Löwenheim-Skolem theorem

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## Idea

The **Löwenheim-Skolem theorem** is a basic result in the _[[model theory]]_ of [[first-order logic]] and is part of a family of closely related theorems that concern the relation between structures or [[models]] of [[first-order theories]] of different [[cardinality]].

One idea behind it is that finitary first-order logic by itself cannot pin down the cardinality of infinite structures or models: roughly speaking, Löwenheim-Skolem says that given a such structure $M$, one can construct elementary substructures or elementary extensions $N$ of different cardinalities. This gives an elementary reason why one cannot expect absolute [[categoricity]] for first-order theories; after this theorem was introduced, attention shifted to the question of categoricity in powers $\kappa$ (i.e., how many models there are of a given cardinality $\kappa$), leading up eventually to Morley's categoricity results and, further down the road and in much greater generality, to Shelah's introduction of [[stability theory]].

Historically, the theorem was one of the first results in model theory with papers published by Leopold Löwenheim (1915) and [[Thoralf Skolem]] (1920) which helped establishing first-order logic in the center of [[mathematical logic]]. It was however conceived by Skolem in view of a problem he saw with first-order set theory, now called "Skolem's [[paradox]]", where one could construct a *countable* model of ZFC, with the paradoxical consequence that (externally speaking) the model admits no uncountable sets but yet (internally speaking) it has them. As usually happens with "paradoxes" (cf. [[Banach-Tarski paradox]]), this phenomenon was over time digested and accepted as the ordinary fact of the matter without qualms, and just one more illustration of how one must simply retrain one's intuition to fit the facts.

Although the theorem fails in [[second-order logic]] (for example, note one has absolute categoricity for the concept of complete ordered field), variants hold in infinitary logic with countably infinite disjunctions and in [[categorical logic]] for [[sketches]]. The theorem plays an important role in [[abstract model theory]] e.g. in [[Lindström's theorems]] on the characterization of standard [[first-order logic]].[^fine]

[^fine]: As a theorem that fails to hold is hardly a theorem at all, one prefers in the context of abstract model theory to speak of the _Löwenheim-Skolem property_.

## Statements and proofs

One frequently sees the Löwenheim-Skolem theorem separated into an "upward" version (which produces [[elementary embedding|elementary extensions]]) and a "downward" part (producing elementary substructures). The upward version can be derived just by a clever application of the [[compactness theorem]]; the downward version involves a construction called "Skolemization". Both proofs involve modifying the language.

Let $\Sigma$ be a first-order [[signature]] and let $T$ be a set of sentences of cardinality $\beta$. The upward version states the following: suppose $T$ has a model of cardinality $\kappa \geq \aleph_{0}$, then for every cardinal $\lambda \geq \max(\aleph_{0}, \beta)$, $T$ has a model of cardinality $\lambda$. The downward version, on the other hand, states the following: suppose $T$ has a model of infinite cardinality, then $T$ has a model of cardinality $\aleph_{0}$.

## With paraconsistent logic

According to [Bremer](#Bremer), given a [[paraconsistent logic|paraconsistent]] [[first order logic]], the Löwenheim-Skolem theorem could be strengthened to this result:

> Any mathematical theory presented in first order logic has a finite paraconsistent model.

## Related entries

* [[Skolem's paradox]]

* [[compactness theorem]]

* [[Lindström's theorem]]

* [[Craig interpolation theorem]]

* [[Thoralf Skolem]]

## Link

* Wikipedia, _[Löwenheim-Skolem theorem](https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem)_

## References

* {#AdamekRosicky94}[[Jiří Adámek]], [[Jiří Rosický]], _Locally presentable and accessible categories_ ,  Cambridge UP 1994.

* J. van Heijenoort (ed.), _From Frege to Gödel - A Source Book in Mathematical Logic 1879-1931_ , Harvard UP 1967.

