24

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Exceptional structures
+-- {: .hide}
[[!include exceptional structures -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

24 is the [[natural number]] that follows 23 and precedes 25. It occurs in modern systems of measurement for historical reasons, thanks ultimately to its having many [[prime factors]]

$$
  24
  \,=\,
  2 \cdot 2 \cdot 2 \cdot 3
$$

and it arises in many circumstances in [[mathematics]] and [[physics]], the relations between which are not always immediately apparent.

## Examples

### Modular arithmetic

Fix a [[natural number]] $n$, and consider those numbers $x$ whose squares are congruent to 1 [[modulo]] $n$:

\[ x^2 \equiv 1 (\mod n) \, . \]

A necessary condition for satisfying this is for $x$ and $n$ to be [[coprime integer|coprime]]. For some choices of $n$, this condition is sufficient, i.e., every $x$ coprime to $n$ satisfies this congruence. In fact, this congruence is equivalent to $gcd(x,n) = 1$ if and only if $n$ divides 24.

### Cannonball problem

The cannonball problem asks what size square pyramids can be made by stacking spheres such that the total number of spheres is a square number. Apart from the trivial solutions involving 0 spheres or 1 sphere, the only solution is a pyramid 24 layers tall:

\[ 0^2 + 1^2 + 2^2 + \cdots + 24^2 = 70^2 \, . \]

### Leech lattice

The [[Leech lattice]] is a "universally optimal" packing of hyperspheres in $\mathbb{R}^{24}$ originally discovered in [[coding theory]]. Surprisingly, it is closely related to the cannonball problem, since it can be constructed from a Lorentzian lattice in two higher dimensions by a method that uses the fact that

\[ v = (0, 1, 2, \ldots, 24, 70) \]

has zero norm under that metric.

### Mathieu groups

The largest [[Mathieu group]], $M_{24}$, is the [[automorphism group]] of a [[Steiner system]] containing 24 blocks. $M_{24}$ is also the automorphism group of the [[binary Golay code]], an [[abelian group]] which is a 12-dimensional subspace of the [[vector space]] $\mathbb{F}_2^{24}$.

The binary Golay code can be used to construct the Leech lattice. Essentially, each codeword defines a point in 24-dimensional space, and we can fill in and build out this set. In more detail: Let $c$ be a Golay codeword, scale it by a factor 2, and add either $4x$, where $x$ is a vector in $\mathbb{Z}^{24}$ whose components sum to an even number, or $1 + 4y$ where $y \in \mathbb{Z}^{24}$ and its components sum to an odd number.

### 24-cell and the binary tetrahedral group

The [[24-cell]] is a four-dimensional [[regular polytope]] with 24 vertices. Interpreting these vertices as [[quaternion]]s, they form a [[group]] under quaternion multiplication, and this group is [[isomorphism|isomorphic]] to the [[binary tetrahedral group]].

### Third stable homotopy group of spheres

The [[third stable homotopy group of spheres]] is the [[cyclic group]] of order 24.

### String theory

Critical [[bosonic string theory|Bosonic string theory]] requires 26 [[spacetime]] [[dimension of a manifold|dimensions]] of [[target spacetime]] (for vanishing [[dilaton]], at least), which can be broken down into 2 dimensions for the string [[worldsheet]] itself and 24 transversal directions for its oscillations.

This is also related to the conventional choice of normalization for the [[central charge]] of the [[Virasoro algebra]]:

\[ [L_m, L_n] = (m - n) L_{m+n} + \frac{c}{12}(m^3 - m) \delta_{m+n,0} \, . \]

Several classes of [[string theory vacua]] require the presence of exactly [[24 branes transverse to K3|24 branes of codimension 4]] transverse to a [[K3-surface]]-[[fiber]].

## Related concepts

Appearances of 24 and its factors, particularly 8 and 12, are a meta-pattern in mathematics; another such meta-pattern is [[ADE classification]]. Sometimes these meta-patterns overlap.

For example, [[tesselation|tessellating]] $\mathbb{R}^4$ with regular 24-cells creates the 24-cell honeycomb, in which the centers of the 24-cells are the points of the $D_4$ lattice. Meanwhile, the [[binary tetrahedral group]] corresponds to the $E_6$ [[Dynkin diagram]] (see *[[McKay correspondence]]*).

Families of two-dimensional [[conformal field theory|conformal field theories]], in which the [[Virasoro algebra]] plays a key role, are among the examples falling into an ADE classification.

* [[Bernoulli number]]

* [[moduli space of curves]]

* [[Dwyer-Wilkerson H-space]], which would be the automorphism group of a [[normed division algebra]] if one could exist in 24 dimensions

* [[Dedekind eta function]]

## References

Blog discussion:

* [More Mysteries of the Number 24](https://golem.ph.utexas.edu/category/2007/06/more_mysteries_of_the_number_2.html) at the $n$-Category Café

* [The Binary Octahedral Group](https://golem.ph.utexas.edu/category/2021/12/the_binary_octahedral_group.html) at the $n$-Category Café

* [John Baez on the Number 24](https://math.ucr.edu/home/baez/numbers/#24)

Discussion in relation to the [[Leech lattice]]:

* {#Sloane80} N. J. Sloane, *A note on the Leech lattice as a code for the Gaussian channel*. Information and Control, **46&** 3 (1980). 270--272 ([pdf](http://neilsloane.com/doc/Me75.pdf))

* {#Gannon06} [[Terry Gannon]], section 2.5.1 of: *Moonshine Beyond the Monster*, Cambridge University Press, 2006  ([doi:10.1017/CBO9780511535116](https://doi.org/10.1017/CBO9780511535116))

* {#CohnKumarMIllerRadschenkoViazovska19} H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. Viazovska, *Universal optimality of the $E_8$ and Leech lattices and interpolation formulas*,  ([arXiv:1902.05438](https://arxiv.org/abs/1902.05438)).

Discussion in relation to the [[J-homomorphism]]:

* [Bernoulli Numbers and the J-homomorphism](https://golem.ph.utexas.edu/category/2020/12/bernoulli_numbers_and_the_jhom.html) at the $n$-Category Café

For an introduction to the [[third stable homotopy group of spheres]], see "Week 102" in

* J. Baez, *This Week's Finds in Mathematical Physics: Weeks 101 to 150*, ([arXiv:2303.02785](https://arxiv.org/abs/2303.02785)).


24 branes transverse to K3

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

#Contents#
* table of contents
{:toc}

## Idea

Several classes of [[string theory vacua]] require the presence of exactly [[24]] [[branes]] of [[codimension]] 4 transverse to a [[K3-surface]]-[[fiber]]; this happens notably:

* in [[F-theory]] on an [[elliptically fibered K3-surface]], which requires/implies the presence of 24 [[D7-branes]] ([Sen 96, p. 5](#Sen96)),

* in [[HET-theory]] [[KK-compactification|KK-compactified]] on a [[K3-surface]] with vanishing [[gauge field]] [[instanton number]], which requires/implies the presence of 24 [[NS5-branes]] ([Schwarz 96, p. 50](#Schwarz96)).

In both cases the condition arises as a kind of [[tadpole cancellation]]-condition, where the charge of the 24 branes in the [[compact topological space|compact]] [[K3]]-fiber space, which naively would be 24 in natural units, cancels out to zero, due to some subtle effect.

Despite the superficial similarity, this subtle effect is, at the face of it, rather different in the two cases:

* for [[F-theory]] it results from the [[Kodeira classification]] of [[elliptic fibrations]], which implies that an [[elliptically fibered K3]] has exactly 24 singular points, counted with multiplicity,

* for [[HET-theory]] it results, via the [[Green-Schwarz mechanism]], from the [[Euler characteristic]] of a [[K3-surface]] being 24, and hence vanishing, via the [[Poincare-Hopf theorem]], after the locus of 24 [[NS5-brane]] loci are cut out.

An argument that these two different-looking mechanisms are in fact equivalent, under suitable [[duality in string theory]], is given in [Braun-Brodie-Lukas-Ruehle 18, Sec. III](#BraunBrodieLukasRuehle18), following detailed analysis due to [Aspinwall-Morrison 97](#AspinwallMorrison97).

### In F-theory on K3
 {#InFTheoryOnK3}

In passing from [[M-theory]] to [[type IIA string theory]], the locus of any [[Kaluza-Klein monopole]] in 11d becomes the locus of [[D6-branes]] in 10d. The locus of the [[Kaluza-Klein monopole]] in turn (as discussed there) is the locus where the $S^1_A$-circle fibration degenerates. Hence in F-theory this is the locus where the fiber of the $S^1_A \times S^1_B$-[[elliptic fibration]] degenerates to the [[nodal curve]]. Since the [[T-duality|T-dual]] of [[D6-branes]] are [[D7-branes]], it follows that [[D7-branes]] in F-theory "are" the singular locus of the elliptic fibration.

Now, considering [[F-theory on K3]], an [[elliptically fibered complex K3-surface]]

$$
  \array{
    T &\longrightarrow& K3
    \\
    && \downarrow
    \\
    && \mathbb{C}\mathbb{P}^1
  }
$$

may be parameterized via the [[Weierstrass elliptic function]] as the solution locus of the equation

$$
  y^2 = x^3 + f(z) x + g(z)
$$

for $x,y,z \in \mathbb{C}\mathbb{P}^1$, with $f$ a [[polynomial]] of degree 8 and $g$ of degree twelve. The [[j-invariant]] of the complex [[elliptic curve]] which this parameterizes for given $z$ is

$$
  j(\tau(z)) = \frac{4 (24 f)^3}{27 g^2 + 4 f^3}
  \,.
$$

The [[poles]] $j\to \infty$ of the [[j-invariant]] correspond to the [[nodal curve]], and hence it is at these poles that the [[D7-branes]] are located.

\begin{imagefromfile}
        "file_name": "K3CobordismBetween24ThreeSpheres.jpg",
        "web": "nlab",
        "width": 260,
        "unit": "px",
        "float": "right",
        "margin": {
            "top": -40,
            "right": 0,
            "bottom": 20,
            "left": 20,
            "unit": "px"
        },
        "alt": "homotopy pasting diagram exhibiting the homotopy Whitehead integral",
        "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

Since the order of the poles is 24 (the polynomial degree of the [[discriminant]] $\Delta = 27 g^2 + 4 f^3$, see at _[[elliptically fibered K3-surface]]  -- [singular points](elliptic+fibration+of+a+K3-surface#SingularPoints)_) there are necessarily _24 D7-branes_ ([Sen 96, page 5](#Sen96), [Lerche 99, p. 6](#Lerche99) , see also [Morrison 04, sections 8 and 17](#Morrison04), [Denef 08, around (3.41)](#Denef08), [Douglas-Park-Schnell 14](duality+between+M%2FF-theory+and+heterotic+string+theory#DouglasParkSchnell14)).

Notice that the _net charge_ of these 24 D7-branes is supposed to vanish, due to [[S-duality]] effects (e.g. [Denef 08, below (3.41)](#Denef08)).

### In IIA-theory on K3

Under [[T-duality]] the [above](#InFTheoryOnK3) discussion in [[F-theory]] translates to 24 [[D6-branes]] in [[type IIA string theory]] on [[K3]] ([Vafa 96, Footnote 2 on p. 6](#Vafa96)).

### In HET-theory on K3

In [[heterotic string theory]] [[KK-compactification|KK-compactified]] on [[K3]] with vanishing [[gauge fields]]-[[instanton number]], the existence of exactly 24 [[NS5-branes]] is implied by the [[Green-Schwarz mechanism]]: This requires that the 3-flux density $H_3$ measuring the [[NS5-brane]] charge satisfies $ d H_3 = \trac{1}{2} p_1(\nabla)$; and using that on [[K3]] we have $\int_{K3} \tfrac{1}{2} p_1(\nabla)  = \int_{K3} \chi_4(\nabla) = \chi_4[K3] = 24$ this implies, with [[Stokes' theorem]], that the $H_3$-flux through the [[3-spheres]] around transversal [[NS5-brane]]-punctures of the K3 equals 24
(e.g. [Schwarz 96, around p. 50](duality+in+string+theory#Schwarz97), [Aspinwall-Morrison 97, Sec. 4-5](#AspinwallMorrison97), [Johnson 98, p. 30](#Johnson98), [Braun-Brodie-Lukas-Ruehle 18, Section III.A](#BraunBrodieLukasRuehle18), [Choi-Kobayashi 19, Sec. 1.1](#ChoiKobayashi19)).

