projective line]] $\mathbb{P}^1(\mathbb{H}) \cong S^4$, with each [[fiber]] a [[torsor]] parameterized by quaternionic [[scalars]] $\lambda$ of unit [[norm]] (so $\lambda \in S^3$).  This canonical $S^3$-bundle (or $SU(2)$-bundle) is classified by a map $S^4 \to \mathbf{B} SU(2)$.

{#HopfParameterization} There are other useful ways to parameterize the quaternionic Hopf fibration, such as the original _[[Hopf construction]]_, see there the section _[Hopf fibrations](Hopf+construction#HopfFibrations)_. By this parameterization $S^4$ is identified as $S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$.

#### The Calabi-Penrose fibration

See at _[[Calabi-Penrose fibration]]_.

#### The complex projective plane
 {#AsAQuotientOfTheComplexProjectivePlane}

+-- {: .num_prop}
###### Proposition
**([[Arnold-Kuiper-Massey theorem]])**

The 4-sphere is the [[quotient space]] of the [[complex projective plane]] by the [[action]] of [[complex conjugation]] (on homogeneous coordinates):

$$
  \mathbb{C}P^2 / (-)^*
  \simeq
  S^4
$$

=--

### Exotic smooth structures

It is open whether the 4-sphere admits an [[exotic smooth structure]]. See ([Freedman-Gompf-Morrison-Walker 09](#FreedmanGompfMorrisonWalker09)) for review.

### $SU(2)$ action
 {#QuaternionAction}

If we identify $\mathbb{R}^5 \simeq_{\mathbb{Q}} \mathbb{R} \oplus \mathbb{H}$ with the [[direct sum]] of the [[real line]] with the [[real vector space]] underlying the [[quaternions]], so that

$$
 S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})
$$

as in the discussion of the quaternionic Hopf fibration [above](#HopfParameterization), then there is induced an [[action]] of the group [[special unitary group|SU(2)]] on the 4-sphere, by identifying

$$
  SU(2) \simeq S(\mathbb{Q})
$$

and then acting by left multiplication.

#### Circle action
  {#CircleAction}

+-- {: .num_prop}
###### Proposition

Given an continuous [[action]] of the [[circle group]] on the [[topological space|topological]] [[4-sphere]], its [[fixed point]] space is of one of two types:

1. either it is the [[0-sphere]] $S^0 \hookrightarrow S^4$

1. or it has the [[rational homotopy theory|rational homotopy type]] of an even-dimensional sphere.

=--

([Félix-Oprea-Tanré 08, Example 7.39](#FelixOpreaTanre08))

For more see at _[[group actions on spheres]]_.

As a special case of the $SU(2)$-action from [above](#QuaternionAction), we discuss the induced circle action via the embedding

$$
  S^1 \simeq U(1) \hookrightarrow SU(2)
  \,.
$$

Consider the following [[circle group|circle]] [[group action on an n-sphere|group action on the 4-sphere]]:

+-- {: .num_defn #CircleActionOn4Sphere}
###### Definition
**($SU(2)$-action on 4-sphere)**

Regard

$$
  S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})
$$

as the [[unit sphere]] inside the [[direct sum]] (as [[real vector spaces]]) of the [[real numbers]] with the [[quaternions]], and regard the [[special unitary group]] $SU(2)$ as the group of unit-norm quaternions

$$
  SU(2) \simeq S(\mathbb{H},\cdot)
$$

In particular this restricts to an [[action]] of the [[circle group]]

$$
  S^1 \simeq U(1) \hookrightarrow SU(2)
$$

(as the [[diagonal matrices]] inside $SU(2)$) on the 4-sphere.

=--

The resulting ordinary [[quotient]] is $S^4/_{ord} S^1 \simeq S^3$ and the [[projection]] $S^4 \to S^3$ is the [[suspension]] of the [[complex Hopf fibration]] $S^3 \to S^2$.

The [[fixed point]] set of the action is the two poles

$$
  S^0 \;=\; \{(\pm 1, 0,0,0,0)\} \;\in\; \mathbb{R} \oplus \mathbb{H}
$$

introduced by the suspension, hence forms the [[0-sphere]] space. Since this is not the [[empty set]], the [[homotopy quotient]] $S^4 // S^1$ of the [[circle action]] differs from $S^3$, but there is still the canonical [[projection]]

$$
  S^4//S^1 \longrightarrow S^4 / S^1 \simeq S^3
  \,.
$$

Hence both $S^4$ and $S^4 // S^1$ are canonically [[homotopy types]] over $S^3$.

A [[minimal dg-module]] presentation in [[rational homotopy theory]] for these projections is given in [Roig & Saralegi-Aranguren 00, second page](#RoigSaralegiAranguren00):

+-- {: .num_prop #FourSphereOverThreeSphereMinimalDgModel}
###### Proposition
**([Roig & Saralegi-Aranguren 00, p. 2](#RoigSaralegiAranguren00))**

Write

$$
  CE(\mathfrak{l}(S^3))) = Sym^\bullet \langle \underset{\text{deg 3}}{\underbrace{h_3}} \rangle
$$

for the [[minimal Sullivan model]] of the [[3-sphere]]. Then [[rational homotopy theory|rational]] [[minimal dg-modules]] for the maps (via Def. \ref{CircleActionOn4Sphere})

$$
  \array{
    S^4
    \\
    \downarrow
    \\
    S^3
  }
  \,,\phantom{AA}
  \array{
    S^4//S^1
    \\
    \downarrow
    \\
    S^3
  }
  \,,\phantom{AA}
  \array{
    S^0
    \\
    \downarrow
    \\
    S^3
  }
$$

as [[dg-modules]] over $CE(\mathfrak{l}(S^3))$ are given as follows, respectively:

$$
  \label{FourSphereAndRelatedOverThreeSphereMinimalDGModels}
  \array{
  \text{fibration} & \array{\text{vector space underlying} \\ \text{minimal dg-model}} & \array{ \text{differential on} \\ \text{minimal dg-model} }
   \\
  \array{
    S^4
    \\
    \downarrow
    \\
    S^3
  }
  &
  Sym^\bullet \langle
    \underset{
      deg = 3
    }{
    \underbrace{
      h_3
    }}
  \rangle
  \otimes
  \left\langle
     \underset{deg = 2p}{
     \underbrace{
       \tilde\omega_{2p}
     }},
     \underset{deg = 2p + 4}{
     \underbrace{
       \omega_{2p + 4}
     }} \,\vert\, p \in \mathbb{N}
  \right\rangle
  &
  d \colon
  \left\{
    \begin{aligned}
      \tilde\omega_0 & \mapsto 0
      \\
      \tilde\omega_{2p+2} &\mapsto h_3 \wedge \tilde \omega_{2p}
      \\
      \omega_4 & \mapsto 0
      \\
      \omega_{2p+6} & \mapsto h_3 \wedge \omega_{2p + 4}
    \end{aligned}
    \right.
  \\
  \array{
    S^0
    \\
    \downarrow
    \\
    S^3
  }
  &
  Sym^\bullet \langle
    \underset{
      deg = 3
    }{
    \underbrace{
      h_3
    }}
  \rangle
  \otimes
  \left\langle
     \underset{deg = 2p}{
     \underbrace{
       \tilde \omega_{2p}
     }},
     \underset{
       deg = 2p
     }{
     \underbrace{
       \omega_{2p}
     }} \,\vert\, p \in \mathbb{N}
  \right\rangle
  &
  d \colon
  \left\{
    \begin{aligned}
      \tilde \omega_0 & \mapsto 0
      \\
      \tilde \omega_{2p+2} &\mapsto h_3 \wedge \tilde \omega_{2p}
      \\
      \omega_0 & \mapsto 0
      \\
      \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p}
    \end{aligned}
    \right.
  \\
  \array{
    S^4//S^1
    \\
    \downarrow
    \\
    S^3
  }
  &
  Sym^\bullet  \langle
    \underset{
      deg = 3
    }{
    \underbrace{
      h_3
     }}
    ,
    \underset{
      deg = 2
    }{
    \underbrace{
      \omega_2
    }}
  \rangle
   \otimes
  \left\langle
    \underset{deg = 2p}{
    \underbrace{
      \tilde \omega_{2p}
    }},
    \underset{
      deg =2p + 4
    }{
    \underbrace{
      \omega_{2p + 4}
    }} \,\vert\, p \in \mathbb{N}
  \right\rangle
  &
  d \colon
  \left\{
    \begin{aligned}
      \tilde \omega_0 & \mapsto 0
      \\
      \tilde \omega_{2p+2} &\mapsto h_3 \wedge \tilde \omega_{2p}
      \\
      \omega_2 & \mapsto 0
      \\
      \omega_{2p+4} & \mapsto h_3 \wedge \omega_{2p + 2}
    \end{aligned}
  \right.
  }
$$

=--

Beware that in the model for $S^4//S^2$ the element $\omega_2$ induces its entire polynomial algebra as generator of the dg-module.