* L. Löwenheim, _Über die Möglichkeiten im Relativkalkül_ , Math. Ann. **76** (1915) pp.447-470. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/pdf/?PPN=GDZPPN002266121); English transl. in van Heijenoort (1967) pp.228-251)

* [[Michael Makkai]], [[Robert Paré]], _Accessible categories: The foundations of categorical model theory_ , Contemporary Mathematics 104. American Mathematical
Society, Rhode Island, 1989.
{#MakkaiPare}

* [[Thoralf Skolem]], *[[Some remarks on axiomatized set theory|Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre]]*, Mathematikerkongressen i Helsingfor 4-7 Juli 1922. English transl. pp.290-301 of van Heijenoort (ed.), _From Frege to Gödel_ , Harvard UP 1967.

* M. Zawadowski, _The Skolem-Löwenheim theorem in toposes_ , Studia Logica **42** (1983) pp.461-475.

* M. Zawadowski, _The Skolem-Löwenheim theorem in toposes II_ , Studia Logica **44** (1985) pp.25-38.

* M. Manzano, _Model Theory_, Clarendon Press (1999)

In [[paraconsistent mathematics]]:

* {#Bremer} Manuel Bremer, *Inconsistent Mathematics*, ([slides](https://www.mbph.de/Logic/Para/InconsistentMathematics.pdf))

[[!redirects Lowenheim-Skolem theorem]]
[[!redirects Löwenheim-Skolem property]]
[[!redirects Lowenheim-Skolem property]]


M M Postnikov

* [MacTutor biography](https://mathshistory.st-andrews.ac.uk/Biographies/Postnikov/)

__Михаил Михайлович Постников__, (that is _Mihail Mihajlovič Postnikov_ (in a slavistic transliteration as in ISO9 or G. Shevelov, A prehistory of Slavic, Columbia 1965) or _Mikhail Mikhailovich Postnikov_) is a former Russian (Soviet) topologist (27.10.1927--27.5.2004) and a professor at MGU until his sudden death.

His thesis under the mentorship of [[L. S. Pontrjagin]] ([ru.wiki](http://ru.wikipedia.org/wiki/Понтрягин,_Лев_Семёнович)), contained a construction of a homotopical decomposition of a topological space into a tower of [[fibration]]s (now called a [[Postnikov tower]]) and cohomological invariants ([[Postnikov invariant]]s) which together with the tower form a "system" ([[Postnikov system]]). The use of these data enabled him to effectively encode the full [[homotopy type]] of a space.

After his early work, Postnikov dedicated himself to teaching, leading a famous seminar at MGU, and advising students (among most successful of his students is Rudyak) and largely neglected his own research. As a clear and superb lecturer, he also wrote several volumes of advanced textbooks in geometry and topology, and a booklet on Galois theory. Unfortunately, he also dedicated much of his efforts and time to an alternative history, considered pseudoscience by the mainstream. Some of his mathematical memoirs may be found in the book _Golden Years of Moscow Mathematics_, by Smilka Zdravkovska (AMS, 2nd ed. 2007).

See also [Postnikov at ru.wikipedia](http://ru.wikipedia.org/wiki/Постников,_Михаил_Михайлович) and [msu page](http://higeom.math.msu.su/people/postnikov)

## Selected writings

Introducing [[Postnikov systems]]:

* [[M. M. Postnikov]], _Determination of the homology groups of a space by means of the homotopy invariants_, Doklady Akad. Nauk SSSR (N.S.) 76: 359–362 (1951)

* [[M. M. Postnikov]], _Issledovaniya po gomotopičeskoĭ teorii nepreryvnyh otobraženiĭ. I. Algebraičeskaya teoriya sistem. II. Naturalʹnaya sistema i gomotopičeskiĭ tip_. (Russian) $[$_Investigations in homotopy theory of continuous mappings. I. The algebraic theory of systems. II. The natural system and homotopy type._$]$ Trudy Mat. Inst. Steklov. no. 46. Izdat. Akad. Nauk SSSR, Moscow, 1955. ([mathnet:tm1182](http://mi.mathnet.ru/eng/tm1182))

## Related $n$Lab entries

* [[compactly generated topological space]]

category: people

[[!redirects Постников]]
[[!redirects Михаил Михайлович Постников]]
[[!redirects Михаил Постников]]
[[!redirects Postnikov]]
[[!redirects Mikhail Postnikov]]
[[!redirects Mikhail Mikhailovich Postnikov]]
[[!redirects Mihail Postnikov]]
[[!redirects Michael Postnikov]]
[[!redirects M. M. Postnikov]]


M-brane

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## Idea

There are two types of [[black brane]] solutions in [[11-dimensional supergravity]], one of dimension $2+1$, one of dimension 5+1. These are thought to correspond to a fundamental 2-brane and its [[electric-magnetic duality|EM dual]], called the

* [[M2-brane]] ([[fractional M2-brane]])

* [[M5-brane]] ([[half M5-brane]])

* [[MK6-brane]]

* [[MO9-brane]]

* [[M-wave]]

The M2-brane carries [[electric charge]] under the [[supergravity C-field]]. The M5-brane is the dual magnetic charge.