The [[duality in string theory|duality]] of this HET-phenomenon with that in F-theory [above](#InFTheoryOnK3) is discussed in [Braun-Brodie-Lukas-Ruehle 18, Section III](#BraunBrodieLukasRuehle18).

### Under Hypothesis H

The vanishing of the [[Euler characteristic]] of K3 after cutting out the [[complement]] of 24 points is precisely the mechanism which witnesses the [[order of a group|order]] 24 of the [[third stable homotopy group of spheres]], seen under [[Pontryagin's theorem]] as the existence of a [[framed manifold|framed]] [[cobordism]] $K3 \setminus 24 \cdot D^4$ between 24 [[3-spheres]]:

\begin{imagefromfile}
    "web": "schreiber",
    "file_name": "24BranesTransversalToK3ViaHypothesisH-20210207.jpg",
    "width": 700,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "from [SS21](https://ncatlab.org/schreiber/show/M-Theory+as+Mf-Theory)"
\end{imagefromfile}

This relates the number of 24 branes transverse on K3 to [[Hypothesis H]]:

\begin{imagefromfile}
    "web": "schreiber",
    "file_name": "KKOn24PuncturedK3ViaHypothesisH-20210207.jpg",
    "width": 760,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "from [SS21](https://ncatlab.org/schreiber/show/M-Theory+as+Mf-Theory)"
\end{imagefromfile}

## Related concepts

* [[duality between M/F-theory and heterotic string theory]]

## References

### In F-theory

Discussion in [[F-theory]] via the [[Kodeira classification]] of [[elliptically fibered K3s]]:

* {#Sen96} [[Ashoke Sen]], _F-theory and Orientifolds_, Nucl. Phys. B475:562-578, 1996 ([arXiv:hep-th/9605150](http://arxiv.org/abs/hep-th/9605150))

* {#Lerche99} [[Wolfgang Lerche]], [p. 6](https://arxiv.org/pdf/hep-th/9910207.pdf#page=6) of: _On the Heterotic/F-Theory Duality in Eight Dimensions_, In: Baulieu L., Green M., Picco M., Windey P. (eds.) _Progress in String Theory and M-Theory_, NATO Science Series (Series C: Mathematical and Physical Sciences), vol 564. Springer 2001 ([arXiv:hep-th/9910207](https://arxiv.org/abs/hep-th/9910207), [doi:10.1007/978-94-010-0852-5_2](https://doi.org/10.1007/978-94-010-0852-5_2))

* {#Morrison04} [[David Morrison]], _TASI Lectures on Compactification and Duality_ ([arXiv:hep-th/0411120](http://arxiv.org/abs/hep-th/0411120))

* {#Denef08} [[Frederik Denef]], [p. 34](https://arxiv.org/pdf/0803.1194.pdf#page=34) of: _Les Houches Lectures on Constructing String Vacua_, in: _[[String theory and the real world]]_ ([arXiv:0803.1194](http://arxiv.org/abs/0803.1194), [spire:780946](https://inspirehep.net/literature/780946))

* {#DouglasParkSchnell14} [[Michael Douglas]], Daniel S. Park, Christian Schnell, _The Cremmer-Scherk Mechanism in F-theory Compactifications on K3 Manifolds_, JHEP05 (2014) 135 ([arXiv:1403.1595](https://arxiv.org/abs/1403.1595))

### In HET-theory

Discussion in [[heterotic string theory]] via the [[Green-Schwarz mechanism]] on K3 (see also at _[[small instantons]]_):

* {#Schwarz96} [[John Schwarz]], around [p. 50](https://arxiv.org/pdf/hep-th/9607201.pdf#page=51) of: _Lectures on Superstring and M Theory Dualities_, Nucl. Phys. Proc. Suppl. 55B:1-32, 1997 ([arXiv:hep-th/9607201](https://arxiv.org/abs/hep-th/9607201))

* {#Johnson98} [[Clifford Johnson]], _Études on D-Branes_, in: [[Mike Duff]] et. al. (eds.) _Nonperturbative aspects of strings, branes and supersymmetry_, Proceedings, Trieste, Italy, March 23-April 3, 1998 ([arXiv:hep-th/9812196](https://arxiv.org/abs/hep-th/9812196), [spire:481393](https://inspirehep.net/literature/481393))

* {#ChoiKobayashi19} Kang-Sin Choi, Tatsuo Kobayashi, _Transitions of Orbifold Vacua_,   JHEP07 (2019) 111 ([arXiv:1901.11194](https://arxiv.org/abs/1901.11194))

### In F- and HET-theory

Joint discussion in F- and [[duality in string theory|dual]] HET-theory:

* {#AspinwallMorrison97} [[Paul Aspinwall]], [[David Morrison]], _Point-like Instantons on K3 Orbifolds_, Nucl. Phys. B503 (1997) 533-564 ([arXiv:hep-th/9705104](https://arxiv.org/abs/hep-th/9705104), doi:10.1016/S0550-3213(97)00516-6)

* {#BraunBrodieLukasRuehle18} [[Andreas Braun]], Callum Brodie, [[Andre Lukas]], [[Fabian Ruehle]], _NS5-Branes and Line Bundles in Heterotic/F-Theory Duality_, Phys. Rev. D 98, 126004 (2018) ([arXiv:1803.06190](https://arxiv.org/abs/1803.06190), [doi:10.1103/PhysRevD.98.126004](https://doi.org/10.1103/PhysRevD.98.126004))

  > (in the context of [[heterotic line bundles]])

[[!include swampland cobordism conjecture -- references]]


24-cell

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

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

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

## Definition

The 24-cell is the [[regular polyhedron]] in the [[Cartesian space]]/[[Euclidean space]] $\mathbb{R}^4$ whose [[vertices]] are, under the identification $\mathbb{R}^4 \simeq_{\mathbb{R}} \mathbb{Q}$  with the space of [[quaternions]], the 8 unit [[quaternions]] $\pm 1$, $\pm i$, $\pm j$, $\pm k$ and the 16 unit quaternions given by
$\frac1{2}(\varepsilon_0 1 + \varepsilon_1 i + \varepsilon_2 j + \varepsilon_3 k)$ where $(\varepsilon_0, \ldots, \varepsilon_3) \in \{-1, 1\}^4$.

(These [[24]] quaternions form a [[group]] under quaternion multiplication, and this group is [[isomorphism|isomorphic]] to the [[binary tetrahedral group]].)

## Properties

### Symmetry group

The [[finite rotation group]] inside [[orthogonal group|O(4)]] which is the [[symmetry group]] of the 24-cell is the [[Coxeter group]] [F4](https://ncatlab.org/nlab/revision/Coxeter+group/7#the_group_).

## Related concepts

* [[120-cell]]

* [[600-cell]]

## References

See also

* Wikipedia, _[24-cell](https://en.wikipedia.org/wiki/24-cell)_


2Cat

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### [[categories of categories - contents|categories of categories]]
+-- {: .hide}
[[!include categories of categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_2Cat_ is the [[3-category]] of [[2-categories]].

It has

* [[objects]] are [[2-categories]];

* [[1-morphisms]] are [[2-functors]];

* [[2-morphisms]] are [[pseudonatural transformation|pseudo]]/[[lax natural transformations]];

* [[3-morphisms]] are [[modifications]].

## Related concepts

* [[Cat]], [[Operad]]

* **$2Cat$**

* [[(∞,1)Cat]], [[(∞,1)Operad]]

* [[(∞,2)Cat]]

* [[(∞,n)Cat]]

category: category

[[!redirects BiCat]]
[[!redirects Bicat]]

[[!redirects 2-Cat]]


2d 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 special case of [[higher dimensional Chern-Simons theory]] for [[dimension]] 2.

## Examples

* [[Poisson sigma-model]]

* [[WZW-model|WZW-term]]/[[B-field]]-coupling of the [[string]]

## Properties

* [[extended geometric quantization of 2d Chern-Simons theory]]

## Related concepts

* [[schreiber:∞-Chern-Simons theory]]

* [[higher dimensional Chern-Simons theory]]

  * [[1d Chern-Simons theory]]

  * **2d Chern-Simons theory**

    * [[2d TQFT]]

  * [[3d Chern-Simons theory]]

    * [[2d Wess-Zumino-Witten 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 2d Chern-Simons theories]]
[[!redirects 2-dimensional Chern-Simons theory]]
[[!redirects 2-dimensional Chern-Simons theories]]


2d quantum gravity

#Contents#
* table of contents
{:toc}

## Idea

In the very low number of just 2 dimensions, [[quantum gravity]] is comparatively trivial and can be completely understood. While trivial as a theory of [[quantum gravity]] on a 2d spacetime, when that spacetime is regarded instead as the [[worldsheet]] of a string, then in fact the string's worldsheet [[sigma-model]] theory is a theory of 2d quantum gravity (before the usual gravitational [[gauge fixing]] in 2d, which then makes what remains a [[conformal field theory]], instead).

## Related concepts

* [[world sheets for world sheets]]

* [[3d quantum gravity]]

## References

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


2d SCFT

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[SCFT|Super-conformal field theory]] in [[dimension]] $d = 2$, locally given by a [[super vertex operator algebra]].

For [[central charge]] 15 this is the [[worldsheet]] [[theory (physics)|theory]] of the [[superstring]].

May be regarded as a "[[2-spectral triple]]" (see there for more), the 2-dimensional generalization of [[spectral triples]] describing the quantum mechanics of [[spinning particles]] (super-particles).

## Examples

* [[superstring]] [[sigma-model]]

  * [[spinning string]] [[superstring]]

  * [[heterotic string]], [[type II superstring]]

  * [[Gepner model]]

  * [[Kazama-Suzuki model]]

  * [[(2,1)-dimensional Euclidean field theories and tmf]]

## Properties

### Classification

See at [supersymmetry -- Classification -- Superconformal algebra -- In dimension 2](supersymmetry#ClassificationSuperconformalInDim2).

## Related concepts

* [[perturbative string theory vacuum]]

## References

### General

A basic but detailed exposition focusing on the super-[[WZW model]] (and the perspective of [[2-spectral triples]]) is in [Fröhlich & Gawedzki (1993)](#FroehlichGawedzki93).

Textbook account:

* [[Ralph Blumenhagen]], [[Erik Plauschinn]], *Introduction to Conformal Field Theory -- With Applications to String Theory*, Lecture Notes in Physics **779**, Springer (2009) [[doi:10.1007/978-3-642-00450-6](https://doi.org/10.1007/978-3-642-00450-6)]

Other accounts:

* [[Lance Dixon]], [[Paul Ginsparg]], [[Jeffrey Harvey]], _$\hat c = 1$ Superconformal field theory_ ([pdf](http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-4515.pdf))

* [[Yasuyuki Kawahigashi]], _$\mathcal{N} = 2$ Superconformal Field Theory and Operator Algebras_ ([pdf](http://www.rci.rutgers.edu/~yzhuang/math/papers/kawahigashi.pdf))

Constructing 2d SCFTs from [[error-correcting codes]] and a hint for a relation to [[tmf]], vaguely in line with the lift of the [[Witten genus]] to the [[string orientation of tmf]]:

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], *Holomorphic SCFTs with small index*, Canadian Journal of Mathematics , **74** 2 (2022) 573-601  [[arXiv:1811.00589](https://arxiv.org/abs/1811.00589), [doi:10.4153/S0008414X2100002X](https://doi.org/10.4153/S0008414X2100002X)]

further on the resulting [[elliptic genera]]:

* Kohki Kawabata, Shinichiro Yahagi, *Elliptic genera from classical error-correcting codes* [[arXiv:2308.12592](https://arxiv.org/abs/2308.12592)]

### Relation to 2-spectral triples

Discussion of 2d SCFTs as a higher analog of [[spectral triples]] ("[[2-spectral triples]]", see there for more) is  in terms of [[vertex operator algebras]] in

* {#FroehlichGawedzki93} [[Jürg Fröhlich]], [[Krzysztof Gawędzki]], _Conformal Field Theory and Geometry of Strings_,  extended lecture notes for lecture given at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4-8 ([arXiv:hep-th/9310187](http://arxiv.org/abs/hep-th/9310187))

* {#Soibelman11} [[Yan Soibelman]], _Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry_ , in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS (2001)

and in terms of [[conformal nets]] in

* {#CHKL09} Sebastiano Carpi, Robin Hillier, [[Yasuyuki Kawahigashi]], [[Roberto Longo]], _Spectral triples and the super-Virasoro algebra_, Commun.Math.Phys.295:71-97 (2010) ([arXiv:0811.4128](http://arxiv.org/abs/0811.4128))

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

[[!include elliptic genera as partition functions -- references]]

[[!redirects 2d SCFTs]]

[[!redirects 2d super conformal field theory]]
[[!redirects 2d super conformal field theories]]

[[!redirects 2d superconformal field theory]]
[[!redirects 2d superconformal field theories]]

[[!redirects 2d super-conformal field theory]]
[[!redirects 2d super-conformal field theories]]


2d TQFT

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

#Contents#
* table of contents
{:toc}

## Idea

A _2-dimensional TQFT_ is a [[TQFT|topological quantum field theory]] on [[cobordisms]] of [[dimension]] 2.