Notice that we changed the notation of the generators compared to [Roig & Saralegi-Aranguren 00, second page](#RoigSaralegiAranguren00), to bring out the pattern:

|  $\phantom{A}$Roig$\phantom{A}$  |  $\phantom{A}$here$\phantom{A}$    |
|------------|-----------------|
|   $\phantom{A}a\phantom{A}$    |        $\phantom{A}h_3\phantom{A}$    |
|  $\phantom{A}1\phantom{A}$    |  $\phantom{A}\tilde\omega_0\phantom{A}$     |
|  $\phantom{A}c_{2n}\phantom{A}$  | $\phantom{A}\tilde\omega_{2n+2}\phantom{A}$   |
| $\phantom{A}c_{2n+1}\phantom{A}$ | $\phantom{A}\omega_{2n+4}\phantom{A}$ |
|    $\phantom{A}e\phantom{A}$     |    $\phantom{A}\omega_2\phantom{A}$   |
| $\phantom{A}\gamma_{2n}\phantom{A}$ | $\phantom{A}\tilde\omega_{2n}\phantom{A}$ |
| $\phantom{A}\gamma_{2n+1}\phantom{A}$ | $\phantom{A}\omega_{2n}\phantom{A}$ |

#### M5-brane orbifolds

The [[supersymmetry|supersymmetric]] [[Freund-Rubin compactifications]] of [[11-dimensional supergravity]] which are [[Cartesian products]] of 7-dimensional [[anti-de Sitter spacetime]] with a compact 4-dimensional [[orbifold]]

$$
  AdS_7 \times X_4
$$

(the [[near horizon geometry]] of a [[black brane|black]] [[M5-brane]]) are all of the form

$$
  X_4 \simeq S^4//G
$$

where $G \subset SU(2)$ is a [[finite group|finite]] [[subgroup]] of $SU(2)$ (i.e. an [[ADE classification|ADE group]]), [[action|acting]] via the identification $S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$ as [above](#QuaternionAction), and where the double slash denotes the [[homotopy quotient]] ([[orbifold quotient]]).

See ([AFHS 98, section 5.2](#AFHS98), [MF 12, section 8.3](#MF12)).

### Free and cyclic loop space
  {#FreeLoopSpace}

We discuss the [[rational homotopy theory]] of the [[free loop space]] $\mathcal{L}(S^4)$ of $S^4$, as well as the [[cyclic loop space]] $\mathcal{L}(S^4)/S^1$  using the results from _[[Sullivan models of free loop spaces]]_:

+-- {: .num_example}
###### Example

Let $X = S^4$ be the [[4-sphere]]. The corresponding [[rational n-sphere]] has minimal Sullivan model

$$
  (\wedge^\bullet \langle g_4, g_7 \rangle, d)
$$

with

$$
  d g_4 = 0\,,\;\;\;\; d g_7 = -\tfrac{1}{2} g_4 \wedge g_4
  \,.
$$

Hence  [this prop.](Sullivan+model+of+free+loop+space#SullivanModelForTheFreeLoopSpace) gives for the rationalization of
$\mathcal{L}S^4$ the model

$$
  ( \wedge^\bullet \langle \omega_4, \omega_6, h_3, h_7 \rangle  , d_{\mathcal{L}S^4} )
$$

with

$$
  \begin{aligned}
    d_{\mathcal{L}S^4} h_3 & = 0
    \\
    d_{\mathcal{L}S^4} \omega_4 & = 0
    \\
    d_{\mathcal{L}S^4} \omega_6 & = h_3 \wedge \omega_4
    \\
    d_{\mathcal{L}S^4} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4
    \\
  \end{aligned}
$$

and [this prop](Sullivan+model+of+free+loop+space#ModelForS1quotient)
gives for the rationalization of $\mathcal{L}S^4 / / S^1$ the model

$$
  ( \wedge^\bullet \langle \omega_2, \omega_4, \omega_6, h_3, h_7 \rangle  , d_{\mathcal{L}S^4 / / S^1} )
$$

with

$$
  \begin{aligned}
    d_{\mathcal{L}S^4 / / S^1} h_3 & = 0
    \\
    d_{\mathcal{L}S^4 / / S^1} \omega_2 & = 0
    \\
    d_{\mathcal{L}S^4 / / S^1} \omega_4 & = h_3 \wedge \omega_2
    \\
    d_{\mathcal{L}S^4 / / S^1} \omega_6 & = h_3 \wedge \omega_4
    \\
    d_{\mathcal{L}S^4 / / S^1} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4 + \omega_2 \wedge \omega_6
  \end{aligned}
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

Let $\hat \mathfrak{g} \to \mathfrak{g}$ be a [[central extension|central]] [[Lie algebra extension]] by $\mathbb{R}$ of a finite dimensional Lie algebra $\mathfrak{g}$, and let $\mathfrak{g} \longrightarrow b \mathbb{R}$ be the corresponding [[L-∞ algebra cohomology|L-∞ 2-cocycle]] with coefficients in the [[line Lie n-algebra|line Lie 2-algebra]] $b \mathbb{R}$, hence ([[schreiber:The brane bouquet|FSS 13, prop. 3.5]]) so that there is a [[homotopy fiber sequence]] of [[L-∞ algebras]]

$$
  \hat \mathfrak{g}
    \longrightarrow
  \mathfrak{g}
    \overset{\omega_2}{\longrightarrow}
  b \mathbb{R}
$$

which is dually modeled by

$$
  CE(\hat \mathfrak{g})
    =
  ( \wedge^\bullet ( \mathfrak{g}^\ast \oplus \langle e \rangle ), d_{\hat \mathfrak{g}}|_{\mathfrak{g}^\ast} = d_{\mathfrak{g}},\;  d_{\hat \mathfrak{g}} e = \omega_2)
  \,.
$$

For $X$ a space with [[Sullivan model]] $(A_X,d_X)$ write $\mathfrak{l}(X)$ for the corresponding [[L-∞ algebra]], i.e. for the $L_\infty$-algebra whose [[Chevalley-Eilenberg algebra]] is $(A_X,d_X)$:

$$
  CE(\mathfrak{l}X) = (A_X,d_X)
  \,.
$$

Then there is an [[isomorphism]] of [[hom-sets]]

$$
  Hom_{L_\infty Alg}( \hat \mathfrak{g}, \mathfrak{l}(S^4) )
  \;\simeq\;
  Hom_{L_\infty Alg/b \mathbb{R}}( \mathfrak{g}, \mathfrak{l}( \mathcal{L}S^4 / S^1 ) )
  \,,
$$

with $\mathfrak{l}(S^4)$ from [this prop.](Sullivan+model+of+free+loop+space#SullivanModelForTheFreeLoopSpace)
and $\mathfrak{l}(\mathcal{L}S^4 //S^1)$ from [this prop.](Sullivan+model+of+free+loop+space#ModelForS1quotient),
where on the right we have homs in the [[slice category|slice]] over the [[line Lie n-algebra|line Lie 2-algebra]], via [this prop.](Sullivan+model+of+free+loop+space#ModelForS1quotient)

Moreover, this isomorphism takes

$$
  \hat \mathfrak{g}
    \overset{(g_4, g_7)}{\longrightarrow}
  \mathfrak{l}(S^4)
$$

to

$$
  \array{
    \mathfrak{g}
      &&
      \overset{(\omega_2,\omega_4, \omega_6, h_3,h_7)}{\longrightarrow}
      &&
    \mathfrak{l}( \mathcal{L}X / S^1 )
    \\
    &
    {}_{\mathllap{\omega_2}}\searrow
      &&
    \swarrow_{\mathrlap{\omega_2}}
    \\
    && b \mathbb{R}
  }
  \,,
$$

where

$$
  \omega_4 = g_4 - h_3 \wedge e
  \;\,,
  \;\;\;
  h_7 = g_7 + \omega_6 \wedge e
$$

with $e$ being the central generator in $CE(\hat \mathfrak{g})$ from above, and where the equations take place in $\wedge^\bullet \hat \mathfrak{g}^\ast$ with the defining inclusion $\wedge^\bullet \mathfrak{g}^\ast \hookrightarrow \wedge^\bullet \mathfrak{g}^\ast$ understood.

=--

This is observed in ([FSS 16](Sullivan+model+of+free+loop+space#FiorenzaSatiSchreiber16), [FSS 16b](#FSS16b)), where it serves to formalize, on the level of [[rational homotopy theory]], the [[double dimensional reduction]] of [[M-branes]] in [[M-theory]] to [[D-branes]] in [[type IIA string theory]] (for the case that $\mathfrak{g}$ is type IIA [[super Minkowski spacetime]] $\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}$ and $\hat \mathfrak{g}$ is 11d [[super Minkowski spacetime]] $\mathbb{R}^{10,1\vert \mathbf{32}}$, and the cocycles are those of [[schreiber:The brane bouquet]]).

+-- {: .proof}
###### Proof

By the fact that the underlying graded algebras are free, and since $e$ is a generator of odd degree, the given decomposition for $\omega_4$ and $h_7$ is unique.

Hence it is sufficient to observe that under this decomposition the defining equations

$$
  d g_4 = 0
  \,,\;\;\;
  d g_{7} = -\tfrac{1}{2} g_4 \wedge g_4
$$

for the $\mathfrak{l}S^4$-valued cocycle on $\hat \mathfrak{g}$ turn into the equations for a $\mathfrak{l} ( \mathcal{L}S^4 / S^1 )$-valued cocycle on $\mathfrak{g}$. This is straightforward:

$$
  \begin{aligned}
    & d_{\hat \mathfrak{g}} ( \omega_4 + h_3 \wedge e ) = 0
    \\
    \Leftrightarrow \;\;\;\;
    &
    d_{\mathfrak{g}} (\omega_4 - h_3 \wedge \omega_2) = 0
    \;\;\; and \;\;\;
    d_{\mathfrak{g}} h_3 = 0
    \\
    \Leftrightarrow \;\;\;\;
    &
    d_{\mathfrak{g}} \omega_4 = h_3 \wedge \omega_2
    \;\;\; and \;\;\;
    d_{\mathfrak{g}} h_3 = 0
  \end{aligned}
$$

as well as

$$
  \begin{aligned}
    & d_{\hat \mathfrak{g}} ( h_7 - \omega_6 \wedge e )
      = -\tfrac{1}{2}( \omega_4 + h_3 \wedge e ) \wedge (\omega_4 + h_3\wedge e)
   \\
    \Leftrightarrow \;\;\;\;
    &
    d_\mathfrak{g} h_7 - \omega_6 \wedge \omega_2
    =
    -\tfrac{1}{2}\omega_4 \wedge \omega_4
    \;\;\; and \;\;\;
    - d_\mathfrak{g} \omega_6 = - h_3 \wedge \omega_4
    \\
    \Leftrightarrow \;\;\;\;
    &
    d_\mathfrak{g} h_7 = -\tfrac{1}{2}\omega_4 \wedge \omega_4  + \omega_6 \wedge \omega_2
    \;\;\; and \;\;\;
    d_\mathfrak{g} h_6 = h_3 \wedge \omega_4
  \end{aligned}
$$

=--

The [[unit of an adjunction|unit]] of the [[double dimensional reduction]]-[[adjunction]]

$$
  \infty Grpd
    \underoverset
      {\underset{\mathcal{L}(-)/S^1}{\longrightarrow}}
      {\overset{hofib}{\longleftarrow}}
      {\bot}
  \infty Grpd_{/B S^1}
$$

([this prop.](double+dimensional+reduction#GeneralReduction)) applied to the $S^1$-[[principal infinity-bundle]]

$$
  \array{
    S^4
    \\
    {}^{\mathllap{hofib(c)}}\downarrow
    \\
    S^4 / S^1
    &\underset{c}{\longrightarrow}&
    B S^1
  }
$$

is a natural map

$$
  S^4/S^1 \longrightarrow \mathcal{L}(S^4)/S^1
$$

from the [[homotopy quotient]] by the [[circle action]] (def. \ref{CircleActionOn4Sphere}), to the [[cyclic loop space]] of the 4-sphere.

