M5-brane

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### Context
#### String theory
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[[!include string theory - contents]]
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#Contents#
* table of contents
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## Idea

In [[11-dimensional supergravity]] the [[brane]] electrically charged under the [[supergravity C-field]] is the [[M2-brane]]/[[membrane]]. The dual under [[electric-magnetic duality]] is the M5-brane.

## Definition

### As a Green-Schwarz type sigma-model

As a [[Green-Schwarz sigma-model]]: [BLNPST 97](#BLNPST97)

### As a black $p$-brane

As a [[black brane]] solution of [[11-dimensional supergravity]] the M5-brane is given ([Gueven 92](#Gueven92)) by the [[spacetime]] $\mathbb{R}^{5,1} \times (\mathbb{R}^5-\{0\})$ with [[pseudo-Riemannian metric]] given by

$$
  g = H^{-1/3} g_{\mathbb{R}^{5,1}} \oplus H^{2/3} g_{\mathbb{R}^5-\{0\}}
$$

for $H = 1 + \frac{1}{r}$ and $r$ the distance in $\mathbb{R}^5$ from the origin, and with [[field strength]] of the [[supergravity C-field]] being

$$
  F = \star_{\mathbb{R}^5} \mathbf{d}H
  \,.
$$

This is a $1/2$-[[BPS state]] of 11-dimensional supergravity.

The [[near horizon geometry]] of this spacetime is [[anti de Sitter spacetime|AdS7]]$\times$[[4-sphere|S4]]. For more on this see at _[[AdS-CFT]]_.

[[!include black branes in supergravity -- table]]

### At an orbifold singularity
 {#AtAnOrbifoldSingularity}

{#NearHorizonOrbifold} More generally for 1/2 BPS black M5-branes, the near horizon geometry is $AdS_7 \times S^4/G$, where $G$ is a [[finite subgroup of SU(2)]] ([[ADE classification|ADE subgroup]]) acting by left multiplication on the [[quaternions]] $\mathbb{H}$ in the canonical way, under the identitfication $S^4 \simeq S(\mathbb{R}^5) \simeq S(\mathbb{R}\oplus \mathbb{H})$ ([MFF 12, section 8.3](#MFF12)).

While this geometric discussion in [MFF 12, section 8.3](#MFF12) works for all the [[finite subgroups of SU(2)]], folklore has it that in [[M-theory]] the M5-branes appear only at A-type singularities, while the more general [[6d (2,0)-superconformal field theories]] for all possible [[ADE-singularities]] appear only after passage to [[F-theory]] ([ZHTV 14, p. 3](#ZHTV14)).

On the other hand, when placing the M5 at an MO5-[[orientifold]] ([Witten 95](#Witten95)) its worldvolume theory breaks from $(2,0)$ to $(1,0)$-supersymmetry and all ADE-singularities should be allowed.

(...)

## Properties

### Worldvolume theory

the [[worldvolume]] theory of the M5-brane is the [[6d (2,0)-superconformal QFT]].

This [[worldvolume]] theory involves [[self-dual higher gauge theory]] of the [[nonabelian cohomology|nonabelian]] kind ([Witten07](#Witten07), [Witten09](#Witten09)): the fields are supposed to be [[connections on a 2-bundle]]($\sim$ [[gerbe]]), presumably with structure [[2-group]] the [[automorphism 2-group]] $AUT(G)$ of some [[Lie group]] $G$.

For instance in the proposal of ([SSW11](#SSW11)) one sees in equation (2.1) _almost_ the data of an $\mathfrak{aut}(\mathfrak{g})$-[[2-groupoid of Lie 2-algebra valued forms|Lie 2-algebra valued forms]].

### Branes inside the M5

The M5-brane admits two solitonic excitations ($p$-branes within branes)

* $p = 1$: the [[self-dual string]]

* $p = 3$: the [[3-brane in 6d]] (see there for more)

See also

* [[M2-M5 brane bound state]]

### Dimensional reduction

On [[Kaluza-Klein mechanism|dimensional reduction]] of [[11-dimensional supergravity]] on a circle the M5-brane turns into the [[NS5-brane]] and the [[D-brane|D4-brane]] of [[type II string theory]].

The [[Kaluza-Klein mechanism|compactification]] of the 5-brane on a [[Riemann surface]] yields as [[worldvolume]] [[theory (physics)|theory]] [[N=2 D=4 super Yang-Mills theory]]. See at _[N=2 D=4 SYM -- Construction by compactification of 5-branes](N%3D2+D%3D4+super+Yang-Mills+theory#ConstructionByCompactificationOf5Branes)_.

### Holographic dual

The [[AdS/CFT correspondence]] for the 5-brane is $AdS_7/CFT_6$ and relates the [[6d (2,0)-superconformal QFT]] to [[7-dimensional supergravity]] obtained by [[Kaluza-Klein mechanism|reduction]] of [[11-dimensional supergravity]] on a 4-[[sphere]] to an asymptotically 7d [[anti de Sitter spacetime]].

### Conformal blocks and 7d Chern-Simons dual
 {#7dCSDual}

The [[self-dual higher gauge theory|self-dual 2-connection]]-field (see there for more details) on the 6-dimensional [[worldvolume]] M5-brane is supposed to have a [[holographic principle|holographic]] description in terms of a [[higher dimensional Chern-Simons theory|7-dimensional Chern-Simons theory]] ([Witten 1996](#Witten96)). We discuss the relevant "fractional" [[quadratic form]] on [[ordinary differential cohomology]] that defines the correct [[action functional]].

Let $\hat G$ be the [[circle n-bundle with connection|circle 3-bundle with connection]] on a 7-dimensional manifold $X$ with boundary the M5-brane, thought of as the [[Kaluza-Klein mechanism|compactification]] of the [[supergravity C-field]] from [[11-dimensional supergravity]] down to [[7-dimensional supergravity]].

As discussed there, the [[higher dimensional Chern-Simons theory|7-dimensional Chern-Simons theory]] [[action functional]] on these 3-connections is

$$
  \hat G_4 \mapsto
  \exp(i \int_X \hat G_4 \cup \hat G_4)
  \,,
$$

where

* $\exp(i \int_X (-))$ is the [[higher holonomy]] / [[fiber integration in ordinary differential cohomology]] from $X$ to the point

* of the [[Beilinson-Deligne cup product]] [[circle n-bundle with connection|7-connection]] $\hat G_4 \cup \hat G_4$.

The space of [[states]] of this 7d theory on the M5 worldvolume $\partial X$ would be the space of [[conformal blocks]] of the [[6d (2,0)-supersymmetric QFT]] on the [[worldvolume]].

Except, that it turns out that the [[first Chern class]] of the corresponding [[prequantum line bundle]] is _twice_ that required from [[geometric quantization]].

Therefore the above action functional is not yet the correct one, but only a fractional version of it is. However, the class $G_4 \cup G_4$ in [[integral cohomology]] has in general no reason to be divisible by 2.

This is related to the fact that as a [[quadratic form]] on the [[ordinary differential cohomology]] group $\hat H^4(X)$, the above is not a [[quadratic refinement]] of

$$
  (\hat G, \hat G') \mapsto \exp(i \int_X \hat G \cup \hat G')
  \,,
$$

but of twice that. In ([Witten 1996](#Witten96)) it was argued, and later clarified in ([Hopkins-Singer](#HopkinsSinger)), that instead the action functional should be replaced by a proper [[quadratic refinement]].

This is accomplished by shifting the center of the [[quadratic form]]
by a lift $\lambda \in H^4(X, \mathbb{Z})$ of the degree-4 [[Wu class]] $\nu_4 \in H^4(X, \mathbb{Z}/2)$ from 0 to $\frac{1}{2}\lambda$.

(For that to make sense in [[integral cohomology]], either the [[Wu class]] $\lambda$ happens to be divisible by 2 on $X$, or else one has to regard it itself as a twisted differential character of sorts, as explained in ([Hopkins-Singer](#HopkinsSinger)). For the moment we will assume that $X$ is such that $\lambda$ is divisbible by 2.)

Since $X$, being a [[spacetime]] for [[supergravity]], admits (and is thought to be equipped with) a [[spin structure]], by the discussion at [[Wu class]] it follows that $\lambda$ is the [[first fractional Pontryagin class]] $\frac{1}{2}p_1$

$$
  (\frac{1}{2}p_1 \; mod \; 2)
   \;
     =
   \;
   \nu_4 \in H^4(X, \mathbb{Z}/2)
  \,.
$$

By the very definition of [[Wu class]], it follows that for any $\hat \alpha \in \hat H^4(X)$ the combination

$$
  \hat \alpha \cup \hat \alpha + \hat \alpha \cup \hat \lambda = Sq^4(\hat \alpha) - \hat \alpha \cup \hat \lambda
  \; =\;  0 \;  mod  \; 2
$$

is divisible by 2.

Therefore define then the modified quadratic form

$$
  \exp(i S^\lambda)
   \; : \;
  \hat a \mapsto \exp i \int_X
  \frac{1}{2}
  \left(
    \hat a \cup \hat a
    +
    \hat a \cup \hat \mathbf{\lambda}
  \right)
$$

(see [[differential string structure]] for the definition of the differential refinement $\hat \mathbf{\lambda} = \frac{1}{2}\hat \mathbf{p}_1$),
where, note, we have included a global factor of 2, which is now possible due to the inclusion of the integral lift of the Wu class.

Notice that where the [[equations of motion]] of the original [[action functional]] are $\hat a = 0$, those of this shifted one are $\hat a = - \frac{1}{2}\hat \mathbf{\lambda}$. One may therefor calls $-\frac{1}{2}\lambda$ here a _background [[charge]]_ for the 7-d Chern-Simons theory.

This is now indeed a [[quadratic refinement]] of the [[intersection pairing]]:

$$
  \exp i \left(
    S^\lambda\left(\hat a + \hat b \right)
    -
    S^\lambda\left( \hat a \right)
    -
    S^\lambda\left( \hat b \right)
    +
    S^\lambda\left( 0 \right)
  \right)
  =
  \exp i \int_X ( \hat a \cup \hat b  )
  \,.
$$

To express the correct action functional for the 7d Chern-Simons theory it is useful to define the shifted [[supergravity C-field]]

$$
  \hat a  := \hat G_4 - \frac{1}{2}\hat \mathbf{\lambda}
  \,,
$$

which the object whose [[equations of motion]] with respect to the 7d Chern-Simons theory are still $\hat a = 0$.

Then in terms of the original $\hat G_4$ the action functional for the [[holographic principle|holographic dual]] [[higher dimensional Chern-Simons theory|7d Chern-Simons theory]] reads

$$
  \exp(i S(\hat G_4))
  =
  \exp(i \int_X
    \frac{1}{2}
    (
      \hat G_4 \cup \hat G_4
      -
      (\frac{1}{2}\hat \mathbf{\lambda})^2
    )
  )
  \,.
$$

This is the action as it appears in ([Witten96, (3.6)](#Witten96)).

In terms of [[twisted differential c-structures]] we may summarize the outcome of this reasoning as follows:

_The divisibility of the action functional requires a $2(G_4 - a)$-[[twisted Wu structure]] in $\mathbb{Z}/2$-cohomology. Its lift to integral cohomology is the $2(G_4 - a)$-[[twisted differential string structure]] known as the "Witten quantization condition" on the [[supergravity C-field]]_.

### Restriction of the supergravity $C$-field
 {#RestrictionOfTheCField}

We discuss the conditions on the
restriction of the [[supergravity C-field]] on the ambient
[[11-dimensional supergravity]] [[spacetime]] to the M5-brane.

This is similar to the analogous situation in [[type II string theory]].
The [[Freed-Witten anomaly cancellation]] condition demands that
the restriction of the [[B-field]] $\hat H_3 \hat H^3(X)$
on spacetime $X$ to an
[[orientation|oriented]] [[D-brane]] $Q \hookrightarrow X$ has
to trivialize, up to [[torsion]], relative to the integral
[[Stiefel-Whitney class]] $W_3 = \beta(w_2)$, where
$\beta$ is the [[Bockstein homomorphism]] induced from the
[[short exact sequence]] $\mathbb{Z} \stackrel{\cdot 2}{\to}
\mathbb{Z} \to \mathbb{Z}_2$:

$$
  H_3|_Q \simeq W_3
  \,,
$$

thus defining a [[twisted spin^c-structure]] on the [[D-brane]].

The analog of this for the M5-brane is discussed in
([Witten00, section 5](#Witten2000)). There it is argued that
there is a class

$$
  \theta \in H^3(Q, U(1))
$$

on the 5-brane such that under the [[Bockstein homomorphism]]
$\beta'$ induced by the [[short exact sequence]]
$\mathbb{Z} \to \mathbb{R} \to U(1)$ we have for the
[[supergravity C-field]] $\hat G \in \hat H^4(X)$ the condition

$$
  G|_Q = \beta'(\theta)
  \,.
$$

By the [above](#7dCSDual) quantization condition, this may also be thought of
as witnessing a [[twisted string structure]] on the 5-brane ([Sati](#Sati10)).