## Classification

When formulated as an (only) "globally" as 1-[[functors]] on a 1-[[category of cobordisms]] (see at _[[FQFT]]_ for more), then 2d TQFTs have a comparatively simple classification: the [[bulk field theory]] is determined by a commutative [[Frobenius algebra]] structure on the [[finite dimensional vector space]] assigned to the circle ([Abrams 96](#Abrams96)).

However, such global 2d TQFTs with [[coefficients]] in [[Vect]]
do not capture the 2d TQFTs of most interest in [[quantum field theory]], which instead are "[[cohomological quantum field theories]]" ([Witten 91](#Witten91)) such as the [[topological string]] [[A-model]] and [[B-model]] that participate in [[homological mirror symmetry]].

These richer 2d TQFTs are instead local TQFTs in the sense of _[[extended TQFT]]_, i.e. they are [[(∞,2)-functors]] on a suitable [[(∞,2)-category of cobordisms]] (see at _[[FQFT]]_ for more), typically on "non-compact" 2-d cobordisms, meaning on those that have non-vanishing outgoing bounary. As such they are now classified by [[Calabi-Yau objects]] in an [[symmetric monoidal (infinity,2)-category]] ([Lurie 09, section 4.2](#Lurie09)). For coefficients in the [[(∞,2)-category]] of [[(∞,n)-vector space|(∞,2)-vector space]] (i.e. [[A-∞ algebras]] with [[(∞,1)-bimodules]] between them in the [[(∞,1)-category of chain complexes]]), these theories had been introduced under the name "[[TCFT]]" in ([Getzler 92](#Getzler92), [Segal 99](#Segal99)) following ideas of [[Maxim Kontsevich]], and have been classified in ([Costello 04](#Costello04)), see ([Lurie 09, theorem 4.2.11, theorem 4.2.14](#Lurie09)).

[[!include 2d TQFT -- table]]

## Filtrations of the moduli space of surfaces {#FiltrationOfModuliSpace}

The following study of the behaviour of 2-dimensional TQFTs in terms of the [[topology]] of the [[moduli spaces]] of marked hyperbolic surfaces is due to [[Ezra Getzler]]. It provides a powerful way to read off various classification results for 2d QFTs from the [[homotopy groups]] of the corresponding [[modular operad]].

### $A_\infty$-monoid objects

Let  $Core$([[FinSet]]) be the [[core]] of the [[category]] of finite sets.
Under union of sets this is a [[symmetric monoidal category]]. Then for
$C$ any [[monoidal category]], a [[symmetric monoidal functor]]

$$
 \Phi : Core(FinSet) \to C
$$

is a commutative [[monoid]] in $C$.

Let now $C$ be a [[category with weak equivalences]], then we can speak of a lax symmetric opmonoidal functor

$$
  \Phi : Core(FinSet) \to C
$$

if the structure maps

$$
  \Phi(n+m) \stackrel{\simeq}{\to} \Phi(m) \otimes \Phi(n)
$$

$$
  \phi(m) \stackrel{\simeq}{\to} \Phi(1)^{\otimes m}
$$

are weak equivalences.

Segal called these "$\Delta$-objects". Since [[Carlos Simpson]] they are called
[[Segal object]]s.

There is also [[Jim Stasheff]]'s notion of an [[A-infinity algebra]], given in terms
of [[associahedra]] $K_n$, which are $(n-2)$-dimensional [[polytope]]s.

There is naturally a filtration on these guys with

$$
  F_0 K_n \subset F_1 K_n \subset \cdots
  \,,
$$

where $F_0 K_n$ is the set of vertices, $F_1 K_n$ the set of edges, etc.

The collection

$$
  \{
    S_\bullet(K_n)
  \}
$$

of simplicial realizations of the $K_n$ form an [[sSet]]-[[operad]] $P$.

For $X$ a [[simplicial category]] that is symmetric monoidal, a
$P$-[[algebra over an operad]] $X$ in $C$ is an $A_\infty$-monoid object

$$
  S_\bullet(K_n) \to C_\bullet(X^{\otimes n}, X)
$$

[[Saunders MacLane|MacLane]]'s [[coherence theorem]] says or uses that if $C$ is an [[n-category]],
we may replace $K_m$ here by the $n$-filtration $F_n K_m$.

### Closed 2d quantum field theory

#### Compactified moduli spaces of Riemann surfaces

Let

$$
  (\Sigma, (z_1, \cdots, z_n))
$$

be a [[compact space|compact]] [[orientation|oriented]] surface
with $n$ distinct marked points. Write

$$
  \mathcal{H}(\Sigma, (z_1, \cdots, z_n))
$$

for the [[moduli space]] of [[hyperbolic metric]]s with cusps at the $(z_i)$.

We have

$$
  M(\Sigma, \vec z)
  =
  \mathcal{H}(\Sigma, \vec z)/Diff_+(\Sigma, \vec z)
$$

and

$$
  M_{g,n} = \Tau_{g,n} / \Gamma^ng
  \,,
$$

where $\Tau_{g,n}$ is the [[Teichmüller space]] and $\Gamma$ the [[mapping class group]].

Here we can assume that the [[Euler characteristic]] $\chi(\Sigma without \{z_i\}) \lt 0$
because otherwise this moduli space is empty.

#### Fenchel-Nielson coordinates on moduli space

We want to parameterize Teichmüller space by cutting surfaces into pieces with
geodesic boundaries and [[Euler characteristic]] $\xi = -1$. These building blocks
(of hyperbolic 2d geometry) are precisely

* the 3-holed sphere;

* the 2-holed cusp;

* the 1-holed 2-cusp;

* the 3-cusp

Each surface of [[genus]] $g$ with $n$ marked points will have

* $2g - 2 + n$ generalized pants;

* $3 g - 3 + n$ closed curves.

The boundary lengths $\ell_i \in \mathbb{R}_+$ and twists
$t_i \in \mathbb{R}$  of these pieces
for

$$
  1 \leq i \leq 3g-3+n
$$

constitute the [[Fenchel-Nielsen coordinates]] on
[[Teichmüller space]] $\Tau$.

Also use $\theta_i := t_i/\ell_i \in \mathbb{R}/\mathbb{Z}$

This constitutes is a real analytic [[atlas]] of Teichmüller space. On $M$ this
reduces to coordinates $t_i \in \mathbb{R}/{\ell_i \mathbb{Z}}$, and these
constitute a real analytic atlas of moduli space.

Allow the lengths $\ell_i$ to go to 0, but keep the angles $\theta_i$.
The resulting space is a real analytic [[manifold with corners]]
$\hat \Tau$ (due to [[Bill Harvey]]) and this constitutes a Borel-Serre
[[bordification]] of $\Tau$.

The [[mapping class group]] $\Gamma$ still acts on $\hat \Tau$ and the
quotient $\hat M$ is an [[orbifold]] with corners, inside which still sits
our moduli space $M$.

Kimura-Stasheff-Voronov: add a choice of
directions at each nodal point in $\Sigma$. This removes
all automorphisms and hence we no longer have to deal with an [[orbifold]].

This yields the [[classifying stack]] $\mathcal{P}_{g,n}$ for $\Gamma_{g,n}$

Then the collection

$$
  \{
    \mathcal{P}_{g,n}
  \}
$$

is a [[modular operad]]: the operad that describes gluing of marked surfaces at
marked points together with the informaiton on how to glue marked points of a single
surface to each other.

A 2-dimensional closed [[TQFT]] is an [[algebra over an operad]] over this in
a simplicial category, in the above sense.

This involves either the [[de Rham complex]] on $\mathcal{P}_{g,n}$ or
$S_\bullet(\mathcal{P}_{g,n})$.

Let

$$
  F_k \mathcal{P}_{g,n}
  :=
  \left\{
    [\Sigma] |
    ...
  \right\}
$$

where $\Sigma$ has $\geq 2g-2+n-k$ spheres as components (after cutting along zero-length
closed curves).

So for instance

* $F_0 \mathcal{P}_{g,n}$ is the pants-decomposition;

* $F_1 \mathcal{P}_{g,n}$ is decompositions into pants and one piece being the
  result of either gluing two pants to each other or of gluing two circles
  of a single pant to each other.

  This $F_1 ..$ is a connected space, due to a theorem by Hatcher-Thurston.
**Notice** This is equivalent to the familiar statement that a closed 2d TFT is
a commutative [[Frobenius algebra]].

* $F_2 \mathcal{P}_{g,n} $  is the decomposition into pieces as before
  together with one two-holed torus or one five-holed sphere.

  This space has the space [[fundamental group]] as $\mathcal{P}_{g,n}$.

  This is equivalent to the theorem by Moore and Seiberg about categorified
  2-d TFT.

+-- {: .un_theorem}
###### Theorem
([[Ezra Getzler]])

The inclusion

$$
  F_k \mathcal{P}_{g,n} \hookrightarrow \mathcal{P}_{g,n}
$$

is $k$-[[connected]].

Here a map $X\to Y$ is $k$-connected if

* $\pi_0(X) \to \pi_0(Y)$ is surjective;

* $\pi_i(X,x) \to \pi_i(Y,f(x))$ is a bijection for $i \lt k$ and surjective $i = k$.

This means precisely that the [[mapping cone]] is $k$-connected.

=--

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

Use the cellular decomposition of moduli space $\mathcal{M}_{g,1}$
following Mumford, Thurston, Harer, Woeditch-Epstein, Penner.

=--

Some other versions of this:

$$
  F_k \Tau_{g,n} \to \Tau_{g,n}
$$

is $k$-connected.

One can also use [[Deligne-Mumford compactification]]s

$$
  F_k \bar \mathcal{M}_{g,n} \to \bar \mathcal{M}_{g,n}
$$

and this is also $k$-connected.

### Open-closed case

...

## Related concepts

* [[sewing constraint]]

  * [[Cardy condition]]

* [[TQFT]]

  * **2d TQFT**

    * [[2d Chern-Simons theory]]

    * [[TCFT]]

      * [[A-model]], [[B-model]]

    * [[Landau-Ginzburg model]]

    * [[Levin-Wen model]]

  * [[3d TQFT]]

  * [[4d TQFT]]

## References

### Global

The [[folklore]] result that global closed 2d TQFTs with coefficients in [[Vect]] are equivalent to commutative [[Frobenius algebra]] structures is proven rigorously in

* {#Abrams96} [[Lowell Abrams]], *Two-dimensional topological quantum field theories and Frobenius algebra*, Jour. Knot. Theory and its Ramifications **5**, 569-587 (1996) [[doi:10.1142/S0218216596000333](https://doi.org/10.1142/S0218216596000333), [ps](http://home.gwu.edu/~labrams/docs/tqft.ps)]

The classification result for open-closed 2d TQFTs was famously announced and sketched in

* {#MooreSegal02} [[Greg Moore]], [[Graeme Segal]], _Lectures on branes, K-theory and RR charges, Clay Math Institute Lecture Notes (2002), _ ([web](http://www.physics.rutgers.edu/~gmoore/clay1/clay1.html))

* {#Lazaroiu00} [[Calin Lazaroiu]], _On the structure of open-closed topological field theory in two dimensions_, Nuclear Phys. B 603(3), 497–530 (2001), ([arXiv:hep-th/0010269](http://arxiv.org/abs/hep-th/0010269))

Textbook account:

* {#Kock2004} [[Joachim Kock]], *Frobenius Algebras and 2d Topological Quantum Field Theories*, Cambridge U. Press (2004) [[doi:10.1017/CBO9780511615443](https://doi.org/10.1017/CBO9780511615443), [webpage](http://mat.uab.cat/~kock/TQFT.html), [course notes pdf](http://mat.uab.es/~kock/TQFT/FS.pdf), [[Kock-FrobAlgTQFT-short.pdf:file]]]

A picture-rich description of what's going on:

* {#LaudaPfeiffer05} [[Aaron Lauda]], [[Hendryk Pfeiffer]], _Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras_ , Topology Appl. 155 (2008) 623-666. ([arXiv:math.AT/0510664](http://arxiv.org/abs/math.AT/0510664))

### Local

The local ([[extended TQFT]]) version of 2d TQFT which captures the [[topological string]] was mathematically introduced under the name "[[TCFT]]".