## Properties

### Reduction to strings, D-branes and NS5-branes

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

### M-brane intersections

[[brane intersection|intersecting]] M-branes:

* [[M5-MO9 brane bound state]]

* [[M2-M5 brane bound state]]

  * [[M-string]], [[E-string]]

* [[membrane triple junction]]

## Related concepts

* [[anti M-brane]]

[[!include table of branes]]

## References

The relevant literature until 1999 is collected in

* {#Duff99} [[Mike Duff]], _[[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))

See also the references at _[[M-theory]]_ and at _[[M2-brane]]_, _[[M5-brane]]_, _[[M9-brane]]_.

Discussion of some [[black brane]]-aspects of M-branes includes

* Takeshi Sato, Section 2.3 of: _Superalgebras in Many Types of M-Brane Backgrounds and Various Supersymmetric Brane Configurations_, Nucl. Phys. B548 (1999) 231-257 ([arXiv:hep-th/9812014](https://arxiv.org/abs/hep-th/9812014))

* [[David Berman]], _M-theory branes and their interactions_, Phys. Rept. 456:89-126, 2008 ([arXiv:0710.1707](http://arxiv.org/abs/0710.1707))

Discussion of plenty of [[cohomology|cohomological]] structures involved in M-brane physics is in

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

Discussion of [[M-brane]] physics in terms of [[equivariant rational homotopy theory|rational equivariant]] [[cohomotopy]] is in

* [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_ ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987))

The [[brane intersection]] laws of M-branes are discussed in

* [[Eric Bergshoeff]], [[Mees de Roo]], [[Eduardo Eyras]], [[Bert Janssen]], [[Jan Pieter Schaar]], _Intersections involving waves and monopoles in eleven dimensions_, Class. Quantum Grav. **14** 2757 [[doi:0264-9381/14/10/005](http://iopscience.iop.org/0264-9381/14/10/005)]

* [[Eric Bergshoeff]], [[Joaquim Gomis]], [[Paul Townsend]], _M-brane intersections from worldvolume superalgebras_, Phys.Lett. B421 (1998) 109-118 ([arXiv:hep-th/9711043](http://arxiv.org/abs/hep-th/9711043))

* {#Townsend97} [[Paul Townsend]], section 4 of _M-theory from its superalgebra_ ([arXiv:hep-th/9712004](http://arxiv.org/abs/hep-th/9712004))

Suppposed analogy with D-brane descent in K-theory (see [there](D-brane#OlsenSzabo00)):

* [[Laurent Houart]], [[Yolanda Lozano]], _Brane Descent Relations in M-theory_, Phys.Lett. B **479** (2000) 299-307 [[arXiv:hep-th/0001170](https://arxiv.org/abs/hep-th/0001170), doi:10.1016/S0370-2693(00)00317-8]

  > (non-BPS M-branes such as M8-branes)

On M-[[brane intersections]] such as [[M2-M5 brane bound states]]:

* [[Arkady Tseytlin]], *Harmonic superpositions of M-branes*, Nucl. Phys. B **475** (19960 149-163 [[arXiv:hep-th/9604035](https://arxiv.org/abs/hep-th/9604035), doi:10.1016/0550-3213(96)00328-8]

On [[M-brane]] [[sigma-models]] on [[exceptional geometry|exceptional]] [[exceptional generalized geometry|generalized geometric]] [[target spacetimes]]:

* [[Yuho Sakatani]], [[Shozo Uehara]], _Branes in Extended Spacetime: Brane Worldvolume Theory Based on Duality Symmetry_, Phys. Rev. Lett. **117** 191601 (2016) [[arXiv:1607.04265](https://arxiv.org/abs/1607.04265), [talk slides](http://www2.yukawa.kyoto-u.ac.jp/~qft.web/2016/slides/sakatani.pdf)]

* [[Yuho Sakatani]], [[Shozo Uehara]], _Exceptional M-brane sigma models and $\eta$-symbols_ [[arXiv:1712.10316](https://arxiv.org/abs/1712.10316)]

Further references:

* Chan Park, _2d SCFT from M-branes and its spectral network_ ([pdf](http://theory.caltech.edu/~chan/documents/2013_String_Math_proceedings.pdf))

[[!redirects M-branes]]


M-brane 3-algebra

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## Idea

In the construction of the [[Lagrangian density]] for the [[supersymmetry|supersymmetric]] [[worldvolume]] [[gauge field|gauge]] [[quantum field theory]] on [[intersecting brane|coincident]] [[M2-branes]] (the [[BLG model]]/[[ABJM model]], a [[conformal field theory|conformal]] [[supersymmetry|super]]-[[Chern-Simons theory]] coupled to  [[matter]]-[[field (physics)|fields]]) a certain [[algebra|algebraic]] structure appears -- and at least in the [[BLG model]] it appears at a pivotal stage in the [[proof]] of [[supersymmetry]] -- which is given by a  [[multilinear map|trilinear map]] on an [[inner product space]] $V$

$$
  V \otimes V \otimes V
  \overset{
    \overset{
       \color{blue}
       \;\;
       {
        \text{3-bracket}
        \atop
        \phantom{-}
       }

       \;\;
    }{
      [-,-,-]
    }
  }{\longrightarrow} V
$$

that is subject to some [[axioms]] which are different from but clearly reminiscent to the [[axioms]] on the [[Lie bracket]] of a [[Lie algebra]], (which is of course a bilinear map).

In special cases, but not generally, this algebraic structure is a special case of a structure introduced by Filippov, which could be called a [[Filippov algebra]], but which mostly came to alternatively be called a _3-Lie algebra_ or a _Lie 3-algebra_ or often just a _3-algebra_.

Unfortunately, in [[homotopy theory]] and [[higher algebra]], these terms refer to something different, namely to [[L-infinity algebras]] or [[categorification|categorifications]] of [[associative algebras]] to [[n-algebras]] -- but the structure seen in the [[BLG model]] on [[M2-branes]] is _none_ of these.

Hence to be on the safe side, one may want to speak of _M2-brane 3-algebras_ or maybe _M-brane algebras_, for definiteness

Moreover, while this "M2-brane 3-algebra" appeared to be a central ingredient in the [[BLG model]], it makes _no_ appearance in the [[ABJM model]], which however generalizes and hence subsumes the BLG model, up to slight re-identifications of terms.

Indeed, inspection reveals that "M2-brane 3-algebras" give rise to and may be re-constructed from data in ordinary [[Lie theory]], namely from [[pairs]] consisting of a [[metric Lie algebra]] and a [[dualizable object|dualizable]] [[Lie algebra representation]].

$$
  \big\{
    \text{M2-brane 3-algebras}
  \big\}
  \;\;\overset{\simeq}{\leftrightarrow}
  \left\{
    {
    \text{faithful orthogonal representations}
       \atop
    \text{of metric Lie algebras}
    }
  \right\}
$$

This is due to [dMFFMER 08, Theorem 11](#dMFFMER08), recalled as Prop. \ref{M2Brane3AlgebrasEquivalentToMetricLieRepresentation} below.

(...)

## Definition

+-- {: .num_defn #AxiomsOnM2Brane3Algebra}
###### Definition
**([[M2-brane 3-algebra]])**

An _M2-brane 3-algebra_ is

1. a [[real vector space|real]] [[finite-dimensional vector space]] $V$;

1. a _metric_ $\langle -,-\rangle$ on $V$, hence a non-degenerate [[inner product space]]  (not required to be positive definite);

1. a [[multilinear map|trilinear function]], the _3-bracket_

   $$
     V \otimes V \otimes V \overset{[-,-,-]}{\longrightarrow} V
   $$

such that the following three [[axioms]] hold, for all [[elements]] $w,x,y,z \in V$:

1. (**unitarity**)

   $$
     \big\langle [x,y,z],w \big\rangle
      \;=\;
     -\big\langle z, [x,y,w]\big\rangle
   $$

1. (**symmetry**)

   $$
     \big\langle [x,y,z],w \big\rangle
      \;=\;
     -\big\langle [z,w,x], y \big\rangle
   $$

1. (**fundamental identity**)

   $$
     \begin{aligned}
       \big[x,y, [v,w,z]  \big]
       &
         = \phantom{+}\;
        \big[  [x,y,v], w, z \big]
       \\
       &
       \phantom{=\;} +
        \big[ v, [x,y, w] , z \big]
       \\
       &
       \phantom{=\;} +
        \big[ v, w , [x,y, z ] \big]
     \end{aligned}
   $$

=--

In this generality, and under the name "generalized metric Lie 3-algebras", this is due to [Cherkis-Saemann 08, 41](#CherkisSaemann08), see [dMFFMER 08, Def. 1](#dMFFMER08)

## Properties

### Equivalence to metric Lie representations

For the following, let the [[ground field]] be the [[real numbers]] and take all [[vector spaces]] involved to be [[real vector space|real]] and [[finite-dimensional vector spaces|finite-dimensional]].

For definiteness, we make the following explicit:

+-- {: .num_defn #ModulesOfMetricLieAlgebras}
###### Definition
**([[metric Lie representation]])**

An _orthogonal representation of a metric Lie algebra_ is

1. a [[metric Lie algebra]]

   $$
     \big((\mathfrak{g},[-,-]), g \big)
   $$

1. a metric [[vector space]]

   $$
     (V,k)
     \,,
   $$

   hence a non-degenerate real [[inner product space]];

1. an [[orthogonal Lie algebra|orthogonal]] [[Lie algebra representation]]

   $$
     \mathfrak{g} \otimes V \overset{\rho}{\to} V
   $$

  of $\mathfrak{g}$ on $(V,k)$.

=--

+-- {: .num_remark #LinearDualsAndAdjuncts}
###### Remark
**(linear duals and adjuncts/transposition)**

Given an orthogonal representation of a metric Lie algebra as in Def. \ref{ModulesOfMetricLieAlgebras},
the non-degenerate [[inner products]] $g$  on $\mathfrak{g}$ and $k$ on $V$ exhibit these as [[self-dual objects]] in the [[monoidal category]] of [[finite-dimensional vector spaces]] (with respect to the [[tensor product of vector spaces]]).

This means that by passsing to [[adjuncts]] we may regard [[linear maps]] of the form

$$
  V \otimes W \overset{f}{\longrightarrow} Z
$$

equivalently as linear maps of the form

$$
  W \overset{\overline{f}}{\longrightarrow} Z \otimes V^\ast
  \,,
$$

where $V^\ast$ is the [[dual vector space]]; and similarly for $\mathfrak{g}$.

Once we choose a [[linear basis]] $\{v_i\}_{i \in \{1, \cdots, dim(V)\}}$ of $V$, with respect to which the metric tensor $k$ has components

\[
  \label{ComponentsOfMetricOnV}
  k_{i j} \;\coloneqq\; k(v_i, v_j)
\]

this forming of adjuncts is equivalently the yoga of "raising and lowering of indices via the metric". Similarly for a choice of [[linear basis]] $\{t_a\}_{a \in \{1,\cdots, dim(\mathfrak{g})\}}$ for the Lie algebra, and the induced components

\[
  \label{ComponentsOfMetricOng}
  g_{a b} \;\coloneqq\; g(t_a, t_b) \,.
\]

Specifically, given the basis component $(\rho_a{}^{i}{}_j)_{a,i,j}$ of the [[Lie algebra representation|Lie action]] $\mathfrak{g} \otimes V \overset{\rho}{\longrightarrow} V$, defined by

\[
  \label{ComponentsOfLieAction}
  \rho(t_a, v_i)
  \;\coloneqq\;
  \rho_a{}^{i}{}_j v_i
\]

we get the components $(\tilde \rho^a{}_{i j})_{a,i,j}$ of the [[adjunct]]

\[
  \label{AdjunctOfLieAction}
  V \otimes V \overset{ \tilde \rho }{\longrightarrow} \mathfrak{g}
\]

by contracting the original component (eq:ComponentsOfLieAction) with the components (eq:ComponentsOfMetricOnV) (eq:ComponentsOfMetricOng)  of the metric tensors:

\[
  \label{ComponentsOfAdjunctOfLieAction}
  (\tilde \rho^a{}_{i j})_{a,i,j}
  \;=\;
  g_{l j} k^{a b} \, \rho_b{}^l{}_i
\]