This condition reduces to the above one for the $B$-field under [[double dimensional reduction]] on the circle.

### M5-brane charge

See at _[[M5-brane charge]]_

\linebreak

### Anomaly cancellation
 {#AnomalyCancellation}

Consider a 2-parameter family $X^{(11)}$ of 11-dimensional [[spin structure|spin]]-[[manifolds]]  and a 2-parameter family $Q_{M5} \hookrightarrow X^{(11)}$ of 6-dimensional [[submanifolds]]. When regarded as a family of [[worldvolumes]] of an [[M5-brane]], the family of [[normal bundles]] $N_X Q_{M5}$ of this inclusion carries a [[characteristic class]]

\[
  \label{IM5}
  I^{M5}
  \;\coloneqq\;
  I^{M5}_{\psi}
  +
  I_{C}
  \;\in\;
  H^8(F \times Q_{M5},\mathbb{Z})
\]

where

1. the first summand is the class of the [[chiral anomaly]] of [[chiral fermions]] on $Q_{M5}$ ([Witten 96, (5.1)](#Witten96)),

1. the second term the class of the [[quantum anomaly]] of a [[self-dual higher gauge field]] ([Witten 96, (5.4)](#Witten96))

Moreover, there is the restriction of the [[I8]]-term (see [there](I8#eq:TheTerm)) to $Q_{M5}$, hence to the [[tangent bundle]] of $X^{11}$ to $Q_{M5}$ (the "anomaly inflow" from the [[bulk spacetime]] to the M5-brane)

\[
  \label{I8AnomalyInflow}
  I_8\vert_{M5}
  \;\coloneqq\;
   I_8
   \big(
     T_{Q_{M5}} X
   \big)
  \;\in\;
  H^8(F \times Q_{M5},\mathbb{Z})
  \,.
\]

The sum of these cohomology classes, evaluated on the [[fundamental class]] of $Q_{M5}$ is proportional to the [[second Pontryagin class]] of the [[normal bundle]]

\[
  \label{SumOfM5AndInflowAnomalyIsp2}
  I^{M5}
  \;+\;
  I_8\vert_{M5}
  \;=\;
  \tfrac{1}{24} p_2(N_{Q_{M5}})
\]

([Witten 96 (5.7)](#Witten96))

This result used to be "somewhat puzzling" ([Witten 96, p. 35](#Witten96)) since consistency of the [[M5-brane]] in [[M-theory]] should require its total [[quantum anomaly]] to vanish. But $p_2(N_{Q_{M5}})$ does not in general vanish, and the right conditions to require under which it does vanish were "not clear" ([Witten 96, p. 37](#Witten96)).

(For more details on computations involved in this and the following arguments, see also [Bilal-Metzger 03](#BilalMetzger03)).

A resolution was proposed in [Freed-Harvey-Minasian-Moore 98](#FreedHarveyMinasianMoore98), also [Bah-Bonetti-Minasian-Nardoni 18 (5)](#BahBonettiMinasianNardoni18), [BBMN 19 (2.9) and appendix A.4, A.5](#BBMN19). By this proposal, the anomaly inflow from the bulk would not be just $I_8$, as in (eq:I8AnomalyInflow) but would be all of the following [[fiber integration]]

\[
  \label{FiberIntegration}
  \array{
    \pi_\ast
    \Big(
      -
      \tfrac{1}{6}
      G_4  G_4  G_4
      +
      G_4  I_8
    \Big)
    & =
    - \tfrac{1}{24} p_2 + \tfrac{1}{2}([G^{M5}_4])^2 + I_8
  }
\]

Here  we used (by [this Prop](Spin5#FiberIntegrationOfCupPowersOfChiOver4Sphere)) that

$$
  \pi_\ast\big(
    \chi^3
  \big)
  \;=\;
  2 p_2
  \,,
$$

which would thus cancel against the first term $\tfrac{1}{24} p_2$ in (eq:FiberIntegration). Hence with this proposal, the previously remaining M5-brane anomaly (eq:SumOfM5AndInflowAnomalyIsp2) would be canceled... except for yet one last remaining term (shown as $\propto \gamma_4^2$ in  [BBMN 19b (4.9), (4.13), (5.22)](#BBMN19b)):

\[
  \label{RemainingAnomaly}
  [G^{M5}_4]^2
  \;\in\;
  H^8(Q_{M5}; \mathbb{Z})
  \,,
\]

where

\[
  \label{BasicFormComponent}
  G_4^{M5}
  \;\coloneqq\;
  (G_4)_{basic}
  \;\in\;
  \Omega^4_{dR}\big( Q_{M4}\big)
\]

denotes the [[basic differential form|basic form]]-component of $G_4$ with respect to the given [[spherical fibration]].

This [[basic differential form|basic form]] component $G_4^{M5}$ (eq:BasicFormComponent) had been ignored in [FHMM 98](#FreedHarveyMinasianMoore98) and previous references. That this [[basic differential form|basic form]] component $G_4^{M5}$ (eq:BasicFormComponent) indeed needs to be considered was pointed out in [FSS 19v1, (19)](#FSS19v1), [BBMN 19b, (3.16) & App. C](#BBMN19b) (where it is denoted $\gamma_4$, see also [BBM 20 (2.3)](#BahBonettiMinasian20)) and [SS 20, (3) & p. 5](#SS20).

See also at _[M-theory -- Open problems -- M5-brane anomaly cancellation](M-theory#OpenProblemM5BraneAnomalyCancellation)_.

The observation that the remaing anomaly  (eq:RemainingAnomaly) vanishes if one assumes [[Hypothesis H]] is [SS 20, Cor. 6](#SS20).

\linebreak

[[!include M2-M5 brane bound states in the BMN matrix model -- subsection]]

\linebreak

## Related concepts

* [[Perry-Schwarz action]]

* [[M5-brane instanton]]

* [[M5-brane intersection]]

* [[small instanton]]

* [[half M5-brane]]

* [[string theory]]

* [[11-dimensional supergravity]], [[M-theory]]

* [[supergravity Lie 6-algebra]], [[M-theory super Lie algebra]]

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

[[!include table of branes]]

## References
 {#References}

### Survey

The history as of the 1990 is reviewed in

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

Further reviews and general accounts include

* [[Robbert Dijkgraaf]], _The mathematics of fivebranes_ ([pdf](http://arxiv.org/PS_cache/hep-th/pdf/9810/9810157v1.pdf))

* [[Hisham Sati]], _[[Geometric and topological structures related to M-branes]]_ (2010)

### Black brane description
 {#ReferencesAsBlackBrane}

#### Unwrapped

The M5 was first found as a [[black brane]] of [[11-dimensional supergravity]] (the [[black fivebrane]]) in

* {#Gueven92} [[Rahmi Gueven]], _Black $p$-brane solutions of $D = 11$ supergravity theory_, Phys. Lett. B276 (1992) 49 (doi:10.1016/0370-2693(92)90540-K, [spire:338203](http://inspirehep.net/record/338203))

  also in: [[Mike Duff]] (ed.),  _[[The World in Eleven Dimensions]]_ 135-141

That this metric, as well as that of every black $p$ brane for _odd_ $p$, is completely non-singular was observed in

* [[Gary Gibbons]], [[Gary Horowitz]], [[Paul Townsend]], p. 15 of _Higher-dimensional resolution of dilatonic black hole singularities_, Class. Quant. Grav. 12:297-318, 1995 ([arXiv:hep-th/9410073](https://arxiv.org/abs/hep-th/9410073))

Classification of more general M5-brane [[ADE-singularities]] is in

* S. Ferrara, A. Kehagias, H. Partouche, A. Zaffaroni, _Membranes and Fivebranes with Lower Supersymmetry and their AdS Supergravity Duals_, Phys. Lett. B **431** (1998) 42-48 ([arxiv:hep-th/9803109](https://arxiv.org/abs/hep-th/9803109))

* {#MFF12} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], Section 8.3 of: _Half-BPS M2-brane orbifolds_, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761), [Euclid](https://projecteuclid.org/euclid.atmp/1408561553))

also to some extent in

* Changhyun Ahn, Kyungho Oh, Radu Tatar, _Orbifolds $AdS_7 \times S^4$ and Six Dimensional $(0, 1)$ SCFT_, Phys. Lett. B442 (1998) 109-116 ([arXiv:hep-th/9804093](https://arxiv.org/abs/hep-th/9804093))

but see p. 3 of

* {#ZHTV14} Michele Del Zotto, [[Jonathan Heckman]], [[Alessandro Tomasiello]], [[Cumrun Vafa]], _6d Conformal Matter_, 10.1007/JHEP02(2015)054 ([arXiv:1407.6359](https://arxiv.org/abs/1407.6359))

Identification of the $\mathcal{N} = (2,0)$ black M5-brane sitting at the [[ADE singularity|A-type singularity]] of an $MO5$ $\mathbb{Z}/2$-[[orientifold]] locally of the form $\mathbb{R}^{5,1} \times ( \mathbb{R}^5 \sslash (\mathbb{Z}/2) )$ is due to

* {#Witten95} [[Edward Witten]], _Five-branes And M-Theory On An Orbifold_, Nucl. Phys. B463:383-397, 1996 ([arXiv:hep-th/9512219](https://arxiv.org/abs/hep-th/9512219))

Discussion in terms of [[E11]]-[[U-duality]] and [[current algebra]] is in

* [[Hirotaka Sugawara]], _Current Algebra Formulation of M-theory based on E11 Kac-Moody Algebra_, International Journal of Modern Physics A, Volume 32, Issue 05, 20 February 2017 ([arXiv:1701.06894](https://arxiv.org/abs/1701.06894))

* [[Shotaro Shiba]], [[Hirotaka Sugawara]], _M2- and M5-branes in E11 Current Algebra Formulation of M-theory_ ([arXiv:1709.07169](https://arxiv.org/abs/1709.07169))

#### Wrapped on hyperbolic 3-manifolds

Solutions to [[D=11 N=1 supergravity]] describing [[black brane|black]] [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-manifolds]] (with application to the [[3d-3d correspondence]] and [[proof]] of the [[volume conjecture]]):

* [[Jerome Gauntlett]], [[Nakwoo Kim]], [[Daniel Waldram]], Section 3.1 of: _M-Fivebranes Wrapped on Supersymmetric Cycles_, Phys. Rev. D63 (2001) 126001 ([arXiv:hep-th/0012195](https://arxiv.org/abs/hep-th/0012195))

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

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

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

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

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

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

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

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

Enhanced to a [[defect field theory]]:

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

#### Wrapped on orbifolds

* Pietro Ferrero, Jerome P. Gauntlett, Dario Martelli, James Sparks, *M5-branes wrapped on a spindle* ([arXiv:2105.13344](https://arxiv.org/abs/2105.13344))

### $\sigma$-Model description
 {#ReferencesSigmaModelDescription}

The [[Green-Schwarz action functional]]-type [[sigma-model]] of the (single) M5-brane of  was found in covariant form in

* {#PST97} [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], *Covariant Action for a $D=11$ Five-Brane with the Chiral Field*, Phys. Lett. B **398** (1997) 41 [[arXiv:hep-th/9701037](https://arxiv.org/abs/hep-th/9701037), doi:10.1016/S0370-2693(97)00188-3]

* {#BLNPST97} [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _Covariant Action for the Super-Five-Brane of M-Theory_, Phys. Rev. Lett. **78** (1997) 4332-4334 [[arXiv:hep-th/9701149](http://arxiv.org/abs/hep-th/9701149), [doi:10.1103/PhysRevLett.78.4332](https://doi.org/10.1103/PhysRevLett.78.4332)]

generally following

* [[Paul Townsend]], section 3.3 of: *D-branes from M-branes*, Phys. Lett. B **373** (1996) 68-75 [[arXiv:hep-th/9512062](https://arxiv.org/abs/hep-th/9512062), doi:10.1016/0370-2693(96)00104-9]

  > (which did not yet have the [[Hopf-Wess-Zumino term]])

and using the covariant mechanism for [[self-dual higher gauge fields]] from

* [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _On Lorentz Invariant Actions for Chiral P-Forms_, Phys.Rev. D55 (1997) 6292-6298 ([arXiv:hep-th/9611100](https://arxiv.org/abs/hep-th/9611100))

based on the non-covariant form of the [[self-dual higher gauge field|self-duality mechanism]] ([[Perry-Schwarz action]]) due to

* {#PerrySchwarz96} [[Malcolm Perry]], [[John Schwarz]], _Interacting Chiral Gauge Fields in Six Dimensions and Born-Infeld Theory_, Nucl. Phys. B489 (1997) 47-64 ([arXiv:hep-th/9611065](http://arxiv.org/abs/hep-th/9611065))

* {#Schwarz97} [[John Schwarz]], _Coupling a Self-Dual Tensor to Gravity in Six Dimensions_, Phys. Lett. B395:191-195, 1997 ([cds:317663](http://cds.cern.ch/record/317663), doi:10.1016/S0370-2693(97)00094-4)