### Diffeomorphism group

Counterexamples (via [[graph complexes]]) to the analogue of the [[Smale conjecture]] for the 4-sphere are claimed in [Watanabe 18](diffeomorphism+group#Watanabe18), reviewed in [Watanabe 21](diffeomorphism+group#Watanabe21).

## Related entries

[[spheres -- contents]]

* [[fuzzy 4-sphere]]

* [[2-sphere]]

* [[3-sphere]]

* [[5-sphere]]

* [[6-sphere]]

* [[7-sphere]]

* [[n-sphere]]

## References

### General

* {#FreedmanGompfMorrisonWalker09} [[Michael Freedman]], [[Robert Gompf]], [[Scott Morrison]], [[Kevin Walker]], _Man and machine thinking about the smooth 4-dimensional Poincaré conjecture_, Quantum Topology, Volume 1, Issue 2 (2010), pp. 171-208 ([arXiv:0906.5177](http://arxiv.org/abs/0906.5177))

* {#RoigSaralegiAranguren00} [[Agustí Roig]], [[Martintxo Saralegi-Aranguren]], _Minimal Models for Non-Free Circle Actions_, Illinois Journal of Mathematics, volume 44, number 4 (2000) ([arXiv:math/0004141](https://arxiv.org/abs/math/0004141))

* {#AFHS98} [[Bobby Acharya]], [[José Figueroa-O'Farrill]], [[Chris Hull]], B. Spence, _Branes at conical singularities and holography_ , Adv. Theor. Math. Phys. 2 (1998) 1249–1286

* {#FelixOpreaTanre08} [[Yves Félix]], John Oprea, [[Daniel Tanré]], _Algebraic Models in Geometry_, Oxford University Press 2008

* {#MF12} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], _Half-BPS M2-brane orbifolds_, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761), [Euclid](https://projecteuclid.org/euclid.atmp/1408561553))

* {#BettiolMendes15} Renato G. Bettiol, Ricardo A. E. Mendes, _Flag manifolds with strongly positive curvature_, Math. Z. 280 (2015), no. 3-4, 1031-1046 ([arXiv:1412.0039](https://arxiv.org/abs/1412.0039))

* Selman Akbulut, _Homotopy 4-spheres associated to an infinite order loose cork_ ([arXiv:1901.08299](https://arxiv.org/abs/1901.08299))

* Akio Kawauchi, _Smooth homotopy 4-sphere_ ([arXiv:1911.11904](https://arxiv.org/abs/1911.11904))

* David T. Gay, _Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins_ ([arXiv:2102.12890](https://arxiv.org/abs/2102.12890))

### Branched covers

All [[PL manifold|PL]] [[4-manifolds]] are _simple_ [[branched covers]] of the  [[4-sphere]]:

* {#Piergallini95} [[Riccardo Piergallini]], _Four-manifolds as 4-fold branched covers of $S^4$_, Topology Volume 34, Issue 3, July 1995 (<a href="https://doi.org/10.1016/0040-9383(94)00034-I">doi:10.1016/0040-9383(94)00034-I</a>, [pdf](https://core.ac.uk/download/pdf/82379618.pdf))

* {#IoriPiergallini02} Massimiliano Iori, [[Riccardo Piergallini]], _4-manifolds as covers of the 4-sphere branched over non-singular surfaces_, Geom. Topol. 6 (2002) 393-401 ([arXiv:math/0203087](https://arxiv.org/abs/math/0203087))

Speculative remarks on the possible role of maps from [[spacetime]] to the [[4-sphere]] in some kind of [[quantum gravity]] via [spectral geometry](spectral+triple) (related to the [[Connes-Lott-Chamseddine-Barrett model]]) are in

* {#ChamseddineConnesMukhanov14} [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, _Quanta of Geometry: Noncommutative Aspects_, Phys. Rev. Lett. 114 (2015) 9, 091302 ([arXiv:1409.2471](https://arxiv.org/abs/1409.2471))

* {#ChamseddineConnesMukhanov14} [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, _Geometry and the Quantum: Basics_, JHEP 12 (2014) 098 ([arXiv:1411.0977](https://arxiv.org/abs/1411.0977))

* {#Connes17} [[Alain Connes]], section 4 of _Geometry and the Quantum_, in _Foundations of Mathematics and Physics One Century After Hilbert_, Springer 2018. 159-196 ([arXiv:1703.02470](https://arxiv.org/abs/1703.02470), [doi:10.1007/978-3-319-64813-2](https://www.springer.com/gp/book/9783319648125))

* [[Alain Connes]], from 58:00 to 1:25:00 in _Why Four Dimensions and the Standard Model Coupled to Gravity - A Tentative Explanation From the New Geometric Paradigm of NCG_, talk at IHES, 2017 ([video recording](https://www.youtube.com/watch?v=qVqqftQ92kA))

[[!redirects 4-spheres]]


4d Chern-Simons theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
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## Idea

The special case of [[higher dimensional Chern-Simons theory]] for [[dimension]] 4.

## Examples

* [[Yetter model]]

* [[BF theory]]

* [[semi-holomorphic 4d Chern-Simons theory]]

## Related concepts

* [[schreiber:∞-Chern-Simons theory]]

* [[higher dimensional Chern-Simons theory]]

  * [[1d Chern-Simons theory]]

  * [[2d Chern-Simons theory]]

  * [[3d Chern-Simons theory]]

  * **4d Chern-Simons theory**

    * [[4d TQFT]]

  * [[5d Chern-Simons theory]]

  * [[6d Chern-Simons theory]]

  * [[7d Chern-Simons theory]]

  * [[11d Chern-Simons theory]]

  * [[infinite-dimensional Chern-Simons theory]]

  * [[AKSZ sigma-model]]

## References

* Danhua Song, Mengyao Wu, Ke Wu, Jie Yang, *Higher Chern-Simons based on (2-)crossed modules*, JHEP **2023** 207 (2023) [[arXiv:2212.04667](https://arxiv.org/abs/2212.04667), doi:10.1007/JHEP07(2023)207]

(...)

[[!redirects 4d Chern-Simons theories]]
[[!redirects 4-dimensional Chern-Simons theory]]
[[!redirects 4-dimensional Chern-Simons theories]]

[[!redirects D=4 Chern-Simons theory]]


4d superconformal gauge field theory

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#### String theory
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=--
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There exists [[superconformal field theories]]/[[super Yang-Mills theories]] in 4 dimensions:

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

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

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

and by [[topological twist|topological twisting]] these give rise to

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

according to the [classification of superconformal symmetry](supersymmetry#ClassificationSuperconformal)...

[[!include superconformal symmetry -- table]]

[[!redirects D=4 SYM]]

[[!redirects 4d superconformal gauge field theories]]

[[!redirects 4d super gauge field theory]]
[[!redirects 4d super gauge field theories]]

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


4d supergravity Lie 2-algebra

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## Idea

[[4d supergravity]] is often formulated as a theory of [[gravity]] coupled to scalar-[[vector multiplets]] only, i.e. 1-form [[gauge fields]]. On the other hand, when the theory is thought of as obtained by [[KK-compactification]] from [[11-dimensional supergravity]], then it naturally contains also 2-form fields ([[tensor multiplets]]), i.e. [[higher gauge fields]]. If these are massless, then in 4-dimensions they may be dualized (via [[Hodge duality]] of their [[field strengths]]) to scalars, which is why they often do not explicitly appear in the formulation. However, when they are massive, which happens for instance when the higher dimensional theory is reduced via a [[flux compactification]], then such dualization does not apply (at least not so directly) and the 2-form [[higher gauge fields]] need to be made explicit (see the introduction of [Gunyadin-McReynolds-Zagerman 05](#GunyadinMcReynoldsZagerman05)).

In the [[D'Auria-Fré formulation of supergravity]] such higher form field contributions are reflected by [[L-infinity algebra]] [[L-infinity extensions|extensions]] of the [[super Minkowski spacetime]] [[supersymmetry]] [[super Lie algebra]] (traditionally displayed in terms of dual [[Chevalley-Eilenberg algebras]], called "FDA"s in the supergravity literature).

For the 2-form fields of $N = 2$ [[4d supergravity]] this yields a [[Lie 2-algebra]] ([Andrianopoli-D'Auria-Sommovigo 07 (4.1)-(4.7)](#AndrianopoliDAuriaSommovigo07)), which hence might be called the "4d supergravity Lie 2-algebra". In fact, including the [[moduli]] fields, this is a [[L-infinity algebroid|Lie 2-algebroid]] ([Andrianopoli-D'Auria-Sommovigo 07 (4.8)-(4.9)](#AndrianopoliDAuriaSommovigo07)). See also ([AAST 11, (4.1)-(4.9)](#AAST11)).