The concept is essentially a formalization of what used to be called [[cohomological field theory]] in

* {#Witten91} [[Edward Witten]], _Introduction to cohomological field theory_, InternationalJournal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 ([[WittenCQFT.pdf:file]])

The definition was given independently by

* {#Getzler92} [[Ezra Getzler]], _Batalin-Vilkovisky algebras and two-dimensional topological field theories_ , Comm. Math. Phys. 159(2), 265–285 (1994) ([arXiv:hep-th/9212043](http://arxiv.org/abs/hep-th/9212043))

and

* {#Segal99} [[Graeme Segal]], _Topological field theory_ , (1999), Notes of lectures at Stanford university. ([web](http://www.cgtp.duke.edu/ITP99/segal/)). See in particular [lecture 5](http://www.cgtp.duke.edu/ITP99/segal/stanford/lect5.pdf) ("topological field theory with cochain values").

The classification of [[TCFT]]s (i.e. "non-compact" local ([[extended TQFT|extended]] 2d TQFT)) by [[Calabi-Yau A-infinity categories]] is due to

* {#Costello04} [[Kevin Costello]], _Topological conformal field theories and Calabi-Yau categories_ Advances in Mathematics, Volume 210, Issue 1, (2007), ([arXiv:math/0412149](http://arxiv.org/abs/math/0412149))

* [[Kevin Costello]], _The Gromov-Witten potential associated to a TCFT_ ([arXiv:math/0509264](http://arxiv.org/abs/math/0509264))

following conjectures by [[Maxim Kontsevich]], e.g.

* {#Kontsevich95} [[Maxim Kontsevich]], _Homological algebra of mirror symmetry_ , in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 120–139, Basel, 1995, Birkhäuser.

The classification of local ([[extended TQFT|extended]]) 2d TQFT (i.e. the "compact" but fully local case) is spelled out in

* {#SchommerPries11} [[Chris Schommer-Pries]], _The Classification of Two-Dimensional Extended Topological Field Theories_ ([arXiv:1112.1000](http://arxiv.org/abs/1112.1000))

This classification is a precursor of the full [[cobordism hypothesis]]-theorem. This, and the reformulation of the original TCFT constructions in full generality is in

* {#Lurie09} [[Jacob Lurie]], section 4.2 of _[[On the Classification of Topological Field Theories]]_ ([arXiv:0905.0465](http://arxiv.org/abs/0905.0465))

[[!redirects 2d TQFTs]]

[[!redirects 2-dimensional TQFT]]
[[!redirects 2-dimensional TQFTs]]

[[!redirects 2d topological field theory]]
[[!redirects 2d topological field theories]]

[[!redirects 2d topological quantum field theory]]
[[!redirects 2d topological quantum field theories]]


2d TQFT -- table

| [[2d TQFT]]  ("[[TCFT]]") | [[coefficients]] |   [[algebra]] structure on [[space of quantum states]] |  |
|-------------|-------------|--------------------|--------|
| [[open string|open]] [[topological string]] | [[Vect]]${}_k$ | [[Frobenius algebra]] $A$ | [[folklore]]+([Abrams 96](2d+TQFT#Abrams96)) |
| [[open string|open]] [[topological string]] with [[closed string]] [[bulk  field theory|bulk theory]] | [[Vect]]${}_k$ | [[Frobenius algebra]] $A$ with [[trace]] map $B \to Z(A)$ and [[Cardy condition]] | ([Lazaroiu 00](#2d+TQFT#Lazaroiu00), [Moore-Segal 02](2d+TQFT#MooreSegal02)) |
| non-compact [[open string|open]] [[topological string]] | [[category of chain complexes|Ch(Vect)]] | [[Calabi-Yau A-∞ algebra]] | ([Kontsevich 95](2d+TQFT#Kontsevich95), [Costello 04](2d+TQFT#Costello04)) |
| non-compact [[open string|open]] [[topological string]] with various [[D-branes]]| [[category of chain complexes|Ch(Vect)]] | [[Calabi-Yau A-∞ category]] | " |
| non-compact [[open string|open]] [[topological string]] with various [[D-branes]] and with [[closed string]] [[bulk field theory|bulk]] sector | [[category of chain complexes|Ch(Vect)]] | [[Calabi-Yau A-∞ category]] with [[Hochschild cohomology]] | " |
| [[extended TQFT|local]] [[closed string|closed]] [[topological string]] | [[2Mod]]([[Vect]]${}_k$) over [[field]] $k$ | separable symmetric [[Frobenius algebras]] | ([SchommerPries 11](2d+TQFT#SchommerPries11)) |
| non-compact [[extended TQFT|local]] [[closed string|closed]] [[topological string]] | [[2Mod]]([[category of chain complexes|Ch(Vect)]]) | [[Calabi-Yau A-∞ algebra]] | ([Lurie 09, section 4.2](2d+TQFT#Lurie09)) |
| non-compact [[extended TQFT|local]] [[closed string|closed]] [[topological string]] | [[2Mod]]$(\mathbf{S})$ for a [[symmetric monoidal (∞,1)-category]] $\mathbf{S}$ | [[Calabi-Yau object]] in $\mathbf{S}$ | ([Lurie 09, section 4.2](2d+TQFT#Lurie09)) |


2d Wess-Zumino-Witten theory

* [[WZW model]]

## Related concepts

* [[2d Chern-Simons theory]]


2Mod

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

#Contents#
* table of contents
{:toc}

## Idea

The [[2-category]] of [[2-modules]]/[[2-vector spaces]].

Fix some [[commutative ring]] or more generally an [[E-infinity ring]] $R$. Then a [[basis]] for a [[2-module]]/[[2-vector space]] over $R$ may be taken to be an $R$-[[associative algebra]] or more generally an $R$[[Mod]]-[[enriched category]] $\mathcal{A}$ (an [[algebras]]): the correspponding [[2-vector space]] is the [[category of modules]] $Hom_{R Mod}(\mathcal{A}, R Mod)$. Then $2Mod_R$ is equivalently the 2-category whose

* [[objects]] are $R$-algebras;

* [[1-morphisms]] are [[bimodules]],

* [[2-morphisms]] are [[intertwiners]].

In [[enriched category theory]] this is equivalently the 2-category of [[Mod]]${}_R$-[[enriched categories]] and [[profunctors]] between them. In this context one can write

$$
  2Mod_R = Mod(Mod_R)
$$

or $2 Mod_R = $ [[Prof]]$(Mod_R)$ for this 2-category.

## Related constructions

### Algebra objects in $2Mod$

An [[algebra object]] in [[2Mod]]${}_3$ is equivalently a [[sesquiunital sesquialgebra]] over $R$. This may be taken to be a basis for a [[3-module]]/[[3-vector space]] over $R$.

### Line 2-bundles

(...) [[line 2-bundle]] (...)

## Related concepts

* [[module]],  [[vector space]]

* [[2-module]]/[[2-vector space]], [[2-representation]]

  * [[2-vector bundle]]

  * _[[TwoVect]]_ is a Mathematica software package for computer algebra with 2-vector spaces

* [[(∞,1)-module]], [[(∞,1)Mod]]

* [[n-vector space]]

* [[nMod]]

[[!redirects 2Vect]]


2T relation

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

#Contents#
* table of contents
{:toc}

## Idea

In [[knot theory]] the _2-term relations_ on [[horizontal chord diagrams]] are the following [[relations]] in the [[linear span]] of the set of [[horizontal chord diagrams]]:

For pairwise distinct indices $i,j,k,l$, the $(i,j)$-chord generator commutes with the $(k,l)$-chord generator, for instance:

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

Together with the horizontal [[4T relations]] these are the relations respected by [[horizontal weight systems]].

## Related concepts

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

## References

* {#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))

* {#Kohno02} [[Toshitake Kohno]], _Loop spaces of configuration spaces and finite type invariants_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arXiv:math/0211056](https://arxiv.org/abs/math/0211056))

[[!redirects 2T relations]]

[[!redirects 2T-relation]]
[[!redirects 2T-relations]]

[[!redirects 2-term relation]]
[[!redirects 2-term relations]]


3-brane in 6d

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

#Contents#
* table of contents
{:toc}

## Idea

There is supposed to be a $(p=3)$-[[brane]] in 6-dimensional [[supersymmetry|super]]-[[spacetime]] given by the [[Green-Schwarz action functional]] induced by the exceptional super Lie algebra $(3+2)$-cocycle on $\mathfrak{siso}(5,1)$ ([Hughes-Liu-Polchinski 86](#HughesLiuPolchinski86)).

This is thought to be the [[intersecting brane|intersection]] locus of two [[M5-branes]] ([Papadopoulos & Townsend 1996](#PapadopoulosTownsend96), [Tseytlin 1996](#Tseytlin96), [Howe, Lambert & West 1998, p. 2](#HoweLambertWest98), [Kachru, Oz & Yin 1998](#KachruOzYin98)), hence the [[duality between M-theory and type IIA string theory|M-theory lift]] of [D4/NS5-brane intersection](NS5-brane#D4EndingOnNS5).

Since this brane has [[codimension]] 2, it is a *[[defect brane]]*.

## Related concepts

* [[D3-brane]], [[self-dual string]]

[[!include brane scan]]

## References
 {#References}

The original construction is in

* {#HughesLiuPolchinski86} James Hughes, Jun Liu, [[Joseph Polchinski]], _Supermembranes_, Physics Letters B Volume 180, Issue 4, 20 November 1986, Pages 370–374 ([spire](http://inspirehep.net/record/20685))

Discussion building on that includes

* [[Martin Rocek]], [[Arkady Tseytlin]], _Partial breaking of global D=4 supersymmetry, constrained superfields, and 3-brane actions_,  	Phys.Rev.D59:106001, 1999 ([arXiv:hep-th/9811232](http://arxiv.org/abs/hep-th/9811232))

The relevant cocycle for discussion as a [[Green-Schwarz sigma-model]] is given in

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], (III.7.18) of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific, 1991

  (in [[D'Auria-Fré formulation of supergravity]])

Discussion of the [[3-brane in 6d]] explicitly as a [[black brane]] in an [[M5-brane]]/[[NS5-brane]] [[worldvolume]] is due to

* {#HoweLambertWest97} [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _The Threebrane Soliton of the M-Fivebrane_, Phys. Lett. B419 (1998) 79-83 ([arXiv:hep-th/9710033](http://arxiv.org/abs/hep-th/9710033))

and the understanding of this configuration as resulting from two intersecting M5-branes is due to

* {#PapadopoulosTownsend96} [[George Papadopoulos]], [[Paul Townsend]], _Intersecting M-branes_, Phys. Lett. B380 (1996) 273 ([arXiv:hep-th/9603087](http://arxiv.org/abs/hep-th/9603087))

* {#Tseytlin96} [[Arkady Tseytlin]], _Harmonic superpositions of M-branes_, Nucl. Phys. B475 (1996) 149 ([arXiv:hep-th/9604035](https://arxiv.org/abs/hep-th/9604035), doi:10.1016/0550-3213(96)00328-8)

* [[Hironori Mori]], *M-theory Perspectives on Codimension-2 Defects*, Osaka (2016) [[inspire:1519095](https://inspirehep.net/literature/1519095)]

* [[Hironori Mori]], Yuji Sugimoto, *Surface Operators from M-strings*, Phys. Rev. D **95** 026001 (2017) [[arXiv:1608.02849](https://arxiv.org/abs/1608.02849), [doi:10.1103/PhysRevD.95.026001](https://doi.org/10.1103/PhysRevD.95.026001)]

with a [[matrix model]]-description in:

* {#KachruOzYin98} [[Shamit Kachru]], [[Yaron Oz]], [[Zheng Yin]], *Matrix Description of Intersecting M5 Branes* JHEP 9811:004, (1998) ([arXiv:hep-th/9803050](https://arxiv.org/abs/hep-th/9803050))