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

Discussion of the equivalence of these superficially different action functionals is in

* [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], *On the equivalence of different formulations of the M Theory five--brane*, Phys. Lett. B **408** (1997) 135-141 [[arXiv:hep-th/9703127](http://arxiv.org/abs/hep-th/9703127), doi:10.1016/S0370-2693(97)00784-3]

The [[equations of motion]] in [[super spacetime]] were derived in

* {#HoweSezginWest97} [[Paul Howe]], [[Ergin Sezgin]], [[Peter West]], _Covariant Field Equations of the M Theory Five-Brane_, Phys. Lett. B **399** (1997) 49-59 [[arXiv:hep-th/9702008](https://arxiv.org/abs/hep-th/9702008), doi:10.1016/S0370-2693(97)00257-8]

and using the [[superembedding approach]] in

* {#HoweSezgin97} [[Paul Howe]], [[Ergin Sezgin]], _$D=11$, $p=5$_, Phys. Lett. B **394** (1997) 62-66 [[arXiv:hep-th/9611008](https://arxiv.org/abs/hep-th/9611008), doi:10.1016/S0370-2693(96)01672-3]

see

* {#Sorokin99} [[Dmitri Sorokin]], Section 5.2 of: _Superbranes and Superembeddings_, Phys. Rept. **329** (2000) 1-101 [[arXiv:hep-th/9906142](http://arxiv.org/abs/hep-th/9906142), doi:10.1016/S0370-1573(99)00104-0]

A variant adapted to a 3+3-dimensional split in

* Sheng-Lan Ko, [[Dmitri Sorokin]], [[Pichet Vanichchapongjaroen]], _The M5-brane action revisited_, JHEP11(2013)072 ([arXiv:1308.2231](https://arxiv.org/abs/1308.2231))

The computation of the small fluctuations of this GS-type sigma-model around a solution embedding as the asymptotic boundary of the [[AdS-spacetime]] [[near-horizon geometry]] of a black 5-brane as [above](#ReferencesAsBlackBrane), and the proof, to low order, that the result is the [[6d (2,0)-supersymmetric QFT]] appearing in [[AdS-CFT|AdS7-CFT6 duality]] is due to

* {#ClausKalloshProeyen97} P. Claus, [[Renata Kallosh]], [[Antoine Van Proeyen]], _M 5-brane and superconformal $(0,2)$ tensor multiplet in 6 dimensions_, Nucl.Phys. B518 (1998) 117-150 ([arXiv:hep-th/9711161](http://arxiv.org/abs/hep-th/9711161))

A review with emphasis on the coupling to the [[M2-brane]] is in

* [[Ergin Sezgin]], P. Sundell, _Aspects of the M5-Brane_ ([arXiv:hep-th/9902171](http://arxiv.org/abs/hep-th/9902171))

Further developments:

* [[Pichet Vanichchapongjaroen]], *Covariant M5-brane action with self-dual 3-form* ([arXiv:2011.14384](https://arxiv.org/abs/2011.14384))

* Sukŗti Bansal, *Manifestly Covariant Polynomial M5-brane Lagrangians* [[arXiv:2307.13449](https://arxiv.org/abs/2307.13449)]

[[!include M5-branes in the BMN matrix model -- references]]

### Worldvolume theory
 {#ReferencesWorldvolumeTheory}

The original article suggesting the description of the [[self-dual higher gauge theory]] on the 5-brane [[holographic principle|holographically]] by a dual [[higher dimensional Chern-Simons theory]] is

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

A precise mathematical formulation of the proposal made there is given in

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

A discussion that embeds this argument into the larger context of [[AdS-CFT duality]] is in

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

Discussion of [[S-duality]] in 6d self-dual higher gauge theory via [[non-commutative geometry|non-commutative]]-deformation:

* {#MathaiSati14} [[Varghese Mathai]], [[Hisham Sati]], _Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality_, J. Geom. Phys. 92:240-251, 2015 ([arXiv:1404.2257](https://arxiv.org/abs/1404.2257))

See also the references at _[[6d (2,0)-supersymmetric QFT]]_.

The [[double dimensional reduction]] to the [[D4-brane]] [[D=5 super Yang-Mills theory]] and the relation to [[Khovanov homology]] is discussed in

* {#Witten11} [[Edward Witten]], _Fivebranes and Knots_, Quantum Topology, Volume 3, Issue 1, 2012, pp. 1-137 ([arXiv:1101.3216](http://arxiv.org/abs/1101.3216), [doi:10.4171/QT/26](https://www.ems-ph.org/journals/show_abstract.php?issn=1663-487X&vol=3&iss=1&rank=1))

with further comments in

* Michele Nardelli, _On some equations concerning Fivebranes and Knots, Wilson Loops in Chern-Simons Theory, cusp anomaly and integrability from String theory. Mathematical connections with
some sectors of Number Theory_  (2011) ([pdf](http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/nardelli2011b.pdf))

A proposal for a construction as a [[higher gauge theory]] for [[differential string structure|string 2-connections]] is due to

* {#SaemannSchmidt17b} [[Christian Saemann]], [[Lennart Schmidt]], _Towards an M5-Brane Model I: A 6d Superconformal Field Theory_, J. Math. Phys. 59 (2018) 043502 [[arXiv:1712.06623](https://arxiv.org/abs/1712.06623), [doi:10.1063/1.5026545](https://doi.org/10.1063/1.5026545)]

based on

* {#SaemannSchmidt17a} [[Christian Saemann]], [[Lennart Schmidt]], _The Non-Abelian Self-Dual String and the (2,0)-Theory_ ([arXiv:1705.02353](https://arxiv.org/abs/1705.02353))

### Hopf-Wess-Zumino term
 {#ReferencesHopfWessZuminoTerm}

The [[higher dimensional WZW model|higher WZW term]] of the M5-brane ([[Hopf-Wess-Zumino term]]) was first proposed in

* {#Aharony94} [[Ofer Aharony]], p. 11 of _String theory dualities from M theory_, Nucl. Phys. B476:470-483, 1996 ([arXiv:hep-th/9604103](https://arxiv.org/abs/hep-th/9604103))

and had been settled by the time of

* [BLNPST 97 (1)](#BLNPST97).

The resemblence of the first summand of the term to the [[Whitehead integral formula]] for the [[Hopf invariant]] was noticed in

* {#Intriligator00} [[Kenneth Intriligator]], _Anomaly Matching and a Hopf-Wess-Zumino Term in 6d, N=(2,0) Field Theories_, Nucl.Phys. B581 (2000) 257-273 ([arXiv:hep-th/0001205](https://arxiv.org/abs/hep-th/0001205))

which hence introduced the terminology "Hopf-Wess-Zumino term". Followup to this terminology includes

* {#KalkkinenStelle02} [[Jussi Kalkkinen]], [[Kellogg Stelle]], Section 3.2 of: _Large Gauge Transformations in M-theory_, J. Geom. Phys. 48 (2003) 100-132 ([arXiv:hep-th/0212081](https://arxiv.org/abs/hep-th/0212081))

* Shan Hu, Dimitri Nanopoulos, _Hopf-Wess-Zumino term in the effective action of the 6d, (2, 0) field theory revisted_, JHEP 1110:054, 2011 ([arXiv:1110.0861](https://arxiv.org/abs/1110.0861))

* [[Alex Arvanitakis]], Section 4.1 of _Brane Wess-Zumino terms from AKSZ and exceptional generalised geometry as an $L_\infty$-algebroid_ ([arXiv:1804.07303](https://arxiv.org/abs/1804.07303))

More on the relation to the Hopf invariant in

* {#Sati13} [[Hisham Sati]], _Framed M-branes, corners, and topological invariants_, J. Math. Phys. 59 (2018), 062304 ([arXiv:1310.1060](http://arxiv.org/abs/1310.1060))

Discussion of the full 6d WZ term is in

* {#FSS19WZ} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization]]_

### Anomaly cancellation
 {#ReferencesAnomalyCancellation}

The original computation of the total M5-brane anomaly due to

* [Witten 96](#Witten96)

left a remnant term of $\tfrac{1}{24} p_2$. It was argued in

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

* {#HarveyMinasianMoore98b} [[Jeff Harvey]], [[Ruben Minasian]], [[Greg Moore]], _Non-abelian Tensor-multiplet Anomalies_, JHEP9809:004, 1998 ([arXiv:hep-th/9808060](https://arxiv.org/abs/hep-th/9808060))

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

that this term disappears (cancels) when properly taking into account the singularity of the [[supergravity C-field]] at the locus of the [[black brane|black]] [[M5-brane]].

This argument is recalled in

* {#Harvey05} [[Jeffrey Harvey]], Section 5 of : _TASI 2003 Lectures on Anomalies_ ([arXiv:hep-th/0509097](https://arxiv.org/abs/hep-th/0509097), [spire:692082](http://inspirehep.net/record/692082))

* {#BahBonettiMinasianNardoni18} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Class $\mathcal{S}$ Anomalies from M-theory Inflow_, Phys. Rev. D 99, 086020 (2019) ([arXiv:1812.04016](https://arxiv.org/abs/1812.04016))

* {#BBMN19} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Anomaly Inflow for M5-branes on Punctured Riemann Surfaces_,  J. High Energ. Phys. 2019, 123 (2019)
([arXiv:1904.07250](https://arxiv.org/abs/1904.07250))

The observation that the [[basic differential form|basic form]] component  $G_4^{M5}$ needs to be discussed is due to:

* {#FSS19v1} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], around equation (19) in: _Twisted Cohomotopy implies M-Theory anomaly cancellation_, preprint Version 1 [arXiv:1904.10207v1](https://arxiv.org/abs/1904.10207v1), later split into:

  1. [[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation on 8-manifolds|CMP 377(3) 2020]] (not containing discussion of M5-brane anomalies)

  1. [SS 20](#SS20) (focusing on the M5-brane anomaly cancellation)

* {#BBMN19b} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Anomalies of QFTs from M-theory and Holography_, J. High Energ. Phys. 2020, 125 (2020) ([arXiv:1910.04166](https://arxiv.org/abs/1910.04166))

* {#SS20} [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M5-brane anomaly cancellation]]_, Letters in Mathematical Physics **111** 120 (2021) ([arXiv:2002.07737](https://arxiv.org/abs/2002.07737), [doi:10.1007/s11005-021-01452-8](https://doi.org/10.1007/s11005-021-01452-8))

* {#BahBonettiMinasian20} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], _Discrete and higher-form symmetries in SCFTs from wrapped M5-branes_, J. High Energ. Phys. 2021, 196 (2021) ([arXiv:2007.15003](https://arxiv.org/abs/2007.15003))

### Double dimensional reduction to D4-brane

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

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

* [APPS 97a, Section 6](#APPS97a)

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

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

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

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

See also ([Witten 11](#Witten11)).

### Open M5-branes

Discussion of open M5-branes ending on [[M9-branes]] in a [[Yang monopole]] is in

* [[Eric Bergshoeff]], [[Gary Gibbons]], [[Paul Townsend]], _Open M5-branes_, Phys.Rev.Lett.97:231601 2006 ([arXiv:hep-th/0607193](http://arxiv.org/abs/hep-th/0607193))

### Nonabelian 2-form fields

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

* {#Witten07} [[Edward Witten]], _[[Conformal field theory in four and six dimensions]]_, in [[Ulrike Tillmann]], _Topology, Geometry and Quantum Field Theory: Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal_,  London Mathematical Society Lecture Note Series (2004) ([arXiv:0712.0157](http://arxiv.org/abs/0712.0157))

as well as sections 3 and 4 of

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

Proposals for how to implement this are for instance in

* [[Chong-Sun Chu]], _A Theory of Non-Abelian Tensor Gauge Field with Non-Abelian Gauge Symmetry $G \times G$_ ([arXiv:1108.5131](http://arxiv.org/abs/1108.5131))

* [[Henning Samtleben]], [[Ergin Sezgin]], Robert Wimmer, _(1,0) superconformal models in six dimensions_ ([arXiv:1108.4060](http://arxiv.org/abs/1108.4060))
 {#SSW11}

A formal proposal is [[schreiber:7d Chern-Simons theory and the 5-brane|here]].

### More on the holographic description

* [[Alexei Nurmagambetov]], I. Y. Park, _On the M5 and the AdS7/CFT6 Correspondence_ ([arXiv:hep-th/0110192](http://arxiv.org/abs/hep-th/0110192))

### More on the algebraic topology

* {#Witten2000} [[Edward Witten]], _Duality relations among topological effects in string theory_, J. High Energy Phys. 0005 (2000) 031 ([arXiv:hep-th/9912086](http://arxiv.org/abs/hep-th/9912086))

* {#Sati10} [[Hisham Sati]], _Geometric and topological structures related to M-branes II: Twisted String and String^c structures_ ([arXiv:1007.5419](http://arxiv.org/abs/1007.5419))

* {#Sati} [[Hisham Sati]], _Twisted topological structures related to M-branes II: Twisted $Wu$ and $Wu^c$ structures_ ([arXiv:1109.4461](http://arxiv.org/abs/1109.4461))

[[!redirects M5-branes]]

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

[[!redirects M5]]


M5-brane charge

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## Idea

Where [[D-brane charge]] is the [[charge]] in [[K-theory]] carried by [[D-branes]] and computed as the [[partition function]]/[[index]] of the "[[superparticles]]" on the D-brane being the [[boundary field theory]] of the [[open string|open]] [[superstring]] ending on them, so open [[M2-branes]] ending on [[M5-branes]] should induced a charge given by the [[partition function]] of their bounding [[self-dual strings]] ("M-strings"), which is now an [[elliptic genus]]. Hence _M5-brane charge_ should be in [[elliptic cohomology]] ([Sati 10](#Sati10)).