This Lie 2-algebra is a non-abelian variant of the [[L-infinity extension]] classified by the 3-cocycle $\propto \overline{\psi} \wedge \Gamma^a \psi \wedge e_a$ ([Andrianopoli-D'Auria-Sommovigo 07 (4.5)](#AndrianopoliDAuriaSommovigo07)), which is the [[WZW term]] for the [[Green-Schwarz superstring]] in 4d (see the notation and conventions at _[[super-Minkowski spacetime]]_).

But the [[brane scan]] says that there is also a [[super 2-brane in 4d]] whose [[WZW-term]] is the 4-cocycle $\propto \overline{\psi} \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b$. Accordingly there should also be a 4d supegravity [[Lie 3-algebra]]. This is left to "future investigations" in ([Andrianopoli-D'Auria-Sommovigo 07, p. 19](#AndrianopoliDAuriaSommovigo07)), but the relevant extension formula by the 4-cocycle is shown in ([Andrianopoli-D'Auria-Sommovigo 07 (2.24)](#AndrianopoliDAuriaSommovigo07))

## Related concepts

* [[supergravity Lie 3-algebra]]

* [[supergravity Lie 6-algebra]]

## References

Background discussion (and further pointers) for $p$-form fields in [[4d supergravity]] is in the introduction of

* {#GunyadinMcReynoldsZagerman05} [[Murat Gunaydin]], S. McReynolds, M. Zagermann, _Unified $N=2$ Maxwell-Einstein and Yang-Mills-Einstein Supergravity Theories in Four Dimensions_, JHEP 0509:026,2005 ([arXiv:hep-th/0507227](https://arxiv.org/abs/hep-th/0507227))

The 4d supergravity Lie 2-algebra was given in

* {#AndrianopoliDAuriaSommovigo07} Laura Andrianopoli, [[Riccardo D'Auria]], Luca Sommovigo, _$D=4$, $N=2$ Supergravity in the Presence of Vector-Tensor Multiplets and the Role of higher p-forms in the Framework of Free Differential Algebras_, Adv.Stud.Theor.Phys.1:561-596,2008 ([arXiv:0710.3107](http://arxiv.org/abs/0710.3107))

Further discussion is in

* {#AAST11} Laura Andrianopoli, [[Riccardo D'Auria]], Luca Sommovigo, Mario Trigiante, _$D=4$, $N=2$ Gauged Supergravity coupled to Vector-Tensor Multiplets_, Nucl.Phys.B851:1-29,2011 ([arXiv:1103.4813](http://arxiv.org/abs/1103.4813))

[[!redirects 4d supergravity Lie 2-algebras]]

[[!redirects 4d supergravity Lie 3-algebra]]
[[!redirects 4d supergravity Lie 3-algebras]]

[[!redirects 4d supergravity Lie 2-algebroid]]
[[!redirects 4d supergravity Lie 2-algebroids]]

[[!redirects 4d supergravity Lie 3-algebroid]]
[[!redirects 4d supergravity Lie 3-algebroids]]


4d TQFT

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## Idea

[[TQFT]] of [[dimension]] 4.

## Constructions

In analogy to how [[3d TQFTs]] are induced from [[quantum groups]]/[[Hopf algebras]]/ and generally [[bialgebras]] (hence [[3-modules]], the [[higher geometry|higher]] [[space of quantum states]] assigned to the point which by the [[cobordism theorem]] defines the theory) one may build 4d TQFTs from higher analogs of these, namely models of [[4-modules]] given by algebraic structures such as _[[trialgebras]]_ and _[[Hopf categories]]_.

## Examples

* [[Yetter model]]

* [[Kapustin-Witten TQFT]]

* [[4d Chern-Simons theory]]

* [[Walker-Wang model]]

* [[3d toric code]]

## Related concepts

* [[loop braid group]]

* [[TQFT]]

  * [[2d TQFT]]

  * [[3d TQFT]]

  * **4d TQFT**

    * [[4d Chern-Simons theory]]

## References

Original references:

* [[Louis Crane]], [[Louis Kauffman]], [[David Yetter]], _State-Sum invariants of 4-manifolds I_, Journal of Knot Theory and Its Ramifications Vol. 06, No. 02, pp. 177-234 (1997)  ([arXiv:hep-th/9409167](https://arxiv.org/abs/hep-th/9409167), [pdf](http://arxiv.org/PS_cache/hep-th/pdf/9409/9409167v1.pdf), [doi:10.1142/S0218216597000145](https://doi.org/10.1142/S0218216597000145))

* {#CraneFrenkel} [[Louis Crane]], [[Igor Frenkel]], _Four dimensional topological quantum field theory, Hopf categories, and the canonical bases_, J. Math. Phys. 35 (1994) 5136-5154, ([arXiv:hep-th/9405183](http://arxiv.org/abs/hep-th/9405183))

See also

* {#CarterKauffmanSaito96} [[Scott Carter]], [[Louis Kauffman]], Masahico Saito, _Diagrams, Singularities, and Their Algebraic Interpretations_, in "10th Brazilian Topology Meeting, S. Carlos, July 22 26, 1996 ([pdf](http://homepages.math.uic.edu/~kauffman/cksBrasil.pdf))

Construction via [[factorization homology]] from [[braided monoidal category|braided]] [[tensor categories]] (with relation to [[double affine Hecke algebras]]) is discussed in

* [[David Ben-Zvi]], [[Adrien Brochier]], [[David Jordan]], _Integrating quantum groups over surfaces: quantum character varieties and topological field theory_ ([arXiv:1501.04652](http://arxiv.org/abs/1501.04652))

From [[fully dualizable object|fully dualizable]] [[braided monoidal category|braided]] [[tensor categories]], via the [[cobordism hypothesis]]:

* [[Adrien Brochier]], [[David Jordan]], [[Pavel Safronov]], [[Noah Snyder]], _Invertible braided tensor categories_ ([arXiv:2003.13812](https://arxiv.org/abs/2003.13812))

As descriptions of [[topological insulators]]/[[topological phases of matter]]:

* [[Kevin Walker]], [[Zhenghan Wang]], _(3+1)-TQFTs and Topological Insulators_, Frontiers of Physics volume 7, pages 150–159 (2012) ([arXiv:1104.2632](http://arxiv.org/abs/1104.2632), [doi:10.1007/s11467-011-0194-z](https://doi.org/10.1007/s11467-011-0194-z))

  > (see [[Walker-Wang model]])

* [[Clement Delcamp]], *Excitation basis for (3+1)d topological phases*, Journal of High Energy Physics **2017** (2017) 128 ([arXiv:1709.04924](https://arxiv.org/abs/1709.04924) doi:10.1007/JHEP12(2017)128)

\linebreak

Many more references should eventually go here...

[[!redirects 4-dimensional TQFT]]

[[!redirects 4d TQFTs]]


4d WZW model

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## Idea

The 4-dimensional version of [[higher dimensional WZW theory]], the boundary theory of the [[higher dimensional Chern-Simons theory|5-dimensional Chern-Simons theory]] that is induced from the third [[Chern class]]. Also called _Nair-Schiff theory_ or _Kähler Chern-Simons theory_ .

## Properties

### Equations of motion

The [[Euler-Lagrange equations]] of the 4d WZW model are those of [[self-dual Yang-Mills theory]].

### Finiteness

The model is finite at 1-loop ([Ketov](#Ketov)) and expected to be finite to all orders.

## References

* [[Ali Chamseddine]], Phys. Lett. B233, 291 (1989);

* V.P.Nair, _Kähler-Chern-Simons Theory_ ([arXiv:hep-th/9110042](http://arxiv.org/abs/hep-th/9110042))

* A. Losev, [[Greg Moore]], [[Nikita Nekrasov]], Samson Shatashvili, _Four-Dimensional Avatars of Two-Dimensional RCFT_ ([arXiv:hep-th/9509151](http://arxiv.org/abs/hep-th/9509151))

* Takeo Inami, Hiroaki Kanno, Tatsuya Ueno, Chuan-Sheng Xiong, _Two-toroidal Lie Algebra as Current Algebra of Four-dimensional Kähler WZW Model_ ([arXiv:hep-th/9610187](http://arxiv.org/abs/hep-th/9610187))

* A. V. Ketov, Phys.Lett. B383 (1996) 390-396
 {#Ketov}

* T. Inami, H. Kanno and T. Ueno, Mod. Phys. Lett. A12, 2757 (1997).

* J. Gegenberg , G. Kunstatter, _Boundary Dynamics of Higher Dimensional AdS Spacetime_ ([arXiv:hep-th/9905228](http://arxiv.org/abs/hep-th/9905228))

* J. Gegenberg , G. Kunstatter, _Boundary Dynamics of Higher Dimensional Chern-Simons Gravity_ ([arXiv:hep-th/0010020](http://arxiv.org/abs/hep-th/0010020))

[[!redirects 4d WZW models]]

[[!redirects 4d Wess-Zumino-Witten model]]
[[!redirects 4d Wess-Zumino-Witten models]]

[[!redirects 4d WZW theory]]


4T relation

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## Idea

### For round chord diagrams
 {#ForCircularChordDiagrams}

In [[knot theory]] by the _4-term relations_ or _4T-relations_, for short, one means the following [[relations]] in the [[linear span]] of [[chord diagrams]]:

([Bar-Natan 95, Def. 1.6](#BarNatan95))

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

These are the relations respected by [[weight systems]] on chord diagrams.