For more on this see

* [[Joaquim Gomis]], [[David Mateos]], [[Joan Simón]], [[Paul Townsend]], _Brane-Intersection Dynamics from Branes in Brane Backgrounds_, Phys. Lett. B430 (1998) 231-236 ([arXiv:hep-th/9803040](http://arxiv.org/abs/hep-th/9803040))

See also

* S. Bellucci, N. Kozyrev, S. Krivonos, A Sutulin, *Component on-shell actions of supersymmetric 3-branes: I. 3-brane in $D = 6$,  Class. Quantum Grav. **32** (2015) 035025 ([doi:10.1088/0264-9381/32/3/035025](https://iopscience.iop.org/article/10.1088/0264-9381/32/3/035025))

The relation to [[D=4 N=2 super Yang-Mills theory]] is discussed in

* {#HoweLambertWest98} [[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))

On [[quantum Seiberg-Witten curves]] in relation to [[class S-theories]] and M3-[[defect branes]] inside [[M5-branes]]:

* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], *Elliptic Quantum Curves of Class $\mathcal{S}_k$*,  J. High Energ. Phys. **2021** 28 (2021) [[arXiv:2008.05155](https://arxiv.org/abs/2008.05155), doi:10.1007/JHEP03(2021)028]

As [[M5-brane|M5]]-[[probe branes]] in an [AdS7-CFT6](AdS-CFT+correspondence#AdS7CFT6) [[background field|background]] (i.e. in the [[near horizon limit]] of [[black brane|black]] [[M5-branes]]):

* [[Varun Gupta]], *Holographic M5 branes in $AdS_7 \times S^4$*, J. High Energ. Phys. **2021** 32 (2021) [[arXiv:2109.08551](https://arxiv.org/abs/2109.08551), doi:10.1007/JHEP12(2021)032]

* [[Varun Gupta]], *More Holographic M5 branes in $AdS_7 \times S^4$* [[arXiv:2301.02528](https://arxiv.org/abs/2301.02528)]

[[!redirects super 3-brane in 6d]]

[[!redirects M5-brane intersection]]
[[!redirects intersecting M5-branes]]

[[!redirects M3-brane]]
[[!redirects M3-branes]]

[[!redirects M3]]


3-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A $3$-category is any of several concepts that generalize $2$-[[2-category|categories]] one step in [[higher category theory]]. The original notion is that of a globular [[strict 3-category]], but the one most often used here is that of a [[tricategory]]. The concept generalizes to $n$-[[n-category|categories]].

## Definition

Fix a meaning of $\infty$-[[infinity-category|category]], however weak or strict you wish. Then a __$3$-category__ is an $\infty$-category such that every 4-morphism is an [[equivalence]], and all parallel pairs of $j$-morphisms are equivalent for $j \geq 4$. Thus, up to equivalence, there is no point in mentioning anything beyond $3$-morphisms, except whether two given parallel $3$-morphisms are equivalent. This definition may give a concept more general than your preferred definition of $3$-category, but it will be equivalent; basically, you may have to rephrase equivalence of $3$-morphisms as [[equality]].

## Specific versions
* globular [[strict 3-category]]
* [[Gray-category]]
* [[tricategory]]

* [[3-groupoid]]

## Examples

* [[3-category of fermionic conformal nets]] ([[2-Clifford algebras]])

## Related concepts

* [[0-category]], [[(0,1)-category]]

* [[category]]

* [[2-category]]

* **3-category**

* [[n-category]]

* [[(∞,0)-category]]

* [[(∞,1)-category]]

* [[(∞,2)-category]]

* [[(∞,n)-category]]

* [[(n,r)-category]]

[[!redirects 3-categories]]


3-category of fermionic conformal nets

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Super-Algebra and Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### AQFT
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[categorification]] of the [[2-category]] of [[Clifford algebras]] and [[bimodules]] between them. The "2-Clifford algebra" objects are [[spinning string]] theories, modeled as fermionic [[conformal nets]].

This is supposedly related to the [[String 2-group]] and [[superstrings]] as Clifford algebras are related to the [[spin group]] and [[spinning particles]].

(...)

## References

A somewhat novel take on fermionic [[conformal nets]] is presented and studied in

* [[Arthur Bartels]], [[Chris Douglas]], [[Andre Henriques]], _Conformal nets and local field theory_ ([arXiv:0912.5307](http://arxiv.org/abs/0912.5307))

* [[Arthur Bartels]], [[Chris Douglas]], [[Andre Henriques]], _Conformal nets I: coordinate-free nets_ ([arXiv:1302.2604](http://arxiv.org/abs/1302.2604))

where fermionic conformal nets are arranged into a [[tricategory]]  with [[symmetric monoidal category|symmetric monoidal structure]] that is a [[delooping]] of the [[bicategory]] of [[von Neumann algebra]]s and [[bimodule]]s between these. Among other things, this work connects the [[AQFT]] notion of conformal nets with the [[FQFT]] notion of [[cobordism]] representations.

More on this in

* [[Chris Douglas]], [[André Henriques]], _Topological modular forms and conformal nets_, in: [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_

* {#DouglasHenriquesGeometric} [[Chris Douglas]], [[André Henriques]], _Geometric string structures_ ([pdf](http://www.staff.science.uu.nl/~henri105/PDF/TringWP.pdf))

[[!redirects 2-Clifford algebra]]
[[!redirects 2-Clifford algebras]]


3-category with contravariance

# 3-categories with contravariance

* table of contents
{: toc}

## Idea

A *3-category with contravariance* is a [[categorification]] of a [[2-category with contravariance]].  It is an abstract structure that generalizes the structure present on the collection of [[2-categories]] and all four sorts of [[contravariant 2-functor]].

## No mixed-variance transformations

As for [[2-category with contravariance]], the simplest case is when we only consider transformations between functors of the same variance.  Thus for each pair of objects $x,y$ we have four disjoint hom-2-categories $hom^{++}(x,y)$, $hom^{+-}(x,y)$, $hom^{-+}(x,y)$, and $hom^{--}(x,y)$, with sixteen composition 2-functors that act like $\mathbb{Z}/2\times \mathbb{Z}/2$ on the gradings and are appropriately variant on their inputs.

Like [[2-categories with contravariance]], these can be described abstractly (at least in the strict case of [[strict 3-categories]], or the [[semistrict n-category|semistrict]] case of [[Gray-categories]]) as [[enriched categories]], in terms of an action of $\mathbb{Z}/2\times \mathbb{Z}/2$ on $2Cat$.  See [Shulman 2016](#Shulman2016) for a few details.  Depending on whether we use the strict cartesian product on $2 Cat$, the usual "pseudo" Gray tensor product, or its lax or colax versions, we would obtain a structure including strict, pseudo, lax, or colax [[2-natural transformations]] (all between 2-functors of the same variance only).

## Mixed-variance transformations

We can also attempt to include transformations between 2-functors of different variance.  In this case our hom-2-categories will have a $\mathbb{Z}/2\times\mathbb{Z}/2$-grading on their objects, but include morphisms between objects of different variance, yielding sixteen different kinds of natural transformation.  We could also choose to make these transformations strict, lax, colax, or pseudo.  All these kinds of transformation can also be regarded as special cases of [[2-dinatural transformations]]; see there for an example.

It is natural to then conjecture that there is a tensor product on an appropriate category of $\mathbb{Z}/2\times\mathbb{Z}/2$-graded 2-categories giving rise to a notion of enriched category that described this structure.

## References

* [[Mike Shulman]], *Contravariance through enrichment*, [arXiv](https://arxiv.org/abs/1606.05058), 2016
 {#Shulman2016}

[[!redirects 3-categories with contravariance]]


3-group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A _3-group_ is equivalently

1. a [[2-truncated]] [[∞-group]];

1. a [[2-groupoid]] $G$ equipped with the structure of a [[loop space object]] of a connected [[3-groupoid]] $\mathbf{B}G$ (its [[delooping]]);

1. a [[monoidal 2-category]] in which every [[object]] has an weak inverse under the tensor product, every 1-morphism has a weak inverse, and every 2-morphism has an inverse.

## Properties

### Presentation by crossed complexes

Some classes of 3-groups are modeled by  [[2-crossed modules]] or  [[crossed squares]].

## Related concepts

* [[group]]

* [[2-group]], [[crossed module]], [[differential crossed module]]

* **3-group**, [[2-crossed module]] / [[crossed square]], [[differential 2-crossed module]]

  * [[braided 3-group]], [[sylleptic 3-group]], [[symmetric 3-group]]

* [[n-group]]

* [[∞-group]], [[simplicial group]], [[crossed complex]], [[hypercrossed complex]]

[[!include homotopy n-types - table]]

[[!redirects 3-groups]]


3-groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of **3-groupoid** is the next higher generalization in [[higher category theory]] of [[groupoid]] and [[2-groupoid]].

## Definition

A **3-groupoid** is an [[∞-groupoid]] such that all [[parallel pair]]s of [[k-morphism]] are [[equivalence|equivalent]] for $k \geq 4$: a 3-[[truncated]] [[∞-groupoid]].

Thus, up to [[equivalence of categories|equivalence]], there is no point in mentioning anything beyond $3$-morphisms, except whether two given parallel $3$-morphisms are equivalent. This definition may give a concept more general than your preferred definition of $3$-groupoid, but it will be equivalent; basically, you may have to rephrase equivalence of $3$-morphisms as [[equality]].

See also [[n-groupoid]].

## Models

A general 3-groupoid is [[geometric definition of higher categories|geometrically modeled]] by a 4-[[simplicial skeleton|coskeletal]] [[Kan complex]]. Equivalently -- via the [[homotopy hypothesis]]-theorem -- by a [[homotopy 3-type]].

A small model of this is a 3-[[hypergroupoid]], where all [[horn]]-filelrs in dimension $\geq 4$ are _unique_ .

A 3-groupoid is [[algebraic definition of higher categories|algebraically modeled]] by a [[tricategory]] in which all morphisms are invertible, and by a 3-[[truncated]] [[algebraic Kan complex]].

A [[semi-strict infinity-category|semistrict]] [[algebraic definition of higher categories|algebraic model]] for 3-groupoids is provided by the notion of [[Gray-groupoid]]. These in turn are encoded by [[2-crossed module]]s.

An entirely strict algebraic model for 3-groupoids (which no longer models all [[homotopy 3-type]]s) is a 3-[[truncated]] [[strict omega-groupoid]].

## Related concepts

[[!include homotopy n-types - table]]

## References ##

* [[Simona Paoli]], Semistrict models of connected 3-types and Tamsamani's weak 3-groupoids, _Journal of Pure and Applied Algebra_ **211** (2007), 801-820. ([arXiv](http://arxiv.org/abs/math/0607330))

* [[Carlos Simpson]], _Homotopy types of strict 3-groupoids_  ([arXiv](http://arxiv.org/abs/math/9810059))

On the [[homotopy hypothesis]] for [[Grothendieck infinity-groupoid|Grothendieck 3-groupoids]]:

* {#HenryLanari} [[Simon Henry]], [[Edoardo Lanari]], *On the homotopy hypothesis in dimension 3*, Theory and Applications of Categories **39** 26 (2023) 735-768 [[arxiv/1905.05625](https://arxiv.org/abs/1905.05625), [tac:39-26](http://www.tac.mta.ca/tac/volumes/39/26/39-26abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/39/26/39-26.pdf)].

[[!redirects 3-groupoids]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[smooth manifold]] $U$ and a [[Lie 3-algebra]] $\mathfrak{g}$, the 3-groupoid of Lie 3-algebra valued forms over $U$ has as objects [[∞-Lie algebroid valued differential forms]] with values in $\mathfrak{g}$, as morphisms gauge transformations of these, as 2-morphisms 2-gauge transformations and so on.

This can be understood as the 3-groupoid of _trivial_ $G$-[[principal ∞-bundle|principal 3-bundle]]s over $U$ with nontrivial connection, for $G$ the [[∞-Lie groupoid|3-Lie group]] related to $\mathfrak{g}$ by [[Lie integration]].

Regarded as a presheaf of 3-groupoids over all suitable manifolds $U$, this is a non-concrete [[∞-Lie groupoid|3-Lie groupoid]].

A [[cocycle]] with coefficients in this 3-groupoid is a [[connection on a 3-bundle]].