## Related concepts

* [[Cohomotopy charge]]

[[!include genera and partition functions - table]]

## References

That the [[elliptic genus]] of the [[self-dual string]] on the [[M5-brane]] should be the M5-brane charge in [[elliptic cohomology]] in higher analogy to [[D-brane charge]] in [[K-theory]] is due to

* {#Sati10} [[Hisham Sati]], _[[Geometric and topological structures related to M-branes]]_, part I, Proc. Symp. Pure Math. 81 (2010), 181-236 [arXiv:1001.5020](http://arxiv.org/abs/1001.5020),

Discussion of computation of the [[elliptic genus]]/[[Witten genus]] of the self-dual string on the M5-brane includes

* {#HoheneggerIqbal13} Stefan Hohenegger, Amer Iqbal, _M-strings, Elliptic Genera and N=4 String Amplitudes_ ([arXiv:1310.1325](http://arxiv.org/abs/1310.1325))

[[!redirects M5-brane charges]]


M5-brane elliptic genus

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## Idea

The _M5-brane elliptic genus_ is the [[elliptic genus]] imagined to arise as the [[M5-brane]] [[partition function]] when the [[M5-brane]] [[worldvolume]]-theory is seen as a [[2d SCFT]] after [[KK-compactification]] on a [[4-manifold]]/[[complex surface]] (as in the [[AGT correspondence]]).

## Related concepts

* [[Witten genus]]

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

* [[D=6 N=(1,0) SCFT]]

## References

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

[[!redirects M5-brane elliptic genera]]

[[!redirects M-string elliptic genus]]
[[!redirects M-string elliptic genera]]

[[!redirects E-string elliptic genus]]
[[!redirects E-string elliptic genera]]


M5-brane instanton

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## Idea

M5-brane [[instantons]] are constituted by [[M5-branes]] [[wrapped brane|wrapped]] on [[spacelike]] [[cycles]].

## Related concepts

* [[membrane instanton]]

[[!include table of branes]]

## References

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

M5-brane instantons [[wrapped brane|wrapping]] entire [[Calabi-Yau manifold]] fibers are discussed in

* [[Gottfried Curio]], Axel Krause, _S-Track Stabilization of Heterotic de Sitter Vacua_, Phys.Rev.D75:126003,2007 ([arXiv:hep-th/0606243](http://arxiv.org/abs/hep-th/0606243))

* Lilia Anguelova, Konstantinos Zoubos, _Five-brane Instantons vs Flux-induced Gauging of Isometries_ ([arXiv:hep-th/0606271](http://arxiv.org/abs/hep-th/0606271))

[[!redirects M5-brane instantons]]


M5-branes in the BMN matrix model -- references

### M5-branes in the BMN matrix model
 {#ReferencesM5BranesInBMNMatrixModel}

On [[light cone transversal]] [[M5-branes]] and [[M2-M5 brane bound states]] in the [[BMN matrix model]]:

The proposal of passing to horizontal/vertical limits of sizes of partitions is due to:

* {#MSJVR02} [[Juan Maldacena]], [[Mohammad Sheikh-Jabbari]], [[Mark Van Raamsdonk]], *Transverse Fivebranes in Matrix Theory*, JHEP 0301:038 (2003) $[$[arXiv:hep-th/0211139](https://arxiv.org/abs/hep-th/0211139), [doi:10.1088/1126-6708/2003/01/038](https://doi.org/10.1088/1126-6708/2003/01/038)$]$

A detailed check is in:

* {#AIST17b} [[Yuhma Asano]], [[Goro Ishiki]], [[Shinji Shimasaki]], [[Seiji Terashima]], _Spherical transverse M5-branes from the plane wave matrix model_, JHEP 02 (2018) 076 $[$[arXiv:1711.07681](https://arxiv.org/abs/1711.07681), doi:10.1007/JHEP02(2018)076$]$

Review in:

* {#AIST17a} [[Yuhma Asano]], [[Goro Ishiki]], [[Shinji Shimasaki]], [[Seiji Terashima]], _On the transverse M5-branes in matrix theory_, Phys. Rev. D **96** 126003 (2017) $[$[arXiv:1701.07140](https://arxiv.org/abs/1701.07140), [doi:10.1103/PhysRevD.96.126003](https://doi.org/10.1103/PhysRevD.96.126003)$]$

In a limit where aspects of [[little string theory]] on [[NS5-branes]] becomes visible:

* [[Hai Lin]], [[Juan Maldacena]], _Fivebranes from gauge theory_, Phys. Rev. D **74** 084014 (2006) $[$[arXiv:hep-th/0509235](https://arxiv.org/abs/hep-th/0509235), [doi:10.1103/PhysRevD.74.084014](https://doi.org/10.1103/PhysRevD.74.084014)$]$

* Henry Ling, Ali Reza Mohazab, Hsien-Hang Shieh, Greg van Anders, [[Mark Van Raamsdonk]], *Little String Theory from a Double-Scaled Matrix Model*, JHEP 0610:018 (2006) $[$[arXiv:hep-th/0606014](https://arxiv.org/abs/hep-th/0606014), [doi:10.1088/1126-6708/2006/10/018](https://doi.org/10.1088/1126-6708/2006/10/018)$]$

* [[Yuhma Asano]], [[Goro Ishiki]], [[Takaki Matsumoto]], [[Shinji Shimasaki]], [[Hiromasa Watanabe]], *On the existence of the NS5-brane limit of the plane wave matrix model*, PTEP 2023 (2023) 4, 043B01 $[$[arXiv:2211.13716](https://arxiv.org/abs/2211.13716), [doi:10.1093/ptep/ptad042](https://doi.org/10.1093/ptep/ptad042)$]$


M5-MO9 brane bound state

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## Idea

The [[brane intersection]] of [[M5-branes]] with [[MO9-planes]] -- the "half M5-brane".

Equivalently:

* under [[duality between M-theory and heterotic string theory]]: [[black brane|black]] [[NS5-brane]] [[solitons]] in [[heterotic string theory]].

* under [[duality between M-theory and type IIA string theory]]:

  [[D4-D8-brane bound states]] or [half NS5-brane](NS5-brane#NSHalfBranes) in [[type I' string theory]]

> graphics grabbed from [HSS18, Example 2.2.7](#HSS18)

> graphics grabbed from [SS 19, Section 4](#SatiSchreiber19)

## In string theory

### As Yang-Mills instantons in heterotic string theory
 {#AsYangMillsInstantons}

In ([[dual heterotic string theory|dual]]) [[heterotic string theory]] [[wrapped brane|wrapped]] [[NS5-branes]] are equivalently [[Yang-Mills instantons]] in the transversal [[Yang-Mills theory]]  ([Strominger 90](#Strominger90), [Witten 96, p. 10](#Witten96)) --
as is generally the case for  [[Dp-D(p+4)-brane bound states]]  (see [there](DpDpPlus4BraneBoundState#AsYangMillsInstantons)).

### As half NS5-branes in type I'

(For more see at [half NS5-brane](NS5-brane#NSHalfBranes).)

If a [[black brane|black]] [[NS5-brane]] sits at an [[O8-plane]], hence at the [[orientifold]] [[fixed point]]-locus, then in the ordinary $\mathbb{Z}/2$-[[quotient]] it appears as a "[[half-brane]]" with only one copy of [[D6-branes]] ending on it:

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

(In [Hanany-Zaffaroni 99](NS5-brane##HananyZaffaroni99) this is interpreted in terms of the [['t Hooft-Polyakov monopole]].)

{#LiftToMTheortyOfNS5Halfbrane} The lift to [[M-theory]] of this situation is an [[M5-brane]] [[brane intersection|intersecting]] an [[M9-brane]] (see at _[[MO5-plane]]_ and at _[[heterotic M-theory on ADE-orbifolds]]_):

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

Alternatively the [[O8-plane]] may [[brane intersection|intersect]] the [[black brane|black]] [[D6-branes]] away from the [[black brane|black]] [[NS5-brane]]:

> from [HKLY 15](NS5-brane##HKLY15)

In general, some of the NS5 sit away from the [[O8-plane]], while some sit on top of it:

> from [Hanany-Zaffaroni 98](NS5-brane##HananyZaffaroni98)

> graphics grabbed from [Apruzzi-Fazzi 17, p. 18](NS5-brane##ApruzziFazzi17)

## Related concepts

* [[heterotic M-theory on ADE-orbifolds]]

* [[E-string]]

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

## References

### As heterotic Yang-Mills instantons

Discussion as [[Yang-Mills instantons]] ("small instantons"):

* {#Strominger90} [[Andrew Strominger]], _Heterotic solitons_, Nucl. Phys. B343 (1990) 167-184 (doi:10.1016/0550-3213(90)90599-9) Erratum: Nucl. Phys. B353 (1991) 565-565 (doi:10.1016/0550-3213(91)90349-3) ([spire:27900](http://inspirehep.net/record/27900))

* {#Witten96} [[Edward Witten]], _Small Instantons in String Theory_,  	Nucl.Phys.B460:541-559, 1996  (doi:10.1016/0550-3213(95)00625-7)

### As half M5-branes

* Fabio Apruzzi, [[Marco Fazzi]], Section 2.1.2 of: _AdS7/CFT6 with orientifolds_, JHEP 01 (2018) 124 ([arXiv:1712.03235](https://arxiv.org/abs/1712.03235))

* {#HSS18} [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], Example 2.2.7 of: _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_, CMP (2019) ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987), [doi:10.1007/s00220-019-03442-3](http://link.springer.com/article/10.1007/s00220-019-03442-3))

* {#FSS19c} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 4 of: _[[schreiber:Super-exceptional embedding construction of the M5-brane|Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5]]_ ([arXiv:1908.00042](https://arxiv.org/abs/1908.00042))

* {#SatiSchreiber19} [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], Section 4.1 of: _[[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation]]_ ([arXiv:1909.12277](https://arxiv.org/abs/1909.12277))

[[!redirects M5-MO9 brane bound states]]

[[!redirects half M5-brane]]
[[!redirects half M5-branes]]

[[!redirects heterotic M5-brane]]
[[!redirects heterotic M5-branes]]


M9-brane

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

#Contents#
* table of contents
{:toc}

## Idea

The $\mathbb{Z}_2$-[[orbifold]]/[[higher orientifold]] [[fixed point]] in [[Hořava-Witten theory]] might be called it _[[O-plane|O9-plane]]_ but is often called the _M9-brane_ (e.g. [GKSTY 02, section 3](#GKSTY02), [Moore-Peradze-Saulina 04](#MoorePeradzeSaulina04)), even though it is on a different conceptual footing than the genuine [[M2-brane]] and [[M5-brane]].

In ([Hull 97, pages 8-9](#Hull97)) the M9-brane was argued to be the object whose charge is the [[Poincaré duality|Poincaré dual]] to the time-component of the [[M2-brane]] charge as it appears in the [[M-theory super Lie algebra]] via

$$
  \wedge^2 (\mathbb{R}^{10,1})^\ast
    \simeq
  \wedge^2 (\mathbb{R}^{10})^\ast \oplus \wedge^{9} \mathbb{R}^{10}
  \,.
$$

(In the same way the time component of the [[M5-brane]] charge is argued to be dual to the charge of the [[KK-monopole]], see there.)