### For horizontal chord diagrams
 {#ForHorizontalChordDiagrams}

For [[horizontal chord diagrams]] the 4T relations is the following:

([Bar-Natan 96, p. 3](#BarNatan96))

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

When the [[linear span]] of [[horizontal chord diagrams]] is regarded as an [[associative algebra]] under concatenation of strands ([here](horizontal+chord+diagram#AlgebraOfHorizontalChordDiagrams)), this relation is the [[infinitesimal braid relation]]

$$
  \big[
    t_{i k} + t_{j k}
    \,,\,
    t_{i j}
  \big]
  \;=\;
  0
  \,.
$$

## Properties

### Relation between horizontal and round 4T relations
 {#RelationBetweenHorizontalAndRound4TRelations}

The 4T-relations for [[round chord diagrams]] are the image of the
4T relations for [[horizontal chord diagrams]] under [[tracing horizontal to round chord diagrams]]:

### Relation to STU-relations

Under the embedding of the set of [[round chord diagrams]] into the set of [[Jacobi diagrams]], the [[STU-relations]] imply the [[4T relations]] on [[round chord diagrams]]:

Using this, one finds that [[chord diagrams modulo 4T are Jacobi diagrams modulo STU]]:

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

## Related concepts

* [[infinitesimal braid relation]]

* [[STU relation]]

[[!include chord diagrams and weight systems -- table]]

## References

Original articles

* {#BarNatan95} [[Dror Bar-Natan]], _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))

* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))

  (for [[horizontal chord diagrams]])

Textbook accounts

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Section 4 of:  _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv/1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))

* [[David Jackson]], [[Iain Moffat]], Section 11 of: _An Introduction to Quantum and Vassiliev Knot Invariants_, Springer 2019 ([doi:10.1007/978-3-030-05213-3](https://link.springer.com/book/10.1007/978-3-030-05213-3))

Lecture notes:

* {#BarNatanStoimenow97} [[Dror Bar-Natan]], Alexander Stoimenow, _The Fundamental Theorem of Vassiliev Invariants_ ([arXiv:q-alg/9702009](https://arxiv.org/abs/q-alg/9702009))

The graphics above are taken from [Sati-Schreiber 19]().

[[!redirects 4T relations]]

[[!redirects 4T-relation]]
[[!redirects 4T-relations]]

[[!redirects 4 term relation]]
[[!redirects 4 term relations]]
[[!redirects 4-term relation]]
[[!redirects 4-term relations]]

[[!redirects horizontal 4T relation]]
[[!redirects horizontal 4T relations]]

[[!redirects horizontal 4T-relation]]
[[!redirects horizontal 4T-relations]]

[[!redirects horizontal 4 term relation]]
[[!redirects horizontal 4 term relations]]
[[!redirects horizontal 4-term relation]]
[[!redirects horizontal 4-term relations]]


5-manifold

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## Idea

A [[manifold]] of [[dimension]] 5

## Related concepts

* [[Sasakian manifold]], [[SU(2)-structure]]

[[!include low dimensional manifolds -- table]]

## References

* {#CadekVanzura93} [[Martin Čadek]], [[Jiří Vanžura]], _On the classification of oriented vector bundles over 5-complexes_, Czechoslovak Mathematical Journal, Vol. 43 (1993), No. 4, 753–764 ([dml:128427](https://dml.cz/handle/10338.dmlcz/128427))

[[!redirects 5-manifolds]]


5-sphere

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## Idea

The [[n-sphere]] for $n = 5$.

## Properties

### Free group actions

\begin{example}
There are, up to [[isomorphism]] of smooth $\mathbb{Z}/2$-actions, 4 distinct free involutions on the [[5-sphere|$S^5$]].
\end{example}
([López de Medrano, Sec. V.6.1](group+actions+on+spheres#LopezdeMedrano71)).

See at *[[group actions on spheres]]*.

## Related concepts

* [[AdS5/CFT4]]

[[!redirects 5-spheres]]


6-sphere

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## Idea

The [[n-sphere]] of [[dimension]] $n = 6$.

## Properties

### Coset structure

The 6-sphere, as a [[smooth manifold]] is [[diffeomorphism|diffeomorphic]] to the [[coset space]]

$$
  S^6 \simeq G_2/ SU(3)
$$

of [[G2]] ([[automorphism group]] of the [[octonions]]) by [[special unitary group|SU(3)]] ([Fukami-Ishihara 55](#FukamiIshihara55)).

For more see at _[[G2/SU(3) is the 6-sphere]]_.

The induced [[action]] of [[G2]] on $S^6$ induces an [[almost Hermitian structure]] which makes it a [[nearly Kaehler manifold]].

Review in is in [Agrikola-Borowka-Friedrich 17](#AgrikolaBorowkaFriedrich17)

[[!include coset space structure on n-spheres -- table]]

### Complex structure

A famous open problem is the question whether the 6-sphere admits an actual [[complex structure]]. For review see [Bryant 14](#Bryant14).

## Related entries

* [[fuzzy 6-sphere]]

* [[2-sphere]]

* [[3-sphere]]

* [[4-sphere]]

* [[7-sphere]]

* [[n-sphere]]

## References

* {#FukamiIshihara55} T. Fukami, S. Ishihara, _Almost Hermitian structure on $S^6$_ , Tohoku Math J. 7 (1955), 151–156.

* {#AgrikolaBorowkaFriedrich17} [[Ilka Agricola]], Aleksandra Borówka, [[Thomas Friedrich]], _$S^6$ and the geometry of nearly Kähler 6-manifolds_ ([arXiv:1707.08591](https://arxiv.org/abs/1707.08591))

* {#Bryant14} [[Robert Bryant]], _S.-S. Chern's study of almost-complex structures on the six-sphere_ ([arXiv:1405.3405](https://arxiv.org/abs/1405.3405))

* [[Robert Bryant]], _Remarks on the geometry of almost complex 6-manifolds_ ([arXiv:math/0508428](https://arxiv.org/abs/math/0508428))

[[!redirects 6-spheres]]


600-cell

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## Idea

...one of the [[regular polytopes]] in [[dimension]] 4...

...hence a higher dimensional analog of the [[Platonic solids]]...

The 120 vertices of the 600-cell form the [[binary icosahedral group]].

## Related concepts

* [[24-cell]]

* [[120-cell]]

## References

* [[John Baez]], _The 600-Cell_ ([Part I](https://johncarlosbaez.wordpress.com/2017/12/16/the-600-cell/), [Part II](https://johncarlosbaez.wordpress.com/2017/12/24/the-600-cell-part-2/), [Part III](https://johncarlosbaez.wordpress.com/2017/12/28/the-600-cell-part-3/))

See also

* Wikipedia, _[600-cell](https://en.wikipedia.org/wiki/600-cell)_


6d Chern-Simons theory

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## Idea

The special case of [[higher dimensional Chern-Simons theory]] for [[dimension]] 6.

## Examples

(...)

## Related concepts

* [[schreiber:∞-Chern-Simons theory]]

* [[higher dimensional Chern-Simons theory]]

  * [[1d Chern-Simons theory]]

  * [[2d Chern-Simons theory]]

  * [[3d Chern-Simons theory]]

  * [[4d Chern-Simons theory]]

  * [[5d Chern-Simons theory]]

  * **6d Chern-Simons theory**

  * [[7d Chern-Simons theory]]

  * [[11d Chern-Simons theory]]

  * [[infinite-dimensional Chern-Simons theory]]

  * [[AKSZ sigma-model]]

[[!redirects 6d Chern-Simons theories]]
[[!redirects 6-dimensional Chern-Simons theory]]
[[!redirects 6-dimensional Chern-Simons theories]]


6j symbol

[[!redirects Wigner 6-j symbol]]
[[!redirects Wigner 6-j symbol]]

#Contents#
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## Idea

$6j$ symbols are pieces of combinatorial data which can be associated to [[fusion categories]], encoding their [[associator]] with a finite amount of algebraic data. The $6j$ symbols along with the [[fusion ring]] is enough data to uniquely reconstruct the category.

$6j$ symbols are only defined for [[multiplicity free fusion category|multiplicity free fusion categories]]. That is, fusion categories in which all of the fusion coefficients are $0$ or $1$. For fusion categories with larger coefficients, one has to replace $6j$ symbols (which are complex numbers) with higher-dimensional matrices.

The name "$6j$ symbol" comes from the fact that the symbol has $6$ indicies associated to it.

The $6j$ symbols depend on a local choice of basis of hom-spaces of your fusion category, and hence they are not bona-fide invariants. Choosing a different choice of local basis corresponds to a [[gauge transformation]].

## Definition

Let $\mathcal{C}$ be a [[multiplicity free fusion category]]. Let

$$\alpha: (- \otimes -)\otimes - \to - \otimes (-\otimes -)$$

be its associator. We wish to encode the associator with a finite amount of algebraic data. That is, with a finite quantity of complex numbers. We do this by studying the action of $\alpha$ with respect to the (finite) set $\mathcal{L}$ of isomorphism classes of simple objects of $\mathcal{C}$.

For every triple $A,B,C$ of simple objects of $\mathcal{C}$, choose a distinguished generator

\begin{imagefromfile}
"file_name": "distinguished-basis-element.jpg",
"width": 300
\end{imagefromfile}

This will be a non-zero morphism whenever $\dim \text{Hom}(A\otimes B,C)\geq 1$ and $0$ otherwise. Here, we are using the graphical language of [[string diagrams]]. There generators are chosen arbitrarily, except for the conditions that

\begin{imagefromfile}
"file_name": "basis-element-conditions.jpg",
"width": 300
\end{imagefromfile}

where $\rho_A$ denotes the right [[unitor]] and $\lambda_A$ denotes the left unitor.

Note the key use of the fact that $\mathcal{C}$ is multiplicity free. If fusion coefficients could be $\geq 2$ then we would not have one-element bases of the spaces $\text{Hom}(A\otimes B,C)$. Instead, we would have to choose larger bases of these spaces are get matrices for our $6j$ symbols instead of complex numbers.