## Related concepts

* [[groupoid of Lie-algebra valued forms]]

* [[2-groupoid of Lie 2-algebra valued forms]]

* **3-groupoid of Lie 3-algebra valued forms**

* [[∞-groupoid of ∞-Lie-algebra valued forms]]

## References

For Lie 3-algebras coming from [[differential 2-crossed module]]s, at least parts of this data have been discussed in

* {#MartinsPicken11} [[João Faria Martins]], [[Roger Picken]], _The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module_, Differential Geometry and its Applications **29** 2 (2011) 179-206  [[arXiv:0907.2566](http://arxiv.org/abs/0907.2566), [doi:10.1016/j.difgeo.2010.10.002](https://doi.org/10.1016/j.difgeo.2010.10.002)]


3-manifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _3-manifold_ is a [[manifold]] of [[dimension]] 3. (Our default meaning of "manifold" is [[topological manifold]], unless a qualifier is added, e.g., *smooth* manifold.)

## Properties

### Triangulability and smoothing

The following is taken from [Hatcher](#Hatcher):

> A pleasant feature of 3-manifolds, in contrast to higher dimensions, is that there is no essential difference between smooth, piecewise linear, and topological manifolds. It was shown by Bing and [Moise](#Moise52) in the 1950s that [[triangulation theorem|every topological 3-manifold can be triangulated]] as a simplicial complex whose combinatorial type is unique up to subdivision. And every triangulation of a 3-manifold can be taken to be a smooth triangulation in some differential structure on the manifold, unique up to diffeomorphism.  Thus every topological  3-manifold has a unique smooth structure, and the classifications up to diffeomorphism and homeomorphism coincide.

Thus it makes no essential difference if we consider 3-manifolds as mere topological manifolds, or as [[piecewise-linear manifolds]] or [[smooth manifolds]]. It's often technically convenient to work in the smooth category.

### Poincaré conjecture

+-- {: .num_theorem}
###### Theorem
**([[Poincaré conjecture]])**

Every [[simply connected]] [[compact space|compact]] 3-manifold without boundary is [[homeomorphism|homeomorphic]] to the 3-sphere.

=--

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

A proof strategy was given by [[Richard Hamilton]]: imagine the manifold is equipped with a [[metric]]. Follow the [[Ricci flow]] of that metric through the space of metrics. As the flow proceeds along parameter time, it will from time to time pass through metrics that describe singular geometries where the compact metric manifold pinches off into separate manifolds. Follow the flow through these singularities and then continue the flow on each of the resulting components. If this process terminates in finite parameter time with the metric on each component stabilizing to that of the round 3-sphere, then the original manifold was a 3-sphere.

The hard technical part of this program is to show that the passage through the  singularities can be controlled. This was finally shown by [[Grigori Perelman]].

=--

### Geometrization conjecture

The _[[geometrization conjecture]]_ says that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure.

### Virtually fibered conjecture

The _[[virtually fibered conjecture]]_ says that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle.

## Related concepts

* [[irreducible 3-manifold]]

* [[atoroidal 3-manifold]]

* [[Seifert 3-manifold]]

* [[hyperbolic 3-manifold]]

* [[Dehn surgery]]

* [[Kirby calculus]]

* [[low dimensional topology]]

* [[knot theory]]

* [[arithmetic topology]]

* [[Chern-Simons theory]]

* [[Atiyah 2-framing]]

* [[lens space]]

* [[associative submanifold]]

* [[2-manifold]], [[4-manifold]], [[8-manifold]]

## References

### General

Review:

* {#Thurston80} [[William Thurston]], _Geometry and topology of three-manifolds_ (1980), electronic version 1.1 (2002) available from MSRI ([web](http://library.msri.org/books/gt3m/))

* {#Hatcher}  [[Allen Hatcher]], _The classification of 3-manifolds -- a brief overview_, ([pdf](https://www.math.cornell.edu/~hatcher/Papers/3Msurvey.pdf)).

* {#Martelli16} Bruno Martelli, _An Introduction to Geometric Topology_ ([arXiv:1610.02592](https://arxiv.org/abs/1610.02592))

The [[triangulation theorem]] for [[3-manifolds]]:

* {#Moise52} [[Edwin E. Moise]],  *Affine Structures in 3-Manifolds: V. The Triangulation Theorem and Hauptvermutung*, Annals of Mathematics Second Series, Vol. 56, No. 1 (Jul., 1952), pp. 96-114 ([doi:10.2307/1969769](https://doi.org/10.2307/1969769), [jstor:1969769](https://www.jstor.org/stable/1969769))

3-manifolds as [[branched covers]] of the [[3-sphere]]:

*  J. Montesinos, _A representation of closed orientable 3-manifolds as  3-fold branched coverings of $S^3$_, Bull. Amer. Math. Soc. 80 (1974), 845-846 ([Euclid:1183535815](https://projecteuclid.org/euclid.bams/1183535815))

See also

* {#BottCattaneo98} [[Raoul Bott]], [[Alberto Cattaneo]], _Integral Invariants of 3-Manifolds_, J. Diff. Geom., 48 (1998) 91-133 ([arXiv:dg-ga/9710001](https://arxiv.org/abs/dg-ga/9710001))

### Hyperbolic 3-manifolds

On [[hyperbolic 3-manifolds]]:

* [[William Thurston]], _Hyperbolic Structures on 3-manifolds, I: Deformation of acylindrical manifolds_, Annals of Math, 124 (1986), 203--246 ([jstor:1971277](https://www.jstor.org/stable/1971277), [arXiv:math/9801019](https://arxiv.org/abs/math/9801019))

* [[William Thurston]], _Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle_ ([arXiv:math/9801045](https://arxiv.org/abs/math/9801045))

* [[William Thurston]], _Three dimensional manifolds, Kleinian groups and hyperbolic geometry_, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 3 (1982), 357-381 ([euclid.bams/1183548782](https://projecteuclid.org/euclid.bams/1183548782))

### Vafa-Witten theory

Computations of Vafa-Witten invariants of 3-manifolds are given in

* [[Sergei Gukov]], Artan Sheshmani, [[Shing-Tung Yau]], _3-manifolds and Vafa-Witten theory_ ([arXiv:2207.05775](https://arxiv.org/abs/2207.05775)).

[[!redirects 3-manifolds]]


3-morphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

* [[object]]

* [[morphism]]

* [[2-morphism]]

* **3-morphism**

* [[k-morphism]]

***

#Contents#
* table of contents
{:toc}

## Definition

A _3-morphism_ in an [[n-category]] is a morphism between [[2-morphism]]s.

## Examples

For $A$ an [[abelian group]], the 3-fold [[delooping]] $\mathbf{B}^3 A$ is a [[3-groupoid]] whose 3-morphisms are given by $A$.

[[!redirects 3-morphisms]]


3-poset

A $3$-poset is any of several concepts that generalize $2$-[[2-poset|posets]] one step in [[higher category theory]]. One does not usually here about $3$-posets by themselves but instead as special cases of $3$-[[3-category|categories]].

$3$-posets can also be called $(2,3)$-[[(n,r)-category|categories]].  The concept generalizes to $n$-[[n-poset|posets]].

# Definition

Fix a meaning of $\infty$-[[infinity-category|category]], however weak or strict you wish. Then a __$3$-poset__ is an $\infty$-category such that all parallel pairs of $j$-morphisms are [[equivalence|equivalent]] for $j \geq 3$. Thus, up to [[equivalence of categories|equivalence]], there is no point in mentioning anything beyond $3$-morphisms, not even whether two given parallel $3$-morphisms are equivalent.


3-sphere

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

#Contents#
* table of contents
{:toc}

## Idea

The $n$-[[sphere]] for $n = 3$.

## Properties

### Isomorphisms

The underlying manifold of the [[special unitary group]] [[SU(2)]] happens to be [[isomorphism|isomorphic]] to the 3-sphere, hence also that of [[spin group|Spin(3)]].

The [[quotient]] of that by the [[binary icosahedral group]] is the [[Poincaré homology sphere]].

### Homotopy groups
 {#HomotopyGroups}

The first few [[homotopy groups]] of the 3-sphere:

| $n =$ |  $0$ |  $1$ |  $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ |
|-----|---|---|---|--|--|---|---|---|--|--|--|--|--|
| $\pi_n(S^3) =$ |  $\ast$ | $0$ | $0$ | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ |  $\mathbb{Z}_{12}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_3$ | $\mathbb{Z}_{15}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ |

e.g. [Calabrese 16](#Calabrese16), for more see at _[[homotopy groups of spheres]]_.

## Related concepts

* [[Hopf fibration]]

* [[fuzzy 3-sphere]]

* [[2-sphere]]

* [[6-sphere]]

* [[7-sphere]]

* [[n-sphere]]

## References

Discussion of [[homotopy groups of spheres]] for the 3-sphere:

* {#Calabrese16} John Calabrese, _The fourth homotopy group of the sphere_, 2016 ([pdf](https://math.rice.edu/~jrc9/stuff/sss.pdf))

Discussion of [[3-manifolds]] as [[branched covers]] of the 3-sphere:

*  J. Montesinos, _A representation of closed orientable 3-manifolds as  3-fold branched coverings of $S^3$_, Bull. Amer. Math. Soc. 80 (1974), 845-846 ([Euclid:1183535815](https://projecteuclid.org/euclid.bams/1183535815))

Classification of [[Riemannian orbifolds]] whose coarse underlying topological space is a 3-sphere:

* [[William Dunbar]], _Geometric orbifolds_, Rev. Mat. Univ. Complutense Madr. 1, No.1-3, 67-99 (1988)

[[!redirects 3-spheres]]


3-strand-braid-1-SVG

<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="403.2646pt" height="143.25488pt" viewBox="-18.65686 -71.62744 403.2646 143.25488 ">

  <defs
     id="defs4">
    <inkscape:perspective
       id="perspective10"
       inkscape:persp3d-origin="372.04724 : 350.78739 : 1"
       inkscape:vp_z="744.09448 : 526.18109 : 1"
       inkscape:vp_y="0 : 1000 : 0"
       inkscape:vp_x="0 : 526.18109 : 1"
       sodipodi:type="inkscape:persp3d" />
  </defs>
  <sodipodi:namedview
     id="base"
     pagecolor="#ffffff"
     bordercolor="#666666"
     borderopacity="1.0"
     inkscape:pageopacity="0.0"
     inkscape:pageshadow="2"
     inkscape:zoom="0.35"
     inkscape:cx="-295"
     inkscape:cy="520"
     inkscape:document-units="px"
     inkscape:current-layer="layer1"
     showgrid="false"
     inkscape:window-width="1440"
     inkscape:window-height="814"
     inkscape:window-x="0"
     inkscape:window-y="0"
     inkscape:window-maximized="1" />
  <metadata
     id="metadata7">
    <rdf:RDF>
      <cc:Work
         rdf:about="">
        <dc:format>image/svg+xml</dc:format>
        <dc:type
           rdf:resource="http://purl.org/dc/dcmitype/StillImage" />
        <dc:title></dc:title>
      </cc:Work>
    </rdf:RDF>
  </metadata>
  <g
     inkscape:label="Layer 1"
     inkscape:groupmode="layer"
     id="layer1">
    <g
       transform="translate(161.57484,168.76493)"
       id="layer1-6"
       inkscape:label="Layer 1">
      <path
         sodipodi:nodetypes="cccccc"
         id="path2383"
         d="m 188.32014,243.16382 97.08878,55.47931 0.11012,84.9171 m -168.47362,-251.3076 1.92559,69.30171 41.60947,27.73966"
         style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:0.80512744px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
         inkscape:connector-curvature="0" />
      <path
         id="path2387"
         d="m 202.18997,132.20522 0,83.21895 -83.21896,83.21896 0,83.21895"
         style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:0.69349128px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
         inkscape:connector-curvature="0" />
      <path
         sodipodi:nodetypes="ccccccc"
         id="path2389"
         d="m 285.40892,215.42417 -49.5351,47.55369 m -8.9163,11.88842 -24.76755,23.77685 0,83.21895 m 83.21895,-249.65686 0,83.21895"
         style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:0.69349128px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
         inkscape:connector-curvature="0" />
    </g>
  </g>
</svg>


3-theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

* table of contents
{: toc}

## Idea

In the context of an account of [[logic]] described in terms of $n$-theories ([Shulman 18a](#Shulman18a)), a *3-theory* specifies a kind of [[2-category]].
It does so by prescribing the allowable judgment structures of the [[2-theories]] that can be expressed within it.