Under the [[duality in string theory|duality]] between [[M-theory]] and [[type IIA string theory]] the M9-brane may be identified with the [[O8-plane]]:

> from [GKSTY 02, section 3](#GKSTY02)

This may be used to understand the [[gauge enhancement]] to [[E8]]-[[gauge groups]] at the [[heterotic string theory|heterotic]] boundary of [[Horava-Witten theory]]:

[[ViaO8GaugeEnhancementOnM9.jpg:file]]

> from [GKSTY 02, section 3](#GKSTY02)

## Related concepts

* [[ninebrane group]], [[ninebrane structure]]

* [[MO5]]

* [[KK monopole]]

[[!include table of branes]]

## References

Early discussion of the possibility of a kind of 9-brane in [[11d supergravity]]:

* [[Mike J. Duff]], p. 28 of: *Supermembranes*, TASI lectures (1996) [[arXiv:hep-th/9611203](https://arxiv.org/abs/hep-th/9611203)]

* [[Paul S. Howe]], [[Ergin Sezgin]], p. 9 in: *Superbranes*, Phys. Lett. B **390** (1997) 133-142 [[arXiv:hep-th/9607227](https://arxiv.org/abs/hep-th/9607227), doi:10.1016/S0370-2693(96)01416-5]

* [[Mees de Roo]], p. 5 of: *Intersecting branes and Supersymmetry*, in: *Supersymmetry and Quantum Field Theory*, Proceedings of the D. Volkov Memorial Seminar Held in Kharkov, Ukraine, 5–7 January 1997, Springer (1998) [[arXiv:hep-th/9703124](https://arxiv.org/abs/hep-th/9703124), [doi:10.1007/BFb0105225](https://doi.org/10.1007/BFb0105225)]

* {#Hull97} [[Chris Hull]], _Gravitational Duality, Branes and Charges_, Nucl. Phys. B **509** (1998) 216-251 [[arXiv:hep-th/9705162](http://arxiv.org/abs/hep-th/9705162)]

On M9-branes (and apparently introducing that terminology) as lifts of the [[D8-branes]] in [[massive type IIA string theory]]:

* [[Eric Bergshoeff]], [[Yolanda Lozano]], [[Tomas Ortin]], (4.8) in: *Massive Branes*, Nucl. Phys. B **518** (1998) 363-423 [[arXiv:hep-th/9712115](https://arxiv.org/abs/hep-th/9712115), doi:10.1016/S0550-3213(98)00045-5]

* [[Eric Bergshoeff]], [[Jan Pieter van der Schaar]], *On M-9-branes*, Class. Quant. Grav. **16** (1999) 23-39 [[arXiv:hep-th/9806069](http://arxiv.org/abs/hep-th/9806069), [doi:10.1088/0264-9381/16/1/002](https://doi.org/10.1088/0264-9381/16/1/002)]

Further discussion:

* M. P. Garcia del Moral, P. Leon, A. Restuccia, *Wordsheet description of a* massive *type IIA superstring in 10D* [[arXiv:2306.16620](https://arxiv.org/abs/2306.16620)]

A proposal for a description of the M9 as an [[schreiber:infinity-Wess-Zumino-Witten theory|higher WZW theory]] is in

* Takeshi Sato, _A 10-form Gauge Potential and an M-9-brane Wess-Zumino Action in Massive 11D Theory_, Phys. Lett. B477 (2000) 457-468 ([arXiv:hep-th/9912030](http://arxiv.org/abs/hep-th/9912030))

Discussion of how the [[M2-brane]] and the [[M5-brane]] may arise from this by [[tachyon condensation]] is in

* [[Laurent Houart]], [[Yolanda Lozano]], _Brane Descent Relations in M-theory_, Phys.Lett. B479 (2000) 299-307 ([arXiv:hep-th/0001170](http://arxiv.org/abs/hep-th/0001170))

Discussion of the M9 as the dual in [[Horava-Witten theory]] of [[O8-planes]] in [[type II string theory]] is in

* {#GKSTY02} E. Gorbatov, V.S. Kaplunovsky, J. Sonnenschein, [[Stefan Theisen]], S. Yankielowicz, section 3 of _On Heterotic Orbifolds, M Theory and Type I' Brane Engineering_, JHEP 0205:015, 2002 ([arXiv:hep-th/0108135](https://arxiv.org/abs/hep-th/0108135))

see also at _[Intersection of D6s with O8s](intersecting+D-brane+model#IntersectionOfD6WithO8)_

Cohomological discussion of [[ninebrane structures]] is in

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

See also

* [[Paul Howe]], A. Kaya, [[Ergin Sezgin]], P. Sundell, _Codimension One Branes_ ([arXiv:hep-th/0001169](http://arxiv.org/abs/hep-th/0001169))

* {#MoorePeradzeSaulina04} [[Gregory Moore]], Grigor Peradze, [[Natalia Saulina]], _Instabilities in heterotic M-theory induced by open membrane instantons_, Nucl.Phys. B607 (2001) 117-154 ([arXiv:hep-th/0012104](http://arxiv.org/abs/hep-th/0012104))

* Takeshi Sato, _On M-9-branes and their dimensional reductions_, Nucl. Phys. Proc. Suppl. 102 (2001) 107-112 ([arXiv:hep-th/0102084](http://arxiv.org/abs/hep-th/0102084))

Discussion of open [[M5-branes]] ending on the [[M9-brane]] is in

* [[Eric Bergshoeff]], [[Gary Gibbons]], [[Paul Townsend]], _Open M5-branes_, Phys. Rev. Lett. 97:231601 2006 ([arXiv:hep-th/0607193](http://arxiv.org/abs/hep-th/0607193), [doi:10.1103/PhysRevLett.97.231601](https://doi.org/10.1103/PhysRevLett.97.231601))

Discussion as [[exotic branes]] in [[exceptional field theory]]:

* {#ObersPioline99} [[Niels Obers]], [[Boris Pioline]], *U-duality and M-Theory*, Phys. Rept. **318** (1999) 113-225 [[arXiv:hep-th/9809039](https://arxiv.org/abs/hep-th/9809039), doi:10.1016/S0370-1573(99)00004-6]

> p. 109: "conjectured M9-brane which should more properly be called M8-brane."

* {#BermanMusaevOtsuki18} [[David S. Berman]], [[Edvard T. Musaev]], [[Ray Otsuki]], *Exotic Branes in Exceptional Field Theory: $E_{7(7)}$ and Beyond*,  J. High Energ. Phys. **2018** 53 (2018) [[arXiv:1806.00430](https://arxiv.org/abs/1806.00430), doi:10.1007/JHEP12(2018)053]

  > [Berman, Musaev & Otsuki (2018), p. 65](#BermanMusaevOtsuki18): "However, as remarked in [1](#ObersPioline99), [the M9 brane] should more properly be called an M8-brane or perhaps KK8 following its mass formula designation 8(1,0). It is, perhaps, to be understood as an object that exists only as a lift of the D8-brane of Type IIA."

[[!redirects M9-branes]]

[[!redirects O9 plane]]
[[!redirects O9 planes]]

[[!redirects O9-plane]]
[[!redirects O9-planes]]

[[!redirects MO9]]
[[!redirects MO9-brane]]
[[!redirects MO9-branes]]

[[!redirects MO9-plane]]
[[!redirects MO9-planes]]

[[!redirects M8-brane]]
[[!redirects M8-branes]]
[[!redirects M8 brane]]
[[!redirects M8 branes]]


Maarten Bergvelt

* [website](http://www.math.uiuc.edu/~bergv/)

category: people

[[!redirects MaartenBergvelt]]


Maarten de Rijke

* [Research group page](https://staff.fnwi.uva.nl/m.derijke/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Maarten_de_Rijke)

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

## Selected writings

On [[modal logic]]:

* {#BlackburnDeRijkeVenema01} [[Patrick Blackburn]], [[Maarten de Rijke]], [[Yde Venema]], *Modal Logic*, Cambridge Tracts in Theoretical Computer Science **53** (2001) [[doi:10.1017/CBO9781107050884](https://doi.org/10.1017/CBO9781107050884)]

category: people


Maarten Solleveld

* [webpage](https://www.math.ru.nl/~solleveld/)

## Related $n$Lab entries

* [[Lie algebra cohomology]]

category:  people


Maarten van de Meent

* [webpage](https://www.aei.mpg.de/person/52146/1589487)

## related $n$Lab entries

* [[piecewise flat spacetime]]

category: people


Maarten Van den Nest

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

## Selected writings

On [[measurement-based quantum computation]]:

* [[Hans J. Briegel]], [[Dan E. Browne]], [[Wolfgang Dür]], [[Robert Raussendorf]], [[Maarten Van den Nest]], *Measurement-based quantum computation*, Nature Physics **5** 1 (2009) 19-26 [[arXiv:0910.1116](https://arxiv.org/abs/0910.1116), [doi:10.1038/nphys1157](https://doi.org/10.1038/nphys1157)]

category: people


Mac Lane's proof of the coherence theorem for monoidal categories

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* the following line creates the automatic table of contents
{: toc}

+-- {: .standout}
Throughout, we will use the notation and terminology of Ch. VII of [CWM](#CWM), with the exception that the product bifunctor of a
monoidal category will be denoted by $\otimes$, instead of Mac Lane's
$\square$ notation.
=--

## Introduction and statement ##

Let $\langle B, \otimes_B, \alpha^{(B)}, e_B, \lambda^{(B)}, \varrho^{(B)}, \rangle$ be some [[monoidal category]].  At first, we might like the _[[coherence theorem for monoidal categories]]_ to state that
_every_ [[diagram]] in $B$ built up from instances of $\alpha^{(B)}$,
$\lambda^{(B)}$, $\varrho^{(B)}$ and identity arrows by multiplications
$\otimes_B$ (such as the pentagon diagram) commutes. This, however, is not possible, because some formally different vertices of such a diagram
might turn out to be the same in our particular
monoidal category $B$, in a way that makes the diagram
non-commutative.

As a concrete example of the problem in a particular
monoidal category, consider Isbell's famous example for showing that
moving to skeleta does not assure that all monoidal categories are
strict (see p. 164 of [CWM](#CWM)). In this example, one shows that if
$D$ is the denumerable set in a skeleton $\mathbf{Set}_0$ of $\mathbf{Set}$, then,
although $D\times(D\times D)$ equals $(D\times D)\times D$, the
associator $\alpha_{D,D,D}$ (defined as usual to commute with the
projections) cannot be the identity.

So, in order for coherence to work, we must change it to the statement
that every "formal" diagram commutes.
The formal diagrams that commute will be described through certain
graphs $\mathcal{G}'_n$ ($n\in \mathbb{N}$).  The vertices of
$\mathcal{G}'_n$ will be _binary words_ of _length_ $n$, and its
arrows will represent the sort of arrows that participate in the
coherence theorem.

We begin by defining the set $W$ of binary words inductively, as the
set of all terms in one variable (denoted by "$-$") in the signature
$\{e_0, \otimes\}$, where $e_0$ is an 0-ary function symbol (a
constant), and $\otimes$ is a binary function symbol.  So, $W$ is the
smallest set of strings in the alphabet
$\{e_0,\otimes,-\}$ satisfying  $-\in W$, $e_0\in W$, and if $u,v\in
W$ then $\otimes u v$ (usually written as $(u) \otimes (v)$ or even
$u\otimes v$ for convenience) is also in $W$.

Next, we define a set $S'$ of certain arrow symbols as
follows (the reason for the tag in $S'$ will be clear later).
Let $S'$ be the smallest set of strings in the alphabet
$W\cup\{1,\alpha,\alpha^{-1},\lambda,\lambda^{-1},\varrho,\varrho^{-1},\displaystyle{\hat{\otimes}\}}$
satisfying $\alpha u v w, \alpha^{-1} u v w\in S'$ for all
$u,v,w\in W$, $\lambda w, \lambda^{-1} w, \varrho w,
\varrho^{-1} w \in S'$ for all $w\in W$, and for all $w\in W$, if
$\beta\in S'$ then also $(\beta)\hat{\otimes} (1w), (1w)\hat{\otimes} (\beta)\in
S'$ (more precisely, the last terms should be called,
$\hat{\otimes} \beta 1 w$ and $\hat{\otimes} 1 w \beta$, but as before, we will
usually use the more convenient infix notation). Note that
elements of $S'$ are comprised of a single associator/unitor symbol
(or their inverses) tensored with one or more strings of the form
$1w$.  Later, when we will
interpret the elements of $S'$
as arrows of a particular monoidal category $B$, we will call them _basic arrows_.
The idea is that every arrow of the "right" kind in
$B$ (the kind considered in the coherence theorem) factors as the
composition of such basic arrows (because $\otimes$ is a bifunctor).

It can be verified that every string of
$S'$ is of one and only one of the following forms:
$\alpha u v w$ for some $u, v, w\in W$, $\alpha^{-1} u v w$ for
some $u, v, w\in W$, $\lambda w$ for some
$w\in W$, ..., $\beta \hat{\otimes} 1w$ for some $w\in W$ and
$\beta \in S'$. Also, in each case the "arguments" are determined uniquely
(for example, both $\beta$ and $w$ are determined from $\beta \hat{\otimes}
1w$), and so we may define functions on $S'$ inductively.

Let $\mathcal{G}'$ be the directed graph (quiver in $n$Lab jargon)
with $W$ as the set of objects, $S'$ as the set of arrows, and with
the obviously defined $dom$ and $cod$ (for example, For $u,v,w \in
W$, $dom(\alpha u v w) := u\otimes (v\otimes w) =:cod(\alpha^{-1}
u v w)$, $dom(\beta \hat{\otimes} 1w) := dom(\beta)\otimes w$, etc.

Define inductively the length $len(w)$ of a binary
word $w$ as follows:

$$
len(e_0):=0,\quad len(-):=1,\quad len(u\otimes v):= len(u) + len(v).
$$

Since for all $s\in S'$, $len(dom(s))=len(cod(s))$ (this can be
verified by induction), $\mathcal{G}'$ is the
union of disjoint graphs $\mathcal{G}'_n$, where for $n\in \mathbb{N}$, the
vertices of $\mathcal{G}'_n$ are the binary words of length $n$, and the
edges are all $s\in S'$ with $len(dom(s)) = n$.