For all quadruples $A,B,C,D$ of simple objects, there is a basis of $\text{Hom}((A\otimes B)\otimes C,D)$ given by the symbols

\begin{imagefromfile}
"file_name": "6j-basis-1.jpg",
"width": 300
\end{imagefromfile}

and there is a basis of $\text{Hom}(A\otimes (B\otimes C),D)$.

\begin{imagefromfile}
"file_name": "6j-basis-2.jpg",
"width": 300
\end{imagefromfile}

where in both cases $N$ ranges over representatives of isomorphism classes of simple objects.

\begin{definition} The associator $\alpha$ induces a map

$$\alpha_{A,B,C}:\text{Hom}((A\otimes B)\otimes C,D)\to \text{Hom}(A\otimes (B\otimes C),D)$$

for all simple objects $A,B,C,D\in\mathcal{C}$. Expressing the source and target of this map in the distinguished bases above, we can canonically identify $\alpha_{A,B,C}$ with a matrix.

The __6j symbol__ $F^{a,b,c}_{d;n,m}$ is defined to be the $(N,M)$th coefficient of this matrix, where $a,b,c,d,n,m$ are isomorphism classes of simple objects. That is, the $6j$ symbols are the unique complex numbers making the identity

\begin{imagefromfile}
"file_name": "6j-identity.jpg",
"width": 300
\end{imagefromfile}

holds.
\end{definition}

## Use of Yoneda perspective

It is a key observation that one does not use the $6j$ symbols to directly encode $\alpha$. Instead, the $6j$ symbols are used to describe the action of the hom-space $\text{hom}(-,D)$.

They key point is that objects are not associated vector spaces in a linear category: hom-spaces are. Hence, when encoding maps as matrices one has to take a dual perspective from objects to morphisms.

The [[Yoneda lemma]] exactly guarantees that the data encoded is enough to recover $\alpha$. Namely, knowing the action of $\alpha$ on the hom-space $\text{hom}((A\otimes B)\otimes C,D)\to \text{hom}(A\otimes (B\otimes C),D)$ for all simple objects $D$ one gets the full data of a natural transformation $\text{hom}((A\otimes B)\otimes C,-)\to \text{hom}(A\otimes (B\otimes C),-)$ by the [[semisimple category | semisimplicity]] of $\mathcal{C}$. Because the Yoneda embedding is fully faithful, we can uniquely determine what map $\alpha$ must have induces this natural transformation.

## Related concepts

* [[Wigner 3-j symbol]] ([[Clebsch-Gordan coefficients]])

* [[Wigner 9j symbol]]

* [[Yoneda lemma]]

[[!redirects Wigner 6-j symbols]]
[[!redirects Wigner 6j symbol]]
[[!redirects Wigner 6j symbols]]

[[!redirects 6-j symbol]]
[[!redirects 6-j symbols]]

[[!redirects 6j symbols]]


7-sphere

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[sphere]] of [[dimension]] 7.

This is one of the [[parallelizable manifold|parallelizable]] spheres, as such corresponds to the [[octonions]] among the [[division algebras]], being the manifold of unit octonions, and is the only one of these which does not carry ([[Lie group|Lie]]) [[group]] structure but just [[Moufang loop]] structure.

## Properties

### Quaternionic Hopf fibration
 {#QuaternionicHopfFibration}

The 7-sphere participates in the [[quaternionic Hopf fibration]], the analog of the complex [[Hopf fibration]] with the field of [[complex numbers]] replaced by the division ring of [[quaternions]] or Hamiltonian numbers $\mathbb{H}$.

$$
  \array{
    S^3 &\hookrightarrow& S^7
    \\
    && \downarrow^\mathrlap{p}
    \\
    && S^4 &\stackrel{}{\longrightarrow}& \mathbf{B} SU(2)
  }
$$

Here the idea is that $S^7$ can be construed as $\{(x, y) \in \mathbb{H}^2: {|x|}^2 + {|y|}^2 = 1\}$, with $p$ mapping $(x, y)$ to $x/y$ as an element in the [[projective line]] $\mathbb{P}^1(\mathbb{H}) \cong S^4$, with each [[fiber]] a [[torsor]] parametrized by quaternionic [[scalars]] $\lambda$ of unit [[norm]] (so $\lambda \in S^3$).  This canonical $S^3$-bundle (or $SU(2)$-bundle) is classified by a map $S^4 \to \mathbf{B} SU(2)$.

### Coset space realizations
 {#CosetSpaceRealization}

+-- {: .num_prop #QuotientOfSpin7ByG2IsS7}
###### Proposition
**([[coset space]] of [[Spin(7)]] by [[G2]] is [[7-sphere]])**

Consider the canonical [[action]] of [[Spin(7)]] on the [[unit sphere]] in $\mathbb{R}^8$ (the [[7-sphere]]),

1. This action is is [[transitive action|transitive]];

1. the [[stabilizer group]] of any point on $S^7$ is [[G2]];

1. all [[G2]]-subgroups of [[Spin(7)]] arise this way, and are all [[conjugate subgroup|conjugate]] to each other.

Hence the [[coset space]] of [[Spin(7)]] by [[G2]] is the [[7-sphere]]

$$
  S^7
  \;\simeq_{diff}\;
  Spin(7)/G_2
  \,.
$$

=--

(e.g [Varadarajan 01, Theorem 3](#Varadarajan01))

Other coset realizations of the usual [[differentiable manifold|differentiable]] 7-sphere ([Choquet-Bruhat, DeWitt-Morette 00, p. 288](#Choquet-Bruhat+DeWitt-Morette00)):

* $S^7 \simeq_{diff} $ [[Spin(6)]]$/SU(3) \simeq_{iso} SU(4)/SU(3)$ (by [this Prop.](sphere#OddDimSphereAsSpecialUnitaryCoset));

* $S^7 \simeq_{diff}  Spin(5)/SU(2)$ ([Awada-Duff-Pope 83](#AwadaDuffPope83), [Duff-Nilsson-Pope 83](#DuffNilssonPope83))

These three coset realizations of 'squashed' 7-spheres together with a fourth

* $S^7 \simeq_{diff}  Spin(8)/Spin(7)$,

the realization of the 'round' 7-sphere, may be seen jointly as resulting from the 8-dimensional representations of even [[Clifford algebras]] in 5, 6, 7, and 8 dimensions (see [Baez](#Baez)) and as such related to the four [[normed division algebras]]. See also [Choquet-Bruhat+DeWitt-Morette00, pp. 263-274](#Choquet-Bruhat+DeWitt-Morette00).

[[!include coset space structure on n-spheres -- table]]

The following gives an [[exotic 7-sphere]]:

* $S^7 \simeq_{homeo} Sp(1)\backslash Sp(2)/Sp(1)$ ([[Gromoll-Meyer sphere]])

\linebreak

### Exotic 7-spheres

A celebrated result of Milnor is that $S^7$ admits [[exotic smooth structures]] (see at _[[exotic 7-sphere]]_), i.e., there are [[smooth manifold]] structures on the [[topological manifold]] $S^7$ that are not [[diffeomorphism|diffeomorphic]] to the standard smooth structure on $S^7$. More structurally, considering smooth structures up to [[orientation|oriented]] diffeomorphism, the different smooth structures form a [[monoid]] under a (suitable) operation of [[connected sum]], and this monoid is isomorphic to the [[cyclic group]] $\mathbb{Z}/(28)$. With the notable possible exception of $n = 4$ (where the question of existence of exotic 4-spheres is wide open), exotic spheres first occur in dimension $7$. This phenomenon is connected to the [[h-cobordism theorem]] (the monoid of smooth structures is identified with the monoid of h-cobordism classes of oriented [[homotopy spheres]]).

One explicit construction of the smooth structures is given as follows (see [Milnor 1968](#Mil2)). Let $W_k$ be the algebraic variety in $\mathbb{C}^5$ defined by the equation

$$z_1^{6 k - 1} + z_2^3 + z_3^2 + z_4^2 + z_5^2 = 0$$

and $S_\epsilon \subset \mathbb{C}^5$ a sphere of small radius $\epsilon$ centered at the origin. Then each of the $28$ smooth structures on $S^7$ is represented by an intersection $W_k \cap S_\epsilon$, as $k$ ranges from $1$ to $28$. These manifolds sometimes go by the name _Brieskorn manifolds_ or _[[Brieskorn spheres]]_ or _[[Milnor spheres]]_.

### $G_2$-structure
 {#G2Structure}

Let $\phi_0 \in \Omega^3(\mathbb{R}^7)$ be the [[associative 3-form]] and let

$$
  \Phi_0 \in \Omega^4(\mathbb{R} \oplus \mathbb{R}^7)
$$

be given by

$$
  \Phi_0 = d x_0 \wedge \phi_0 + \star \phi_0
$$

(where $x_0$ denotes the canonical coordinate on the first factor of $\mathbb{R}$ and $\phi_0$ is pulled back along the projection to $\mathbb{R}^7$)
.

By construction this is its own [[Hodge dual]]

$$
  \Phi = \star \Phi
  \,.
$$

This implies that as we restrict $\Phi_0$ to

$$
  \mathbb{R}^8 - \{0\} \simeq \mathbb{R} \times S^7
$$

then there is a unique 3-form

$$
  \phi \in \Omega^3(S^7)
$$

on the 7-sphere such that

$$
  \Phi_0 = r^3 \wedge \phi + r^4 \star_{S^7} \phi
  \;\;\;\; (on \; \mathbb{R}^8 - \{0\})
  \,.
$$

This 3-form $\phi$ defines a [[G2-structure]] on $S^7$. It is _nearly parallel_ in that

$$
  d \phi = 4 \star \phi
  \,.
$$

(e.g. [Lotay 12, def.2.4](#Lotay12))

## Related concepts

[[!include spheres -- contents]]

## References

* [[Martin Cederwall]], Christian R. Preitschopf, _The Seven-sphere and its Kac-Moody Algebra_, Commun. Math. Phys. 167 (1995) 373-394 ([arXiv:hep-th/9309030](http://arxiv.org/abs/hep-th/9309030))

* Takeshi Ôno, _On the Hopf fibration $S^7 \to S^4$ over $Z$_, Nagoya Math. J.
Volume 59 (1975), 59-64. ([Euclid](http://projecteuclid.org/euclid.nmj/1118795554))

Relation to the [[Milnor fibration]]:

* [[Kenneth Intriligator]], [[Hans Jockers]], [[Peter Mayr]], [[David Morrison]], M. Ronen Plesser, _Conifold Transitions in M-theory on Calabi-Yau Fourfolds with Background Fluxes_, Adv.Theor.Math.Phys. 17 (2013) 601-699 ([arXiv:1203.6662](http://arxiv.org/abs/1203.6662))

An [[ADE classification]] of finite subgroups of $SO(8)$ [[free action|acting freely]] on $S^7$ (see at _[[group action on an n-sphere]]_) such that the quotient is [[spin structure|spin]] and has at least four [[Killing spinors]] (see also at [[ABJM model]]) is in

* [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], Sunil Gadhia, [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, 	JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))

* [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], _Half-BPS M2-brane orbifolds_ ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761))

Discussion of [[subgroups]]:

* {#Varadarajan01} [[Veeravalli Varadarajan]], _Spin(7)-subgroups of SO(8) and Spin(8)_, Expositiones Mathematicae, 19 (2001): 163-177 ([pdf](https://core.ac.uk/download/pdf/81114499.pdf))

Discussion of [[exotic smooth structures]] on 7-spheres includes

* Wikipedia, _Exotic sphere_, [link](https://en.wikipedia.org/wiki/Exotic_sphere).