## Examples

1.  The 3-theory of (totally unstructured) 2-categories.  Syntactically, this corresponds to "unary type (2-)theories" with judgments such as $x:A \vdash b:B$ with exactly one type to both the left and the right of the turnstile ([LS 15](#LS15)).

1. The 3-theory of cartesian monoidal 2-categories (or cartesian 2-multicategories).  Syntactically, this corresponds to "simple type (2-)theories" with judgments such as $x:A, y:B, z:C \vdash d:D$ with a finite list of types to the left but exactly one type to the right, and no dependent types ([LSR 17](#LSR17)).

1. The 3-theory of first-order logic. Syntactically, this corresponds to type 2-theories with one layer of dependency: there is one layer of types which cannot depend on anything, and then there is a second layer of types (“propositions” in the logical reading) which can depend on types in the first layer, but not on each other. 2-theories here are for hyperdoctrines and indexed monoidal categories, and include (typed) first-order classical logic, first-order intuitionistic logic, first-order linear logic, the type theory of indexed monoidal categories, first-order modal logic, etc. Semantic 2-theories in this 3-theory should be a sort of “2-hyperdoctrine”. A [[higher-order logic]] is a 2-theory which adds rules specifying a universe of propositions.

1. The 3-theory that corresponds syntactically to "dependent type (2-)theories" with judgments such as $x:A, y:B(x), z:C(x, y) \vdash d:D(x, y, z)$. The semantic version of this should be some kind of "comprehension 2-category" ([Shulman 18b](#Shulman18b)). [[Homotopy type theory]] is a 2-theory defined in this 3-theory.

1. The 3-theory that corresponds syntactically to "classical simple type (2-)theories" with judgments such as $A,B,C \vdash D,E$ that allow finite lists of types on both sides of the turnstile.  For instance, the 2-theory of $\ast$-autonomous categories (classical linear logic), and also of Boolean algebras (classical nonlinear logic), live in this 3-theory, as does the 2-theory for symmetric monoidal categories described in ([Shulman 19](#Shulman19)). Variants would allow composition along multiple types at once (corresponding to a prop), composition only along one type at a time (corresponding to a polycategory), or both.

1. The classical 3-theory that extends the first-order 3-theory by allowing multiple sequents. (The term 'classical' here and above owes its origin to the contrast between systems LK and LJ of [[Gentzen's]] [[sequent calculus]], where LJ provides a proof system for [[intuitionistic logic]] via restriction to single sequents of the rules of LK for classical logic. This naming practice leads to a certain confusion as it is quite possible to represent classical first-order logic in a single sequent system such as Gentzen's NJ.)

A 3-theory may be said to subsume another 3-theory, in the sense that a [[2-theory]] written in the latter may be formulated in the former. Here the first four 3-theories are ordered in terms of increasing expressivity. (5) subsumes (2), and (6) subsumes (3), but (4) and (5) are not comparable.

Modal variants of 2-theories may be defined in 3-theories where the types appearing in judgments are indexed by a system (2-category) of modes ([Shulman 18b](#Shulman18b)).

## References

* {#Shulman18a} [[Mike Shulman]], _What Is an n-Theory?_, ([blog post](https://golem.ph.utexas.edu/category/2018/04/what_is_an_ntheory.html))

* {#LS15} [[Dan Licata]], [[Mike Shulman]], _Adjoint logic with a 2-category of modes_, in _[Logical Foundations of Computer Science 2016](http://lfcs.info/lfcs-2016/)_ ([pdf](http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint.pdf), [slides](http://dlicata.web.wesleyan.edu/pubs/ls15adjoint/ls15adjoint-lfcs-slides.pdf))

* {#LSR17} [[Daniel Licata]], [[Mike Shulman]], and [[Mitchell Riley]], _A Fibrational Framework for Substructural and Modal Logics (extended version)_, in Proceedings of 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017) ([doi: 10.4230/LIPIcs.FSCD.2017.25](http://drops.dagstuhl.de/opus/volltexte/2017/7740/), [pdf](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/lsr17multi-ex.pdf))

* {#Shulman18b} [[Mike Shulman]], _Type 2-theories_, ([HoTTEST seminar](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html))

* {#Licata18} [[Dan Licata]], _Synthetic Mathematics in Modal Dependent Type Theories_, tutorial at _[Types, Homotopy Theory and Verification](https://www.him.uni-bonn.de/programs/current-trimester-program/types-sets-constructions/workshop-types-homotopy-type-theory-and-verification/)_, 2018 ([pdf](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/him-tutorial.pdf))

* {#Shulman19} [[Mike Shulman]], _A practical type theory for symmetric monoidal categories_, ([arXiv:1911.00818](https://arxiv.org/abs/1911.00818))


3d Calabi-Yau object

#Contents#
* table of contents
{:toc}

## Idea

A [[Calabi-Yau object]] ([[Calabi-Yau manifold]], [[Calabi-Yau category]]) of (complex) [[dimension]] 3.

## Properties

### As exceptional geometry

[[!include Spin(8)-subgroups and reductions -- table]]

### Moduli spaces of line bundles

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

## Related concepts

* [[Calabi-Yau cohomology]]

## References

### Intermediate Jacobian

Discussion of [[intermediate Jacobians]] of Calabi-Yau 3-folds includes

* C. Herbert Clemens, [[Phillip Griffith]], _The intermediate Jacobian of the cubic threefold_, Annals of Mathematics Second Series, Vol. 95, No. 2 (Mar., 1972), pp. 281-356 ([JSTOR](http://www.jstor.org/stable/1970801))

* [[Claire Voisin]] ([pdf](http://www.math.polytechnique.fr/~voisin/Articlesweb/griffithsgroup.pdf))

* [[Andreas Höring]], _Minimal classes on the intermediate Jacobian of a generic cubic threefold_, 2008 ([pdf](http://math.unice.fr/~hoering/articles/a5-intermediate.pdf))

Discussion of the [[Artin-Mazur groups]] of CY3s in [[positive number|positive]] [[characteristic]]:

* {#GeerKatsura03} [[Gerard van der Geer]], T. Katsura, _On the height of Calabi-Yau varieties in positive characteristic_ ([arXiv:math/0302023](http://arxiv.org/abs/math/0302023))

### Hall algebra

Discussion of [[motivic Hall algebras]] of CY 3-folds is in

* {#KontsevichSoibelman08} [[Maxim Kontsevich]], [[Yan Soibelman]], _Stability structures, motivic Donaldson-Thomas invariants and cluster transformations_ ([arXiv:0811.2435](http://arxiv.org/abs/0811.2435))

[[!redirects 3d Calabi-Yau space]]

[[!redirects Calabi-Yau 3-fold]]
[[!redirects Calabi-Yau 3-folds]]


3d 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 special case of [[higher dimensional Chern-Simons theory]] for [[dimension]] 3.

## Examples

* ordinary [[Chern-Simons theory]]

* [[Dijkgraaf-Witten theory]]

* [[Courant sigma-model]]

* [[3d quantum gravity]]

## Related concepts

* [[schreiber:∞-Chern-Simons theory]]

* [[higher dimensional Chern-Simons theory]]

  * [[1d Chern-Simons theory]]

  * [[2d Chern-Simons theory]]

  * **3d Chern-Simons theory**

    * [[3d TQFT]]

    * [[3d quantum gravity]]

    * [[2d Wess-Zumino-Witten 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]]

* [[string field theory]]

* [[AKSZ sigma-model]]

[[!redirects 3d Chern-Simons theories]]
[[!redirects 3-dimensional Chern-Simons theory]]
[[!redirects 3-dimensional Chern-Simons theories]]


### 3d Gravity and Chern-Simons theory
 {#3dGravityAndChernSimonsTheoryReferences}

On 3-dimensional ([[quantum gravity|quantum]]) [[gravity]] ([[general relativity]]) with [[cosmological constant]], and its (non-)relation to [[Chern-Simons theory]] with non-[[compact Lie group|compact]] [[gauge groups]]:

The original articles on [[3d gravity]], discussing its formulation as a [[Chern-Simons theory]] and discovering its [[holography|holographic]] relation to a [[2d CFT]] [[boundary field theory]] (well before [[AdS/CFT]] was conceived from [[string theory]]):

* [[Stanley Deser]], [[Roman Jackiw]], [[Gerard 't Hooft]], _Three-dimensional Einstein gravity: Dynamics of flat space_, Ann. Phys. 152 (1984) 220 (doi:10.1016/0003-4916(84)90085-X)

* [[Stanley Deser]], [[Roman Jackiw]], _Three-dimensional cosmological gravity: Dynamics of constant curvature_, Annals of Physics, Volume 153, Issue 2, 1 April 1984, Pages 405-416 (doi:10.1016/0003-4916(84)90025-3, [spire:192694](http://inspirehep.net/record/192694))

* [[Roman Jackiw]], _Lower dimensional gravity_, Nuclear Physics B Volume 252, 1985, Pages 343-356 (doi:10.1016/0550-3213(85)90448-1, [spire:204694](http://inspirehep.net/record/204694))

* J. D. Brown, [[Marc Henneaux]], _Central charges in the canonical realization of asymptotic symmetries: An example from three dimensional gravity_, Commun. Math. Phys. (1986) 104: 207 ([doi:10.1007/BF01211590](https://doi.org/10.1007/BF01211590))

* A. Achucarro, [[Paul Townsend]], _A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories_, Phys. Lett. B180 (1986) 89 (doi:10.1016/0370-2693(86)90140-1, [spire:21208](http://inspirehep.net/record/21208))

* [[Steven Carlip]], _Inducing Liouville theory from topologically massive gravity_, Nuclear Physics B Volume 362, Issues 1–2, 16 September 1991, Pages 111-124 (doi:10.1016/0550-3213(91)90558-F)

* O. Coussaert, [[Marc Henneaux]], P. van Driel, _The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant_, Class. Quant. Grav. 12 (1995) 2961-2966 ([arXiv:gr-qc/9506019](https://arxiv.org/abs/gr-qc/9506019))

The corresponding [[non-perturbative QFT|non-perturbative]] [[quantization]] of 3-dimensional gravity, via [[quantization of 3d Chern-Simons theory]]:

* {#Witten88} [[Edward Witten]], _(2+1)-Dimensional Gravity as an Exactly Soluble System_ Nucl. Phys. B311 (1988) 46. ([web](http://adsabs.harvard.edu/abs/1988NuPhB.311...46W))

* [[Herman Verlinde]], _Conformal field theory, two-dimensional quantum gravity and quantization of Teichmüller space_, Nuclear Physics B
Volume 337, Issue 3, 25 June 1990, Pages 652-680 (doi:10.1016/0550-3213(90)90510-K)

Review:

* [[Steven Carlip]], _Lectures in (2+1)-Dimensional Gravity_, J. Korean Phys. Soc. 28: S447-S467, 1995 ([arXiv:gr-qc/9503024](https://arxiv.org/abs/gr-qc/9503024))

* [[Steven Carlip]], _Quantum Gravity in 2+1 Dimensions_, Cambridge Monographs on Mathematical Physics (2003) ([publisher](http://www.cambridge.org/de/academic/subjects/physics/cosmology-relativity-and-gravitation/quantum-gravity-21-dimensions))

* [[Steven Carlip]], _Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe_, Living Rev. Rel. 8:1, 2005 ([arXiv:gr-qc/0409039](https://arxiv.org/abs/gr-qc/0409039))

* [[Steven Carlip]], _[My Research -- (2+1)-Dimensional quantum gravity](http://www.physics.ucdavis.edu/Text/Carlip.html#2+1)_

* [[Laura Donnay]], _Asymptotic dynamics of three-dimensional gravity_, Proceedings of *Eleventh Modave Summer School in Mathematical Physics*, POS **271** (2016)  $[$[arXiv:1602.09021](http://arxiv.org/abs/1602.09021), [doi:10.22323/1.271.0001 ](https://doi.org/10.22323/1.271.0001 )$]$

* [[Wout Merbis]], _Chern-Simons-like Theories of Gravity_ ([arXiv:1411.6888](https://arxiv.org/abs/1411.6888))

* Chen-Te Ma, *$AdS_3$ Einstein Gravity and Boundary Description: Pedagogical Review* $[$[arXiv:2310.04665](https://arxiv.org/abs/2310.04665)$]$

Further developments:

* {#Witten} [[Edward Witten]], _Three-dimensional gravity revisited_, (2007) [arxiv/0706.3359](http://arxiv.org/abs/0706.3359)

* [[Paul Townsend]], _Massive 3d (super)gravity_, slides, ([pdf](http://superfields.web.cern.ch/Superfields/docs/Seminars/Townsend.pdf))

* Gaston Giribet, _Black hole physics and AdS3/CFT2 correspondence_, lectures at [[Croatian Black Hole School]] 2010

* Alan Garbarz, Gaston Giribet, Yerko Vásquez, _Asymptotically AdS$_3$ solutions to topologically massive gravity at special values of the coupling constants_, [arxiv/0811.4464](http://arxiv.org/abs/0811.4464)

* Rudranil Basu, Samir K Paul, _Consistent 3D Quantum Gravity on Lens Spaces_ ([arXiv:1109.0793](http://arxiv.org/abs/1109.0793))

* [[Marc Henneaux]], [[Wout Merbis]], Arash Ranjbar, _Asymptotic dynamics of $AdS_3$ gravity with two asymptotic regions_ ([arXiv:1912.09465](https://arxiv.org/abs/1912.09465))

* Viraj Meruliya, [[Sunil Mukhi]], Palash Singh, _Poincaré Series, 3d Gravity and Averages of Rational CFT_ ([arXiv:2102.03136](https://arxiv.org/abs/2102.03136))

* [[Scott Collier]], [[Lorenz Eberhardt]], [[Mengyang Zhang]], *3d gravity and Chern-Simons theory -- references* $[$[arXiv:2304.13650](https://arxiv.org/abs/2304.13650)$]$

See also:

* Nathan Benjamin, Scott Collier, Alexander Maloney, *Pure Gravity and Conical Defects*, Journal of High Energy Physics **2020** 34 (2020) $[$[arXiv:2004.14428](https://arxiv.org/abs/2004.14428), doi:10.1007/JHEP09(2020)034$]$


3d mirror symmetry

## Related concepts

* [[symplectic duality]]

* [[duality in physics]]

## References

The original article is

* [[Ken Intriligator]], [[Nathan Seiberg]], _Mirror Symmetry in Three Dimensional Gauge Theories_ ([arXiv:hep-th/9607207](http://arxiv.org/abs/hep-th/9607207))

based on the [[ADE classification]] of [[ALE spaces]] due to

* [[Peter Kronheimer]], _The construction of ALE spaces as hyper-Kähler quotients_, J. Differential Geom. Volume 29, Number 3 (1989), 665-683. ([Euclid](https://projecteuclid.org/euclid.jdg/1214443066))

Review includes

* Wikipedia, _[3d Mirror symmetry](https://en.wikipedia.org/wiki/3D_mirror_symmetry)_

* Giulia Ferlito, _Coulomb branch and the moduli space of instantons_, 2015 ([pdf](http://www.maths.liv.ac.uk/TheorPhys/RESEARCH/STRING_THEORY/2015/Slides/GiuliaFerlito.pdf))

Further developments include

* [[Mina Aganagic]], [[Kentaro Hori]], [[Andreas Karch]], [[David Tong]], _Mirror Symmetry in 2+1 and 1+1 Dimensions_, JHEP 0107:022,2001 ([arXiv:hep-th/0105075](http://arxiv.org/abs/hep-th/0105075))


3d quantum gravity

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Solving the problem of [[quantization]] of systems including [[gravity]] -- [[quantum gravity]] -- is notoriously hard. It does however simplify drastically in very low [[dimensions]]. While the theory in dimension 4 ([[Kaluza-Klein mechanism|or higher]]) is evidently relevant for the phenomenology of the world that we perceive, the [[Einstein-Hilbert action]] and its variants that defines the theory of [[gravity]] (at least as an [[effective QFT]]) makes sense in any dimension.

The case of dimension 3 is noteworthy, because in this case the quantum theory can be and has been fairly completely understood and is nevertheless non-trivial.  Informally, this is due to the fact that behaviour of [[gravity]] in 3-dimensions is much simpler than in higher dimensions: there cannot be [[gravitational waves]] in 3-dimensions, hence no "local excitations". Accordingly, the theory turns out to have a _finite dimensional_ [[covariant phase space]].

More formally, one finds that in 3-dimensions, the [[Einstein-Hilbert action]], in the [[first order formulation of gravity]] ([[Cartan connection]]), becomes equivalent to the [[action functional]] of a very well studied 3-dimensional field theory, namely [[Chern-Simons theory]] for [[gauge group]] the [[Poincaré group]] $Iso(2,1)$ (for vanishing [[cosmological constant]]) or the [[de Sitter group]]/[[anti de Sitter group]] $Iso(2,2)$ or $Iso(3,1)$ (for non-vanishing cosmological constant). See at _[[de Sitter gravity]]_ and at _[Chern-Simons Gravity](http://ncatlab.org/nlab/show/Chern-Simons+gravity)_.

This means that one can take the [[quantization]] of $Iso(2,1)$-[[Chern-Simons theory]] as the _definition_ of 3d [[quantum gravity]]. This has first been noticed and successfully carried out in ([Witten88](#Witten88)).  One should note here that this means that one allows degenerate [[vielbein]]/[[pseudo-Riemannian metric]] tensors as field configurations of gravity. In fact, as ([Witten88](#Witten88)) discusses in detail, this [[compactification]] of configuration space can be seen as the source of the reasons why 3d quantum gravity makes exists.

Based on this striking situation in 3-dimensions it is natural to wonder if it makes sense to consider $Iso(n,1)$-[[higher dimensional Chern-Simons theory]], or variants thereof, as theories of [[quantum gravity]] in higher dimensions. The for $n \gt 2$ the corresponding [[action functional]]s differ from the [[Einstein-Hilbert action]] by higher curvature terms, but in suitable limits of the theory these can be argued to play a negligible role. For more on this see the entry _[[Chern-Simons gravity]]_ .

## Variants

One can add additional terms arriving at what is called _massive 3d gravity models_ . Very relevant for its study is the [[holographic principle|AdS3/CFT2 correspondence]].

## Related concepts

* [[BTZ black hole]]

* [[3d TQFT]]

* [[3-dimensional supergravity]]

* [[quantum gravity]]

* [[Chern-Simons gravity]]

* [[Boulatov model]]

* [[2d quantum gravity]]

* [[volume conjecture]]

* [[piecewise flat spacetime]]

## References

* Nathan Benjamin, Scott Collier, Alexander Maloney, *Pure Gravity and Conical Defects*, Journal of High Energy Physics **2020** 34 (2020) [[arXiv:2004.14428](https://arxiv.org/abs/2004.14428), doi:10.1007/JHEP09(2020)034]

* Mauricio Leston, et al., *3d Quantum Gravity Partition Function at 3 Loops: a brute force computation* [[arXiv:2307.03830](https://arxiv.org/abs/2307.03830)]

Survey:

* [[Steve Carlip]], *Quantum Gravity in 2+1 Dimensions*, in:  *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.12596](https://arxiv.org/abs/2312.12596)]

[[!include 3d gravity and Chern-Simons theory -- references]]

[[!include Chern-Simons Wilson lines in AdS3-CFT2 -- references]]

[[!redirects 3d gravity]]
[[!redirects D=3 gravity]]


3d superconformal gauge field theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

According to the [classification of superconformal symmetry](supersymmetry#ClassificationSuperconformal), there should exists [[superconformal field theories]] in 3 dimensions

[[!include superconformal symmetry -- table]]

Such superconformal [[gauge field theories]] involving [[Chern-Simons theory]] coupled to matter exist and are thought to describe the [[worldvolume]] theory of [[black brane|black]] [[M2-branes]] located at [[ADE-singularities]]:

[[!include 7d spherical space forms -- table]]

The corresponding story for the [[M5-brane]] is the [[6d superconformal gauge field theory]].

## Related concepts

* [[D=3 N=2 super Yang-Mills theory]]

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

## References

The original article on the $N=8$-case (the [[BLG model]]):

* {#BaggerLambert06} [[Jonathan Bagger]], [[Neil Lambert]], _Modeling Multiple M2's_, Phys. Rev. D75, 045020 (2007). ([hep-th/0611108](http://arxiv.org/abs/hep-th/0611108)).

* [[Jonathan Bagger]], [[Neil Lambert]], Phys. Rev. D77, 065008 (2008). ([arXiv:0711.0955](http://arXiv.org/abs/0711.0955)).

The original article on the $N=6$-case (the [[ABJM model]]):

* {#ABJM08} [[Ofer Aharony]], [[Oren Bergman]], Daniel Louis Jafferis, [[Juan Maldacena]], _$N=6$ superconformal Chern-Simons-matter theories, M2-branes and their gravity duals_, JHEP 0810:091,2008, [DOI:10.1088/1126-6708/2008/10/091](http://iopscience.iop.org/article/10.1088/1126-6708/2008/10/091/meta;jsessionid=FCE6764D4E19F3038C9530E50B057A56.c3.iopscience.cld.iop.org) ([arXiv:0806.1218](http://arxiv.org/abs/0806.1218))

The $\mathcal{N}=5$-case is discussed in

* {#HLLLP08} Kazuo Hosomichi, Ki-Myeong Lee, Sangmin Lee, Sungjay Lee, [[Jaemo Park]], _$\mathcal{N}=5,6$ Superconformal Chern-Simons Theories and M2-branes on Orbifolds_, JHEP 0809:002, 2008 ([arXiv:0806.4977](https://arxiv.org/abs/0806.4977))

* {#BHRSS08} [[Eric Bergshoeff]], [[Olaf Hohm]], Diederik Roest, [[Henning Samtleben]], [[Ergin Sezgin]], _The Superconformal Gaugings in Three Dimensions_, JHEP0809:101, 2008 ([arXiv:0807.2841](https://arxiv.org/abs/0807.2841))

* {#AharonyBergmanJafferis08} [[Ofer Aharony]], Oren Bergman, Daniel Louis Jafferis, _Fractional M2-branes_, JHEP 0811:043, 2008 ([arXiv:0807.4924](https://arxiv.org/abs/0807.4924))

The $N=4$-case is discussed in

* {#HLLLP08b} Kazuo Hosomichi, Ki-Myeong Lee, Sangmin Lee, Sungjay Lee, [[Jaemo Park]], _$\mathcal{N}=4$ Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets_, JHEP 0807:091,2008 ([arXiv:0805.3662](https://arxiv.org/abs/0805.3662))

* {#ChenWu10} Fa-Min Chen, Yong-Shi Wu, _Superspace Formulation in a Three-Algebra Approach to D=3, N=4,5 Superconformal Chern-Simons Matter Theories_, Phys.Rev.D82:106012, 2010 ([arXiv:1007.5157](https://arxiv.org/abs/1007.5157))

Review includes

* {#KlebanovTorri10} [[Igor Klebanov]], Giuseppe Torri, _M2-branes and AdS/CFT_, Int.J.Mod.Phys.A25:332-350, 2010 ([arXiv;0909.1580](https://arxiv.org/abs/0909.1580))

* Neil B. Copland, _Introductory Lectures on Multiple Membranes_ ([arXiv:1012.0459](https://arxiv.org/abs/1012.0459))

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

For more see at _[[ABJM model]]_.

[[!redirects D=3 SYM]]

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


3d toric code

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What has come to be called the "3d toric code" is a [[4d TQFT]] (i.e. with 3 spatial dimensions, whence the name) which is the higher-dimensional analog of the [[3d TQFT]] known as the *[[toric code]]*.

## References

The model is due to:

* Alioscia Hamma, Paolo Zanardi, [[Xiao-Gang Wen]], *String and Membrane condensation on 3D lattices*,  Phys. Rev. B72:035307, 2005 ([arXiv:cond-mat/0411752](https://arxiv.org/abs/cond-mat/0411752), [doi:10.1103/PhysRevB.72.035307](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.72.035307))

Further discussion of [[defect QFT|defects]] in the model (such as [[defect brane|defect strings]]):

* {#KongTianZhang20} [[Liang Kong]], Yin Tian, Zhi-Hao Zhang,  *Defects in the 3-dimensional toric code model form a braided fusion 2-category*, J. High Energ. Phys. 2020, 78 (2020) ([arXiv:2009.06564](https://arxiv.org/abs/2009.06564), doi:10.1007/JHEP12(2020)078)

[[!redirects 3d toric codes]]