Now, let us fix some monoidal category $\langle B, \otimes_B, \alpha^{(B)}, e_B, \lambda^{(B)}, \varrho^{(B)}, \rangle$, and some object
$b\in obj(B)$.  We can interpret vertices and edges of $\mathcal{G}'$
in $B$ by defining inductively functions $(\cdot)_b: W\to obj(B)$,
$\mathcal{I}:S'\to arr(B)$
as follows:

$$
(e_0)_b:=e_B,\text{ } (-)_b:=b, (u\otimes v)_b:= u_b\otimes_B v_b,
$$

$$
\begin{aligned}
&&\mathcal{I}(\alpha u v w) := \alpha^{(B)}_{u_b,v_b,w_b},\quad \mathcal{I}(\alpha^{-1}
u v w):= (\alpha^{(B)}_{u_b,v_b,w_b})^{-1},\\
&& \mathcal{I}(\lambda u) := \lambda^{(B)}_{u_b}, \quad \mathcal{I}(\lambda^{-1} u) :=
(\lambda^{(B)}_{u_b})^{-1},\\
&& \mathcal{I}(\varrho u) := \varrho^{(B)}_{u_b}, \quad \mathcal{I}(\varrho^{-1} u) :=
(\varrho^{(B)}_{u_b})^{-1},\\
&& \mathcal{I}(\beta\hat{\otimes} 1w) := \mathcal{I}(\beta)\otimes_B 1_{w_b},\quad
\mathcal{I}(1w\hat{\otimes} \beta) := 1_{w_b}\otimes_B\mathcal{I}(\beta)
\end{aligned}
$$

(for $u,v,w\in W$ and $\beta\in S'$).

It is straightforward to verify that the pair $\langle (\cdot)_b,\mathcal{I}\rangle$
is a morphism of graphs $\mathcal{G}'\to U B$ (where $UB$ is the
underlying graph of $B$), that is, that for all $s\in S'$, we have

$$
(dom(s))_b = dom_B(\mathcal{I}(s)),\quad (cod(s))_b = cod_B(\mathcal{I}(s)).
$$

Therefore, if $s_1,\ldots,s_m$ form a path in $\mathcal{G}'$ (i.e.,
$dom(s_{i+1})=cod(s_i)$ for all $i$),
then $\mathcal{I}(s_1),\ldots,\mathcal{I}(s_m)$ (in that order) are composable
arrows in $B$.  We can finally state the coherence theorem:

+-- {: .num_theorem #CohA}
###### Theorem
**(Mac Lane's Coherence Theorem, version 1)**. Let $v,w\in W$ be two binary words of the same length $n$.  Suppose
that $s_1,\ldots,s_m$ and $s'_1,\ldots s'_{m'}$ form two paths from $v$
to $w$ in $\mathcal{G}'_n$.  Then the two corresponding sequences of
arrows in $B$ have the same composition in $B$:

$$
\mathcal{I}(s_m)\circ \cdots\circ \mathcal{I}(s_1) = \mathcal{I}(s'_{m'})\circ
\cdots\circ \mathcal{I}(s'_1)
$$
=--

Note that at a first glance, this "single variable coherence" looks
quite limited:  We only prove commutativity of diagrams in $B$ whose
vertices are obtained by tensoring a single element $b$ (e.g.,
vertices such as $b\otimes_B ((b\otimes_B b)\otimes_B b)$).  However, it
turns out that the general case follows almost immediately from this
case.  The trick is to construct an appropriate monoidal category
$It(B)$ in a way that the coherence in one variable for $It(B)$ and
one special object of $It(B)$ is the general coherence for $B$.

The coherence theorem  in [CWM](#CWM) (Theorem VII.2.1) is stated in a
different way, but the main bulk of its proof is actually devoted to
prove the above coherence theorem. To state also the version of Theorem
VII.2.1 of [CWM](#CWM), let $\mathcal{W}$ be the category whose objects are
binary words, with a single arrow $u\to v$ between any two binary
words of the same length and with the obvious composition.  The
monoidal structure on $\mathcal{W}$ is the obvious one: The unit is $e_0$, and
the components of the associator and the unitors are each the
appropriate single arrow of the relevant hom-set.  The coherence
theorem in [CWM](#CWM) is as follows:

+-- {: .num_theorem #CohB}
###### Theorem
**(Mac Lane's Coherence Theorem, version 2)**. For any monoidal category $B$ and any object $b$ of $B$, there is a
unique strict morphism of monoidal categories $\mathcal{W}\to B$ with $
(-)\mapsto b$.
 =--

Version 2 says that $\mathcal{W}$ is the free monoidal category on one
generator $(-)$ (that is, that it is the object part of a universal
arrow from $(-)$ to the forgetful (object) functor $\mathbf{Moncat}_s \to \mathbf{Set}$, where
$\mathbf{Moncat}_s$ stands for monoidal categories with strict morphisms of
monoidal categories) . As we shall see later, this version follows
almost immediately  from version 1.  While this is not clear at a
first glance, it is also
true that version 1 follows from version 2. For this, we will define a
monoidal category $F[1]$ that will be almost evidently free (and
hence, given version 2, isomorphic to $\mathcal{W}$).

## Coherence only for the associator ##

It will be simpler to first prove coherence only for the
associator, forgetting about the unit and the unitors.
Let $\mathcal{G}$ be the full subgraph of $\mathcal{G}'$ whose vertex set
consists only of those binary words that do not involve $e_0$.  We
will write $W_0$ for the subset of $W$ consisting of such binary
words (that is, for the vertex set of $\mathcal{G}$).

Let $S$ be the subset of $S'$ consisting of the strings that
involve only instances of $\alpha$ and $\alpha^{-1}$ (but not
$\lambda$, $\lambda^{-1}$, $\varrho$, and $\varrho^{-1}$) and also do _not_ involve $e_0$
(e.g., as in $\alpha e_0 v w$).  Actually, $S$
could be defined inductively, similarly to $S'$, omitting the reference
to $\lambda$, $\lambda^{-1}$, $\varrho$, and $\varrho^{-1}$, and using $W_0$
instead of $W$.  Note that $S$ is the set of arrows of $\mathcal{G}$:
$S=\{s\in S'|dom(s),cod(s)\in W_0\}$. For simplicity, we will use
the same names $dom,cod$ for the restriction of $dom, cod$ to
$S$.

Let $\mathcal{G}_n$ be the full subgraph of $\mathcal{G}$ on the vertices
of length $n$. The proof will use a certain vertex $w^{(n)}$ of $\mathcal{G}_n$ as
a "pivot."  Specifically, we let $w^{(n)}\in W_0$ be the unique word of
length $n$ in which all pairs of parentheses start in front (e.g.,
$(-\otimes -)\otimes -$ for $n=3$, $((-\otimes -)\otimes -)\otimes -$
for $n=4$, and $(((-\otimes-)\otimes - )\otimes -)\otimes -$ for
$n=5$; a precise inductive definition is $w^{(1)}=-$, $w^{(n)}=
w^{(n-1)}\otimes -$ for $n\geq 1$).

The proof will also use a measure $\rho$ of how "far" is a word
$w\in W_0$ from being $w^{(n)}$. So, we define $\rho$ inductively by

$$
\rho(-) = 0, \quad \rho(v\otimes w) = \rho(v)+\rho(w)+len(w)-1.
$$

It is easily verified that $\rho(u)\geq 0$ for all $u\in W_0$, with
equality iff $u = w^{(n)}$.  In fact,  $\rho(u)$ is the
length of the longest directed path (see ahead for what "directed"
means here) from $u$ to $w^{(n)}$. (Intuitively, the idea is
that if $u=v\otimes w$, then we may first handle $v$, then handle $w$, in
order to reach $w^{(n_1)}\otimes w^{(n_2)}$ (with $n_1:=len(v)$ and
$n_2:=len(w)$) but not yet to $w^{(n)}$. Then, by $n_2-1$ additional
applications of associators, we finally reach $w^{(n)}$.)

We will call any $s\in S$ that involves $\alpha$ a _directed arrow_, and any $s\in S$
that involves $\alpha^{-1}$ an _anti-directed arrow_.  A more precise inductive definition is this:
For all $u,v,w\in W_0$, $\alpha u v w$ is directed, and if $\beta$ is
directed, then $\beta\hat{\otimes} 1 w$ and $1 w\hat{\otimes} \beta$ are
directed for all $w\in W_0$. The definition of anti-directed is
obtained by replacing $\alpha$ by $\alpha^{-1}$ and "directed" by
"anti-directed" in the previous definition. It can be shown by
induction that if $s\in S$ is directed, then $\rho(cod(s))$ is
strictly smaller that $\rho(dom(s))$.

For future reference, define the "formal inverse" of $s'\in S'$ in the
obvious inductive way, by setting, for example,  $inv(\alpha uvw): =
\alpha^{-1}_{u,v,w}$, $inv(\alpha^{-1} uvw):=
\alpha uvw$ $inv(\beta\hat{\otimes} 1w):=
inv(\beta) \hat{\otimes} 1w$, etc.  It can be verified that the formal
inverse has all expected properties (such as being a bijection of arrows,
switching domain and codomain, taking an anti-directed arrow to a
directed one, and being interpreted as the inverse of the
interpretation of the original arrow).

From each $w\in W_0$ of length $n$, there is a
"canonical" path, involving only directed arrows, from
$w$ to $w^{(n)}$ in $\mathcal{G}_n$, obtained by
successively moving the outermost parentheses to the left with the appropriate
$\alpha$.  For an example, consider the
paths $\to\to$ and $\to\uparrow$ in the pentagon (5) on p. 162 of
[CWM](#CWM). Note that if $v,w\in W_0$ are both of length $n$, then
there is a path in $\mathcal{G}_n$ from $v$ to $w$:  Take the canonical
path from $v$ to $w^{(n)}$, and then extend it by reversing (using
$inv$) every edge  of the canonical path from $w$ to
$w^{(n)}$.

Given $v,w\in W_0$ of the same length $n$, our task is to show that
for any path from $v$ to $w$ in $\mathcal{G}_n$, applying $\mathcal{I}$ to
the arrows of the path and composing in $B$ results in the same arrow
$v_b\to w_b$ in $B$, independently of the choice of path.  In
other words, we would like to prove the theorem obtained from
replacing $W$ by $W_0$ and $\mathcal{G}'_n$ by $\mathcal{G}_n$ in the first
version of the coherence theorem.

As a preparation, we start with the following degenerate case, that
can be proved (as usual) by induction on $s$ and $s'$.

+-- {: .un_prop}
###### Proposition
There are no parallel directed arrows in $\mathcal{G}_n$:  If $s,s'\in S$
are directed arrows with $dom(s) = dom(s')$ and $cod(s) =
cod(s')$, then $s=s'$.
=--

Now, take some path $s_1,\ldots,s_m$ from $v$ to $w$, as in

[[Coh_fig_1.gif:pic]]

In $B$, we have the corresponding path (note that both in $\mathcal{G}_n$
and in $B$, it is possible that some of the vertices are in fact equal)

[[Coh_fig_2.gif:pic]]

In our original path, it is possible that some arrows (some $s_i$'s)
are anti-directed.  We can replace each anti-directed arrow by its formal
inverse so that we are left only with directed arrows.
For example, if $s_2$ and $s_3$ are the only
anti-directed arrows in our path, then after formally inverting them we
get the following series of arrows in $\mathcal{G}_n$:

[[Coh_fig_3.gif:pic]]

In $B$, we get

[[Coh_fig_4.gif:pic]]

Let us write $can(u)$ for the canonical path from $u\in W_0$ with
$len(u)=n$ and with $u \neq w^{(n)}$ to $w^{(n)}$.  Let
us also write $\mathcal{I}(can(u))$ for the arrow of $B$ obtained by
composing $\mathcal{I}$ of all arrows of $can(u)$.  Suppose that all
squares in the following diagram of $B$ commute:

[[Coh_fig_5.gif:pic]]

(where we write $w_b^{(n)}$ for $[w^{(n)}]_b$).
This would imply that all squares in

[[Coh_fig_6.gif:pic]]

commute, which also implies that the outer rectangle commutes, that
is:

$$
\mathcal{I}(s_m)\circ \cdots \circ\mathcal{I}(s_1) = \mathcal{I}(can(w))^{-1}\circ \mathcal{I}(can(v)).
$$

The right-hand side  depends only on $w$
and $v$ (and not on the particular choice of the path $s_1,\ldots
s_m$), which is exactly what we would like to prove.  Hence, it is
sufficient to show that all squares in the last but one diagram commute.  For
this, it is clearly sufficient to prove the following
proposition:

+-- {: .un_prop}
###### Proposition
Let $v\in W_0$ be a word of length $n$, different from $w^{(n)}$.  Then
applying $\mathcal{I}$ to
any two directed paths (paths consisting only of directed arrows) from
$v$ to $w^{(n)}$ results in two paths in $B$ that have the same
composition in $B$.
=--

+-- {: .proof}
The proof is by induction on $\rho(v)$. For the basis, if $\rho(v)=0$
then $v = w^{(n)}$ and the assertion is trivial. Suppose, therefore, that
$\rho(v)$>$0$ and the assertion is
true for all $v'\in W_0$ with $\rho(v')$<$\rho(v)$.  For reasons that
will be understood later, we will prove this part by one additional
induction: on the length $len(v)$.