The explicit construction of exotic 7-spheres by intersecting algebraic varieties with spheres is described in

* {#Mil2} [[John Milnor]], "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968).

Discussion of (nearly) [[G2-structures]] on $S^7$ and [[calibrated submanifolds]] includes

* {#Lotay12} [[Jason Lotay]], _Associative Submanifolds of the 7-Sphere_, Proc. London Math. Soc. (2012) 105 (6): 1183-1214 ([arXiv:1006.0361](http://arxiv.org/abs/1006.0361), [talk slides](http://www.homepages.ucl.ac.uk/~ucahjdl/JDLotay_KIAS2011_slides.pdf))

On [[coset]]-realizations:

* {#Kramer98} Linus Kramer, _Octonion Hermitian quadrangles_, Bull. Belg. Math. Soc. Simon Stevin Volume 5, Number 2/3 (1998), 353-362 ([euclid:1103409015](https://projecteuclid.org/euclid.bbms/1103409015))

* {#Choquet-Bruhat+DeWitt-Morette00} [[Yvonne Choquet-Bruhat]], [[Cécile DeWitt-Morette]], *Analysis, manifolds and physics*, Part II, North Holland (1982, 2001) $[$[ISBN:9780444860170](https://www.elsevier.com/books/analysis-manifolds-and-physics-revised-edition/choquet-bruhat/978-0-444-86017-0)$]$

* {#AwadaDuffPope83} M. A. Awada, [[Mike Duff]], [[Christopher Pope]], _$N=8$ Supergravity Breaks Down to $N=1$_, Phys. Rev. Lett. 50, 294 – Published 31 January 1983 ([doi:10.1103/PhysRevLett.50.294](https://doi.org/10.1103/PhysRevLett.50.294))

* {#DuffNilssonPope83} [[Mike Duff]], [[Bengt Nilsson]], [[Christopher Pope]], _Spontaneous Supersymmetry Breaking by the Squashed Seven-Sphere_, Phys. Rev. Lett. 50, 2043 – Published 27 June 1983; Erratum Phys. Rev. Lett. 51, 846  ([doi:10.1103/PhysRevLett.50.2043](https://doi.org/10.1103/PhysRevLett.50.2043))

* {#Baez} [[John Baez]], _Rotations in the 7th Dimension_, ([blog post](https://golem.ph.utexas.edu/category/2007/09/rotations_in_the_7th_dimension.html)), and _TWF 195_, ([webpage](http://math.ucr.edu/home/baez/week195.html))

[[!redirects 7-spheres]]

[[!redirects seven sphere]]
[[!redirects seven spheres]]


7d Chern-Simons theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The case of [[higher dimensional Chern-Simons theory]] in [[dimension]] 7.

## Examples

We discuss

1. [Abelian 7d CS theory](#AbelianTheory) of an abelian 3-form connection;

1. [7d p2 theory on String 2-connections](#OnString2Connections)

1. [2-species cup-product theory on a G2 manifold](#TwoSpeciesCupProductTheoryOnG2Manifold)

### Abelian theory
 {#AbelianTheory}

A basic 7d [[higher dimensional Chern-Simons theory]] is the abelian theory, whose [[extended Lagrangian]] $\mathbf{L}$ is the [[diagonal]] of the [[cup product in ordinary differential cohomology]]

$$
  \mathbf{L}_{\mathbf{DD}\cup \mathbf{DD}}
  \colon
  \mathbf{B}^3 U(1)_{conn}
  \stackrel{\Delta}{\to}
  \mathbf{B}^3 U(1)_{conn}
  \times
  \mathbf{B}^3 U(1)_{conn}
  \stackrel{\widehat {\cup}}{\to}
  \mathbf{B}^7 U(1)_{conn}
  \,.
$$

The [[transgression]] of this to [[codimension]] 0 hence for $\Sigma_7$ a [[closed manifold]] of [[dimension]] 7 is the [[action functional]]

$$
  \exp\left(
    2 \pi i \int_{\Sigma_7}
    [\Sigma_7, \mathbf{L}_{\mathbf{DD}\cup \mathbf{DD}}]
  \right)
  \;\colon\;
  [\Sigma_7, \mathbf{B}^3 U(1)_{conn}]
  \to
  U(1)
  \,.
$$

A [[gauge field]] configuration

$$
  \phi \;\colon\; \Sigma_7 \to \mathbf{B}^3 U(1)_{conn}
$$

here is a [[circle n-bundle with connection|circle 3-bundle with connection]]. In the special case that the underlying [[circle 3-group]] [[principal 3-bundle]] is trivializable and trivialized, this is equivalently a [[differential 3-form]] $C \in \Omega^3(\Sigma_7)$ and the above [[action functional]] takes this to the simple expression

$$
  C
   \mapsto
  \exp\left(
     2 \pi i \int_{\Sigma_7}  C \wedge d C
  \right)
  \in U(1)
  \,,
$$

where in the [[exponent]] we have the [[integration of differential forms]] over the [[wedge product]] of $C$ with its [[de Rham differential]]. On general field configurations the action functional is the suitable globalization of this expression.

In ([Witten 97](#Witten97)), ([Witten 98](#Witten98)) a slight refinement of this construction (a  [[quadratic refinement]] induced by an [[integral Wu structure]]) was argued to be the [[holographic principle|holographic dual]] to the [[self-dual higher gauge theory]] of the abelian self-dual 2-form gauge field in the [[6d (2,0)-superconformal QFT]] on the [[worldvolume]] of the [[M5-brane]]. The issue of the quadratic refinement was discussed in more detail in ([HopkinsSinger](#HopkinsSinger)). A refinement to [[extended Lagrangians]] as above is discussed in ([FSSII](#FSSII)).

By the argument in ([Witten98](#Witten98)) the above relation holds when we interpret the fields $\phi \colon : \Sigma_7 \to \mathbf{B}^3 U(1)_{conn}$ as the [[supergravity C-field]] after [[Kaluza-Klein mechanism|compactification]] on a 4-[[sphere]] in the [[AdS-CFT|AdS7-CFT6]] setup. By the discussion at [[11-dimensional supergravity]] this field is in general not simply a 3-connection as above but receives corrections by a [[Green-Schwarz mechanism]] and "flux quantization" which give it non-abelian components. This, and the resulting non-abelian generalization of the above extended Lagrangian is discussed in ([FSSI](#FSSI), [FSSII](#FSSII)).

The nonabelian 7d action functional this obtained contains the following two examples as summands.

### Nonabelian $p_2$ theory on String 2-connections
 {#OnString2Connections}

The [[second fractional Pontryagin class]]

$$
  [\tfrac{1}{6}p_2] \in H^8(B String, \mathbb{Z})
$$

has a smooth and differential refinement (see at _[[twisted differential fivebrane structure]]_) to an [[extended Lagrangian]]

$$
  \tfrac{1}{2}\hat \mathbf{p}_2
  \;\colon\;
  \mathbf{B}String_{conn}
  \to
  \mathbf{B}^7 U(1)_{conn}
  \,.
$$

where the domain is the [[smooth infinity-groupoid|smooth]] [[moduli infinity-stack|moduli 2-stack]] of [[String 2-group]] [[connection on a 2-bundle|principal 2-connections]] (see at _[[differential string structure]]_ for more). This modulates the [[Chern-Simons circle 7-bundle]] with connection on $\mathbf{B}String_{conn}$.

The [[transgression]] of this to codimension 0 yields an [[action functional]]

$$
  \exp\left(
    2 \pi i \int_{\Sigma_7}
    [\Sigma_7, \tfrac{1}{6}\hat \mathbf{p}_2]
  \right)
  \;\colon\;
  [\Sigma_7, \mathbf{B}String_{conn}]
  \to
  U(1)
$$

on string 2-connection fields. This is part of the quantum-corrected and flux-quantized extended action functional of the [[supergravity C-field]] in [[11-dimensional supergravity]] by the analysis in ([FSSII](#FSSII)).

### Two-species cup-product theory on a $G_2$ manifold
 {#TwoSpeciesCupProductTheoryOnG2Manifold}

For $X$ a [[G2-manifold]] with characteristic [[differential forms]]

$$
  \omega_3 \in \Omega^3(X)
$$

and

$$
  \omega_4 = \star \omega_3\in \Omega^4(X)
$$

and for $G$ a simply connected compact [[semisimple Lie group]] with [[invariant polynomial]] $\langle -,-\rangle$, consider the [[action functional]] on the space of $\mathfrak{g}$-[[Lie algebra valued 1-forms]] $A$ given by the [[integration of differential forms]]

$$
  A
    \mapsto
  \exp\left(
    2 \pi i\int_{X} \omega_4 \wedge CS\left(A\right)
  \right)
  \,,
$$

where $CS(A) \in \Omega^3(X)$ is the [[Chern-Simons form]] of $A$. This, or some suitable globalization of this, has been considered as an [[action functional]] for 7-dimensional Chern-Simons-type theory in ([Donaldson-Thomas](#DonaldsonThomas)) and ([Baulieu-Losev-Nekrasov](#BaulieuLosevNekrasov)). This appears as an action functional in [[topological M-theory]] ([deBoer et al](#deBoerEtAl)).