Hence, we have to prove that if the assertion holds for all $v'$ of
rank smaller than $r := \rho(v)$ and for all $v'$ of rank $r$ and length
smaller than $n$ (for $n$>$1$ -- the basis for $n=1$ is trivial as then
$v' = (-) = w^{(1)}$), then it holds also for all $v'$ of rank $r$
and length $n$.

From $\rho(v)$>$0$ we know that $v\neq - $, and hence
there exist $u,w\in W_0$ such that $v = u\otimes w$.  Consider two
paths from $v$ to $w^{(n)}$. Let $\beta\in S$ be the first edge of
the first path, and let $\gamma\in S$ be the first edge of the second
path (so that in particular $dom(\beta) = v = dom(\gamma)$).
Write $v':=cod(\beta)$ and $v'':=cod(\gamma)$, so that in
$\mathcal{G}_n$ we have

[[Coh_fig_7.gif:pic]]

Since $\rho(v')$<$\rho(v)$ and $\rho(v'')$<$\rho(v)$, by hypothesis any two paths from $v'$ to $w^{(n)}$ have the same composition in $B$, and similarly for $v''$. In particular, if $\beta = \gamma$, then we are done:

[[Coh_fig_8.gif:pic]]

So, we will continue by assuming that $\beta\neq \gamma$. Note that in
this case $v'\neq v''$, because it can verified by induction that
there are no parallel arrows in $\mathcal{G}_n$.  In this
case, it will be sufficient to find some $z\in W_0$ of length $n$ and directed
paths (recall that here a directed path is a path consisting
only of directed arrows) in $\mathcal{G}_n$ from $v'$ to $z$ and from
$v''$ to $z$ such that applying $\mathcal{I}$ to the following diamond
(upper part of the diagram) results in a
commutative diamond in $B$:

[[Coh_fig_9.gif:pic]]

(the outer paths are our two paths from $v$ to $w^{(n)}$). This is indeed
enough, because by hypothesis, the two lower trapezoids are
commutative in $B$.

The existence of $z\in W_0$ and of directed paths from $v'$ and $v''$
to $z$ such that the above diamond is commutative in $B$ will be proved
by considering all possible cases for $\beta$ and $\gamma$.  For
$\beta$, we have three possibilities:

* $\beta = \beta'\hat{\otimes} 1\tilde{w}$ for some $\tilde{w}\in W_0$ and some
directed $\beta'\in S$.  In this case

$$
v = u\otimes w = dom(\beta) = dom(\beta')\otimes \tilde{w}
$$

and therefore $dom(\beta') = u$ and $\tilde{w}=w$ (by standard "unique
decomposition" of terms in a first-order language).  Also,
$v' = cod(\beta) = cod(\beta')\otimes w $.

* $\beta = 1\tilde{u}\hat{\otimes}\beta'' $ for some $\tilde{u}\in W_0$ and some
directed $\beta''\in S$. As above, we get in this case that
$dom(\beta'') =w$ and $\tilde{u} = u$. Also,
$v' = cod(\beta) = u\otimes cod(\beta'')$.

* $\beta = \alpha \tilde{u} s t$ for some $\tilde{u},s,t\in W_0$, in which
case

$$
v = u\otimes w = dom(\beta) = \tilde{u}\otimes (s\otimes t),
$$

so that $\tilde{u} = u$ and $w = s\otimes t$.  Also, $v' = cod(\beta) =
(u\otimes s)\otimes t$.

In summary, we have the following 3 cases for $\beta$:

* Case $1_{\beta}$: $\beta = \beta'\hat{\otimes} 1w$ for some directed
$\beta'\in S$ with $dom(\beta')=u$. Also, $v' =
cod(\beta')\otimes w$.

* Case $2_{\beta}$: $\beta = 1u \hat{\otimes} \beta''$ for some
directed $\beta''\in S$ with $dom(\beta'') = w$. Also, $v' =
u\otimes cod(\beta'')$.

* Case $3_{\beta}$: $\beta = \alpha u s t$ for some $s,t\in W_0$,
and also $w = s\otimes t$ and $v' = (u\otimes s)\otimes t$.

Similarly, we have Case $1_{\gamma}$, Case $2_{\gamma}$ and Case
$3_{\gamma}$ for $\gamma$.  We will now consider each one of the
possible cases for $\beta$ and $\gamma$:

$1_{\beta}$ and $1_{\gamma}$: The situation is as follows:

[[Coh_fig_10.gif:pic]]

Since $len(u)$<$len(v)$ (as $w\in W_0$ cannot be $e_0$) and since

$$
\rho(u) = \rho(v) - (\underbrace{\rho(w)}_{\geq 0} +
\underbrace{len(w) - 1}_{\geq 0}) \leq \rho(v),
$$

it follows from the induction hypothesis that any two directed paths
from $u$ to $w^{(len(u))}$ have the same composition in $B$.  In
particular, applying $\mathcal{I}$ to the following diamond in
$\mathcal{G}_{len(u)}$ results in a commutative diamond in $B$:

[[Coh_fig_11.gif:pic]]

Since $\otimes_B$ is a bifunctor, it follows that the following diagram
is again commutative when interpreted in $B$:

[[Coh_fig_12.gif:pic]]

(where for $x\in W_0$ with $len(x) = len(u)$, we write
$can(x)\hat{\otimes}  1 w$ for the directed path in $\mathcal{G}_n$ obtained by
$\hat{\otimes}$-ing each arrow of $can(x)$ from the right with $1 w$).
This completes the proof for the current case.

$2_{\beta}$ and $2_{\gamma}$:  Proved similarly to the
previous case.

$1_{\beta}$ and $2_{\gamma}$ (or $2_{\beta}$ and $1_{\gamma}$): We
have the following diamond:

[[Coh_fig_13.gif:pic]]

which commutes in $B$, because $\otimes_B$ is a bifunctor.

$3_{\beta}$ or $3_{\gamma}$:  W.l.o.g.,
we will assume $3_{\beta}$. Note that in this case, it is not
possible that also $3_{\gamma}$ holds, that is, that $\gamma = \alpha u s'
t'$ for some $s',t'$ with $w = s'\otimes t'$,
for then $s'\otimes t' = s \otimes t$, which implies that
$s' = s$ and $t' = t$, so that $\gamma = \beta$, contradicting our assumption.  So, we have to consider only
$1_{\gamma}$ and $2_{\gamma}$. For $1_{\gamma}$, we have
$\gamma = \gamma'\hat{\otimes} 1 \underbrace{s\otimes t}_{w}$ and $dom(\gamma') = u$, and we get the following diamond:

[[Coh_fig_14.gif:pic]]

which commutes in $B$ by the naturality of $\alpha^{(B)}$.

For $2_{\gamma}$ ($\gamma =1 u\hat{\otimes} \gamma''$ and
$dom(\gamma'') = w = s\otimes t$), we have to consider the three possible cases for $\gamma''$.

If $\gamma'' = 1\tilde{s}\hat{\otimes} \delta$ for some
$\tilde{s}\in W_0$ and some directed $\delta\in S$, then by comparing
domains we get $s\otimes t = dom(\gamma'') = \tilde{s}\otimes
dom(\delta)$, which implies that $\tilde{s} =s$ and $dom(\delta) = t$,
that is, $\gamma'' = 1 s \hat{\otimes} \delta $ with $dom(\delta) =
t$. Hence we get the diamond

[[Coh_fig_15.gif:pic]]

which commutes in $B$, because $\alpha^{(B)}$ is natural.

The case where $\gamma'' = \delta\hat{\otimes} 1\tilde{t}$ for some $\tilde{t}\in W_0$ and some
directed $\delta\in S$ can be treated similarly.

Finally, if $\gamma'' = \alpha r p q$, then
by comparing domains we get $s\otimes t = dom(\gamma'') =
r\otimes(p\otimes q)$, and therefore $r = s$ and $t = p\otimes
q$. Hence, $\gamma'' = \alpha s p q$ and $t = p\otimes q$.  In this
case, the two first edges $\beta$, $\gamma$ in our two paths from $v$
to $w^{(n)}$ look like part of the coherent pentagon (Eq. (5) on
p. 162 of [CWM](#CWM)):

[[Coh_fig_16.gif:pic]]

All edges in the above pentagon are directed, and the diagram is obviously
commutative in $B$ (as $B$ is monoidal). This completes the proof of
the proposition.
=--

## Including the unit and the unitors ##

Note:  By Kelly's 1964 paper, we know that the pentagon identity and
one triangle identity are all that is required (that the two other triangle
identities follow from the above is shown both in [CWM](#CWM) an in Kelly,
while the fact that $\lambda^{(B)}_{e_B} = \varrho^{(B)}_{e_B}$ follows from the
pentagon and the triangle identities is proved in Kelly).  From this
point on, we will be using freely all identities from pp. 162-163 of
[CWM](#CWM).

+-- {: .query}
Still need to fill this. The following are the highlights.
=--

First, show that any directed path in path in $\mathcal{G}'_n$ may be replaced by
a path with the same starting and ending vertices in which all
associators (if any) are at the end, and with the same composition in
$B$ as the original path.  The main ingredient in the following proposition:

+-- {: .un_prop}
###### Proposition
Let

$$
u \stackrel{s'}{\to} u' \stackrel{s''}{\to} u''
$$

be two directed arrows of $\mathcal{G}'_n$ with an associator  appearing in
$s'$ and a unitor appearing in $s''$ (note: the definition of
"directed" is extended in an obvious way to include unitors). Then
either there exists a directed  arrow $u\stackrel{s_1}{\to} u''$
containing a unitor with $\mathcal{I}(s_1)=\mathcal{I}(s'')\circ\mathcal{I}(s')$, or
there exist directed $s_1$, $s_2$ as in

$$
u \stackrel{s_1}{\to} ? \stackrel{s_2}{\to} u''
$$

with $s_1$ containing a unitor, $s_2$ containing an associator, and
$\mathcal{I}(s_2)\circ\mathcal{I}(s_1) = \mathcal{I}(s'')\circ\mathcal{I}(s')$.
=--

This proposition may be proved by a long and tedious induction on $s'$
and $s''$.

+-- {: .query}
For this I needed the triangle identities and the
naturality of $\alpha^{(B)}$ (as mentioned in [CWM](#CWM)), but also the naturality
of $\varrho^{(B)}$ and $\lambda^{(B)}$, as well as the bifunctoriality of
$\otimes_B$.  For example, consider:

[[Coh_fig_17.gif:pic]]

(where some associator appears in $\beta$).
So -- what am I missing here?
=--

Recall that our original goal is to show that any two paths
in $\mathcal{G}'_n$ with the same starting vertex and the same ending
vertex have the same composition in $B$.  There is still a path from
any vertex of $\mathcal{G}'_n$ to $w^{(n)}$ (first remove all units, then
proceed as in the case of $\mathcal{G}_n$), and we can use exactly the
same method used above (when only the associator was considered) to
reduce the problem to:  Any two _directed_ paths of $\mathcal{G}'_n$
starting at some vertex $v$ and ending at $w^{(n)}$ have the same
composition in $B$. From what we have seen above, any two arbitrary
such paths may be replaced be two paths starting with unitors and
ending with associators, as in

[[Coh_fig_18.gif:pic]]

Here, $clean(v)$ is the word obtained by removing all instances of
the unit from $v$ (this can be defined rigorously by induction), and
$r$ is the total number of units appearing in $v$.
The right part of the last diagram, where there are no units and
unitors, commutes in $B$ by the "coherence
for the associator" that was already proved.  Hence, it remains to
prove that the left part, where all arrows involve unitors, commutes
in $B$ ("coherence for the unitors").

First one must prove that two parallel directed arrows with unitors in
$\mathcal{G}'_n$ have the same interpretation in $B$ .
Recall that in $\mathcal{G}_n$ (no units and no unitors), there are no
parallel directed arrows. However, this is not the case in
$\mathcal{G}'_n$ (for example, consider $\lambda e_0\otimes e_0,1
e_0\hat{\otimes} \lambda e_0, 1 e_0\hat{\otimes} \varrho e_0\colon
e_0\otimes(e_0\otimes e_0)\rightrightarrows e_0\otimes e_0$). Hence,
we must settle for equality of the interpretations in
$B$.

+-- {: .query}
For this, I used induction on the total number of
$\hat{\otimes}$ in both $s$ and $s'$, using $\varrho^{(B)}_{e_B}=\lambda^{(B)}_{e_B}$
and the naturality of $\varrho^{(B)}$ and $\lambda^{(B)}$.  Am I missing
something here?  Is there an obvious one-line argument that I'm
failing to see?
=--

Now, we can finally prove coherence for the unitors, by induction on
the number $r$ of units in $v$. Considering the
left part of the last diagram,  if $u_1=u'_1$, as in

[[Coh_fig_19.gif:pic]]

then the parallel arrows are equal in $B$, while the
"large diamond" on the right commutes in $B$ by the induction
hypothesis.

We can therefore assume that $u_1 \neq u'_1$. As in the case of
coherence for the associator, it is sufficient to
find a commutative diamond, that is, to find some vertex $w$ of
$\mathcal{G}'_n$ and directed arrows $\gamma$ and $\gamma'$ in $S'$
containing unitors, as in

[[Coh_fig_20.gif:pic]]

such that the diamond on the left commutes in $\mathcal{B}$:
If we find such $w$, $\gamma$ and $\gamma'$, then since by the
induction hypothesis the upper and lower parallelograms on the right
commute in $B$, we are done.

Finally, the required commutative diamond may be found by induction on
$v$.

## Proof of version 2 of the coherence (Theorem VII.2.1 of [CWM](#CWM)) ##
+-- {: .query}
I intend to come back and fill this.  The most
difficult part was already proved above.
=--

## The construction of $F[1]$ and proof of universality #
The category $\mathcal{W}$ has one-element hom-sets by construction, but
it was not at all obvious that it is indeed the free monoidal
category on one generator. Alternatively, it is possible to construct
a category $F[1]$ that will be almost obviously a free monoidal
category on one generator. This would mean that $F[1]$ is isomorphic
to $\mathcal{W}$, and in particular has one-element hom-sets. The fact
that $F[1]$ is free and has one-element hom-sets may then serve as
yet a third version of coherence.

+-- {: .query}
This should still be filled.
=--

## Moving from one variable to any number of variables: The category $It(B)$ ##

+-- {: .query}
This should still be filled.
=--

## References ##

* [[Saunders Mac Lane]], _Categories for the Working Mathematician_, 2nd edition, Springer GTM, 1998. {#CWM}

* [[Tom Leinster]], _Monoidal Categories_, Lecture 8 of [_An ABC of Category Theory_](https://www.maths.ed.ac.uk/~tl/ct/), Autumn 2004, [course notes available online](https://www.maths.ed.ac.uk/~tl/ct/eight.pdf)

* [[Max Kelly]], _On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc._, _Journal of Algebra_, 1, pp. 397--402, 1964.

* Ilya Beylin and [[Peter Dybjer]], _Extracting a Proof of Coherence for Monoidal Categories from a Proof of Normalization for Monoids_,
[paper available online](http://www.cse.chalmers.se/~peterd/papers/Hol_alf.pdf)

* [[P. Etingof]], S. Galeki, D. Nikshych, and [[V. Ostrik]], _Tensor Categories_, [course notes available online](http://www-math.mit.edu/~etingof/tenscat.pdf)

* J. Armstrong, [Mac Lane's Coherence Theorem](https://unapologetic.wordpress.com/2007/06/29/mac-lanes-coherence-theorem/)

* Luke Trujillo, [A Coherent Proof of Mac Lane's Coherence Theorem](https://scholarship.claremont.edu/hmc_theses/243/)


Macdonald polynomial

#Contents#
* table of contents
{:toc}

## Idea

__Macdonald polynomials__ are a generalization of a [[Schur function]]s; they unify a theory of Hall-Littlewood and [[Jack polynomial]]s. They form a family of [[orthogonal polynomials]] which are [[symmetric functions]] in $x_1,\ldots,x_n$ with coefficients which are rational functions of two additional variables $q$ and $t$.

Given a [[partition]] $\lambda$, one defines a shift operator $T_{q,x_i}$ which maps $f = f(x_1,\ldots, x_n)$ to $f(x_1,\ldots, x_{i-1}, q x_i, x_{i+1},\ldots,x_n)$ and the operators $D_r$, $r = 0, 1, \ldots, n$ via

$$
D_r = t^{\frac{r(r-1)}{2}} \sum_{I\subset \{1,\ldots,n\}, |I| = r} \prod_{i\in I, j\notin I} \frac{t x_i-x_j}{x_i-x_j}\prod_{i\in I} T_{q, x_i},
$$
and the corresponding generating series $D := \sum_{r=0}^n D_r u^r$.

The __Macdonald polynomial__ $P_\lambda(x;q,t)$ is an eigenfunction of $D$ with the eigenvalue

$$
\prod_{i=1}^n (1 + u t^{n-i} q^{\lambda_i})
$$

In the case $q = t$ we get the [[Schur function]] $P_\lambda(x; t,t) = s_\lambda(t)$. Similarly, shifted Macdonald polynomials generalize shifted Schur functions.

## References

* [[Ian G. Macdonald]],  *A New Class of Symmetric Functions*, Publ. I.R.M.A. **372** (S-20) (1988) 131-171 [[web](https://www.mat.univie.ac.at/~slc/opapers/s20macdonald.html)]

* [[Ian G. Macdonald]], _Symmetric functions and Hall polynomials_, Oxford Math. Monographs, 2nd enlarged ed. (1995) [[pdf](https://math.berkeley.edu/~corteel/MATH249/macdonald.pdf)]

* Wikipedia, *[Macdonald polynomial](http://en.wikipedia.org/wiki/Macdonald_polynomials)*

* A. M. Garsia, C. Procesi, _On certain graded $S_n$-modules and the $q$-Kostka polynomials_, Adv. Math. __94__ (1992) 82-138
* A. Okounkov, _(Shifted) Macdonald polynomials: q-integral representation and combinatorial formula_, Compositio Math. __112__ (1998), 147–182. [MR99h:05120](http://www.ams.org/mathscinet-getitem?mr=1626029), [doi](http://dx.doi.org/10.1023/A:1000436921311), _BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials_, Transform. Groups __3__ (1998) 181–207, [MR99h:33061](http://www.ams.org/mathscinet-getitem?mr=1628453), _Combinatorial formula for Macdonald polynomials and generic Macdonald polynomials_, Transform. Groups __8__ (2003), no. 3, 293–305, [MR2004e:05202](http://www.ams.org/mathscinet-getitem?mr=1996418), [doi](http://dx.doi.org/10.1007/s00031-003-0306-0)
* N. Bergeron, A. M. Garsia, _On certain spaces of harmonic polynomials_, in: Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991), Contemp. Math. __138__, 51–86 (Amer. Math. Soc. 1992)
* A. Yu. Okounkov, _A remark on the Fourier pairing and the binomial formula for the Macdonald polynomials_, Funktsional. Anal. i Prilozhen. 36 (2002), no. 2, 62--68, 96; translation in Funct. Anal. Appl. 36 (2002), no. 2, 134–139, [doi](http://dx.doi.org/10.1023/A:1015670523770)
* G. Felder, L. Stevens, [[A. Varchenko]], _Modular transformations of the elliptic hypergeometric functions, Macdonald polynomials, and the shift operator_, Moscow Math. J. __3__, n. 2 (2003), 457-473, [pdf](http://www.ams.org/distribution/mmj/vol3-2-2003/felder-etal.pdf), [arXiv:math.QA/0203049](http://lanl.arxiv.org/abs/math/0203049), [MR2025269](http://www.ams.org/mathscinet-getitem?mr=2025269)
* Mark Haiman, _Hilbert schemes, polygraphs and the Macdonald positivity conjecture_, J. Amer. Math. Soc. __14__ (2001), no. 4, 941–1006, [MR2002c:14008](http://www.ams.org/mathscinet-getitem?mr=1839919), [doi](http://dx.doi.org/10.1090/S0894-0347-01-00373-3); _Macdonald polynomials and geometry_, in: New perspectives in algebraic combinatorics (Berkeley, CA, 1996–97), 207–254, Math. Sci. Res. Inst. Publ. __38__, Cambridge Univ. Press 1999, [pdf](http://math.berkeley.edu/~mhaiman/ftp/nfact/msri.pdf)
* M. Haiman, _Cherednik algebras, Macdonald polynomials and combinatorics_, Proc. ICM, Madrid 2006, Vol. III, 843-872, [djvu scan](http://www.mathunion.org/ICM/ICM2006.3/Main/icm2006.3.0843.0872.ocr.djvu), [author's pdf](http://math.berkeley.edu/~mhaiman/ftp/icm-2006/comb-mac-chered.pdf)
* M. Haiman, A. Woo, _Geometry of $q$ and $q,t$-analogs in combinatorial enumeration_, in: Geometric combinatorics, 207–248,
IAS/Park City Math. Ser. __13__, Amer. Math. Soc., Providence, RI, 2007, [pdf](http://math.berkeley.edu/~mhaiman/ftp/pcmi-2004/notes.pdf), [ps](http://math.berkeley.edu/~mhaiman/ftp/pcmi-2004/notes.ps)
* A. M. Garsia, M. Haiman, _A graded representation model for Macdonald's polynomials_, Proc. Nat. Acad. Sci. U.S.A. __90__ (1993) 3607–3610, [MR94b:05206](http://www.ams.org/mathscinet-getitem?mr=1214091), [PNAS](http://www.pnas.org/cgi/reprintframed/90/8/3607)
* A. M. Garsia, G. Tesler, _Plethystic formulas for Macdonald $q, t$-Kostka coefficients_, Advances in Math. __123__ (1996) 144–222, [MR1420484](http://www.ams.org/mathscinet-getitem?mr=1420484); A. M. Garsia, J. Remmel, _Plethystic formulas and positivity for $q,t$-Kostka coefficients_, Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996), 245–262, Progr. Math. __161__, Birkhäuser  1998, [MR99j:05189d](http://www.ams.org/mathscinet-getitem?mr=1627327)
* Friedrich Knop, _Integrality of two variable Kostka functions_,
J. Reine Angew. Math. 482 (1997), 177–189, [doi](http://dx.doi.org/10.1515/crll.1997.482.177), [MR99j:05189c](http://www.ams.org/mathscinet-getitem?mr=1427661)
* Siddhartha Sahi, _Interpolation, integrality, and a generalization of Macdonald's polynomials_, Internat. Math. Res. Notices 1996, no. 10, 457–471, [MR99j:05189b](http://www.ams.org/mathscinet-getitem?mr=1399411), [doi](http://dx.doi.org/10.1155/S107379289600030X)
* Anatol N. Kirillov, Masatoshi Noumi, _Affine Hecke algebras and raising operators for Macdonald polynomials_, Duke Math. J. __93__ (1998), no. 1, 1–39, [MR99j:05189a](http://www.ams.org/mathscinet-getitem?mr=1620075), [doi](https://doi.org/10.1215/S0012-7094-98-09301-2)
* Anatol Kirillov Jr., _Traces of intertwining operators and Macdonald's polynomials_, [q-alg/9503012](http://arxiv.org/abs/q-alg/9503012)
* Anton Gerasimov, Dimitri Lebedev, Sergey Oblezin, _Baxter operator formalism for Macdonald polynomials_. [arxiv/1204.0926](http://arxiv.org/abs/1204.0926)
* Persi Diaconis, [[Arun Ram]], _A probabilistic interpretation of the Macdonald polynomials_, [arxiv/1007.4779](http://arxiv.org/abs/1007.4779)
* Anton Khoroshkin, _Highest weight categories and Macdonald polynomials_, [arxiv/1312.7053](http://arxiv.org/abs/1312.7053)
*  E. Carlsson, E. Gorsky, A. Mellit, The $\mathbf{A}_{q,t}$ algebra and parabolic flag Hilbert schemes [arxiv/1710.01407](https://arxiv.org/abs/1710.01407);  A. Garsia, A. Mellit, _Five-term relation and Macdonald polynomials_, [arxiv/1604.08655](https://arxiv.org/abs/1604.08655); A. Mellit, _Plethystic identities and mixed Hodge structures of character varieties_, [arxiv/1603.00193](https://arxiv.org/abs/1603.00193)
* Wy. Chuang, D-E. Diaconescu, R. Donagi, [[T. Pantev]], _Parabolic refined invariants and Macdonald polynomials_, Commun. Math. Phys. __335__, 1323--1379 (2015) [doi](https://doi.org/10.1007/s00220-014-2184-9)

> A string theoretic derivation is given for the conjecture of Hausel, Letellier and Rodriguez-Villegas on the cohomology of character varieties with marked points. Their formula is identified with a refined BPS expansion in the stable pair theory of a local root stack, generalizing previous work of the first two authors in collaboration with Pan. Haiman’s geometric construction for Macdonald polynomials is shown to emerge naturally in this context via geometric engineering. In particular this yields a new conjectural relation between Macdonald polynomials and refined local orbifold curve counting invariants. The string theoretic approach also leads to a new spectral cover construction for parabolic Higgs bundles in terms of holomorphic symplectic orbifolds.

category: algebra, combinatorics

[[!redirects Macdonald polynomials]]

[[!redirects Hall polynomial]]
[[!redirects Hall polynomials]]