To refine this to an [[extended Lagrangian]] and then fully globalize the action functional we can ask for a [[higher geometric quantization|higher geometric prequantization]] of $\omega_4$, regarded as a [[n-plectic structure|3-plectic structure]], by a [[prequantum n-bundle|prequantum 3-bundle]] $\hat \mathbf{G}_2$

$$
  \array{
   && \mathbf{B}^3 U(1)_{\mathrm{conn}}
   \\
   & {}^{\mathllap{\hat \mathbf{G}_2}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}}
   \\
    X
   &\stackrel{\omega_4}{\to}&
   \Omega^4_{cl}
  }
  \,,
$$

where $\mathbf{B}^3 U(1)_{conn} \in $ [[Smooth∞Grpd]] is the smooth [[moduli ∞-stack]] of [[circle n-bundle with connection|circle 3-bundles with connection]].

If moreover we write

$$
  \hat \mathbf{c} \;:\;
  \mathbf{B}G_{conn}
  \to
  \mathbf{B}^3 U(1)_{conn}
$$

for the universal differential characteristic map which is the [[Lie integration]] of $\langle-,-\rangle$ (as discussed at _[[differential string structure]]_), hence the [[extended Lagrangian]] for ordinary [[3d Chern-Simons theory|3d]] $G$-[[Chern-Simons theory]], then an [[extended Lagrangian]] for the above [[action functional]] is given by the [[cup product in ordinary differential cohomology]]

$$
  \exp\left(
   2 \pi i \int_{\Sigma_7} [\Sigma_7, \hat {\mathbf{G}}_4 \hat \cup \hat \mathbf{S}]
  \right)
  \;\colon\;
  X
  \times
  \mathbf{B}G_{conn}
  \stackrel{(\hat \mathbf{G}_2, \hat \mathbf{c})}{\to}
  \mathbf{B}^3 U(1)_{conn}
  \times
  \mathbf{B}^3 U(1)_{conn}
  \stackrel{\hat \cup}{\to}
  \mathbf{B}^7 U(1)_{conn}
  \,.
$$

(This is an cup product extended Lagrangian of the kind considered in ([FSSIII](#FSSIII)).)

Notice that the prequantization lift to [[differential cohomology]] is entirely demanded by the interpretation of $\omega_4$ as the [[field strength]] of the [[supergravity C-field]] in interpretations of this setup in [[M-theory on G2-manifolds]].

Moreover, the above considerations do not really need $X$ to be a [[G2-manifold]] to go through, a manifold with [[weak G2 holonomy]] is just as well, hence equipped with $\phi \in \Omega^3(X)$ such that

$$
  \omega_4 = \lambda \star \phi
$$

and

$$
 d \phi = \omega_4
 \,.
$$

This arises from [[Freund-Rubin compactifications]] with [[cosmological constant]] $\lambda$ ([Bilal-Derendinger-Sfetsos](#BilalDerendingerSfetsos)).

## Properties

### Moduli of fields (abelian case)

[[!include moduli of higher lines -- table]]

## Related concepts

* [[schreiber:∞-Chern-Simons theory]]

* [[higher dimensional Chern-Simons theory]]

  * [[1d Chern-Simons theory]]

  * [[2d Chern-Simons theory]]

  * [[3d Chern-Simons theory]]

  * [[4d Chern-Simons theory]]

  * [[5d Chern-Simons theory]]

  * [[6d Chern-Simons theory]]

  * **7d Chern-Simons theory**

  * [[11d Chern-Simons theory]]

  * [[AKSZ sigma-models]]

  * [[string field theory]]

  * [[infinite-dimensional Chern-Simons theory]]

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

* [[Hitchin functional]]

## References

### Abelian theory

The abelian 7d [[higher dimensional Chern-Simons theory]] of a [[circle n-bundle with connection|circle 3-bundle with connection]] was considered in

* {#Witten97} [[Edward Witten]], _Five-Brane Effective Action In M-Theory_ J. Geom. Phys.22:103-133,1997 ([arXiv:hep-th/9610234](http://arxiv.org/abs/hep-th/9610234))

* {#Witten98} [[Edward Witten]], _AdS/CFT Correspondence And Topological Field Theory_ JHEP 9812:012,1998 ([arXiv:hep-th/9812012](http://arxiv.org/abs/hep-th/9812012))

and argued to be the [[holographic principle|holographic dual]] to the [[self-dual higher gauge theory]] of an abelian 2-form connection on a _single_ [[M5-brane]] in its [[6d (2,0)-supersymmetric QFT]] on the worldvolume.

The precise formulation of this functional in terms of [[differential cohomology]] and [[integral Wu structure]] was given in

* [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_
 {#HopkinsSinger}

### Nonabelian theories
 {#ReferencesNonabelianTheories}

In

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:7d Chern-Simons theory and the 5-brane]]_
 {#FSSI}

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The moduli 3-stack of the C-field]]_
 {#FSSII}

the 7d Chern-Simons action obtained by [[Kaluza-Klein reduction|compactifying]] [[11-dimensional supergravity]] including the quantum corrections of the [[supergravity C-field]] on a 4-sphere (the [[AdS-CFT|AdS7/CFT6]] setup) is considered and refined to an [[extended Lagrangian]]. It contains the [Donaldson-Thomas](#DonaldsonThomas)-functional $\int_X CS(A) \wedge G_4$ as one summand and the [Witten 97](#Witten97)-functional as another.

Further discussion of [[extended Lagrangians]] for 7d CS theories is in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Extended higher cup-product Chern-Simons theories]]_
 {#FSSIII}

### On $G_2$-manifolds

The Chern-Simons type action functionals $A \mapsto \int_X CS(A) \wedge \omega_4$ on a 7d [[G2-manifold]] $(X, \omega_3)$ was first considered in

* [[S. Donaldson]], R. Thomas, _Gauge theory in higher dimensions_ ([pdf](http://www2.imperial.ac.uk/~rpwt/skd.pdf))
 {#DonaldsonThomas}

and around (3.23) of

* L. Baulieu, A. Losev, [[Nikita Nekrasov]], _Chern-Simons and Twisted Supersymmetry in Higher Dimensions_, Nucl.Phys. B522 (1998) 82-104 ([arXiv:hep-th/9707174](http://arxiv.org/abs/hep-th/9707174))
 {#BaulieuLosevNekrasov}

In

* [[Jan de Boer]], [[Paul de Medeiros]], Sheer El-Showk, Annamaria Sinkovics, _Open $G_2$ Strings_ ([arXiv:hep-th/0611080](http://arxiv.org/abs/hep-th/0611080)) {#deBoerEtAl}

this is put into the context of [[topological M-theory]] (see around equation (2) in the introduction).

Discussion for [[weak G2-holonomy]] is in

* A. Bilal, J.-P. Derendinger, K. Sfetsos, _(Weak) $G_2$ Holonomy from Self-duality, Flux and Supersymmetry_, Nucl.Phys. B628 (2002) 112-132 ([arXiv:hep-th/0111274](http://arxiv.org/abs/hep-th/0111274))
 {#BilalDerendingerSfetsos}

### Formulation in extended TQFT

Formulation in [[extended TQFT]] is discussed in

* [[Dan Freed]], _[[4-3-2 8-7-6]]_, talk at _[ASPECTS of Topology](https://people.maths.ox.ac.uk/tillmann/ASPECTS.html)_ Dec 2012

[[!redirects D=7 Chern-Simons theory]]

[[!redirects 7-dimensional Chern-Simons theory]]
[[!redirects 7d Chern-Simons theories]]
[[!redirects 7-dimensional Chern-Simons theories]]


7d spherical space forms -- table

| $N$ [[Killing spinors]] on
 [[spherical space form]] $S^7/\widehat{G}$  | $\phantom{AA}\widehat{G} =$ | [[spin group|spin]]-lift of [[subgroup]] of
[[isometry group]] of [[7-sphere]] | [[3d superconformal gauge field theory]]
 on [[black brane|back]] [[M2-branes]]
 with [[near horizon geometry]] $AdS_4 \times S^7/\widehat{G}$ |
|---------|----------------------------------------------|------|----|
| $\phantom{AA}N = 8\phantom{AA}$ |  $\phantom{AA}\mathbb{Z}_2$ | [[cyclic group of order 2]] | [[BLG model]] |
| $\phantom{AA}N = 7\phantom{AA}$ |  ---   | --- | ---  |
| $\phantom{AA}N = 6\phantom{AA}$ |  $\phantom{AA}\mathbb{Z}_{k\gt 2}$ | [[cyclic group]] | [[ABJM model]] |
| $\phantom{AA}N = 5\phantom{AA}$ |  $\phantom{AA}2 D_{k+2}$
 $2 T$, $2 O$, $2 I$ | [[binary dihedral group]],
 [[binary tetrahedral group]],
 [[binary octahedral group]],
 [[binary icosahedral group]] | ([HLLLP 08a](ABJM+model#HLLLP08), [BHRSS 08](ABJM+model#BHRSS08)) |
| $\phantom{AA}N = 4\phantom{AA}$ |  $\phantom{A}2 D_{k+2}$
 $2 O$, $2 I$ | [[binary dihedral group]],
 [[binary octahedral group]],
 [[binary icosahedral group]] | ([HLLLP 08b](3d+superconformal+gauge+field+theory#HLLLP08b), [Chen-Wu 10](#ChenWu10)) |

* [[José Figueroa-O'Farrill]] et al. 2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))