=--

+-- {: .num_defn #TrilinearMapInducedFromOrthogonalRepOfMetricLieAlgebra}
###### Definition
**(trilinear map induced from [[metric Lie representation]] -- [[Faulkner construction]])**

Given an orthogonal representation $\mathfrak{g} \otimes V \overset{\rho}{\to} V$ of a metric Lie algebra as in Def. \ref{ModulesOfMetricLieAlgebras}, define

1. a [[bilinear map]]

   $$
     V \otimes V
       \overset{
          \;\;D\;\;
       }{\longrightarrow}
     \mathfrak{g}
   $$

   given by

   \[
     \label{DefiningEquationForD}
     (X, D(x,y))
     \;=\;
     \langle \rho(X,x), y \rangle
     \;\;\;\;
     \mathrlap{
       \text{for all}\; X \in \mathfrak{g}\,, x, y \in V
     }
   \]

1. a [[multilinear map|trilinear map]]

   $$
     \label{DefiningEquationForInducedTrilinearBracket}
     V \otimes V \otimes V
       \overset{
          [-,-,-]_{\rho} \coloneqq \rho(D(-,-),-)
       }{\longrightarrow}
     V
    \,,
   $$

   hence

   $$
     [x, y, z]_\rho
     \;\coloneqq\;
     \rho\big(  D(x, y), z \big)
   $$

=--

In this form and with this notation  this is [dMFFMER 08, equation (9) and over Prop. 10](#dMFFMER08) (except that we write "$\rho(X,x)$" for their "$X \cdot x$"). The construction itself, up to one dualization of $V$, was introduced in [Faulkner 73](#Faulkner73).

+-- {: .num_remark}
###### Remark
**(component expression of trilinear bracket)**

After a choice of [[linear bases]] as in Remark \ref{LinearDualsAndAdjuncts}, in terms of which Lie algebra elements $X \in \mathfrak{g}$
are expanded as $X \coloneqq X^a t_a$ and representation vectors $x \in V$ are expanded as $x \coloneqq x^i v_i$, the defining equation (eq:DefiningEquationForD)

$$
  (X, D(x,y))
  \;=\;
  \langle \rho(X,x), y \rangle
  \;\;\;\;
  \text{for all arguments}
$$

reads

$$
  k_{a b} X^a \, D_{i j}{}^b \, x^i y^j
  \;=\;
  g_{l j} \, \rho_a{}^l{}_i \,  X^a x^i y^j
  \;\;\;\;
  \text{for all components}
$$

(where here and in the following we use the [[Einstein summation convention]]), hence reads

$$
  k_{a b} D_{i j}{}^b
  \;=\;
  g_{l j} \rho_a{}^l{}_i
  \;\;\;\;
  \text{for all indices}
$$

hence equivalently reads

$$
  D_{i j}{}^a
  \;=\;
  g_{l j} k^{a b} \, \rho_b{}^l{}_i
  \;\;\;\;
  \text{for all indices}
$$

hence says that the [[tensor]] $D$ is equal to the adjunct (eq:AdjunctOfLieAction) of the Lie action tensor $\rho$, given in components by the evident "raising and lowering of indices via the metrics" as in (eq:ComponentsOfAdjunctOfLieAction):

$$
  D_{i j}{}^a
  \;=\;
  \rho^a{}_{i j}
  \;\;\;\;
  \text{for all indices}
  \,.
$$

With this, the induced trilinear bracket (eq:DefiningEquationForInducedTrilinearBracket)

$$
     V \otimes V \otimes V
       \overset{
          [-,-,-]_{\rho} \coloneqq \rho(D(-,-),-)
       }{\longrightarrow}
     V
$$

is given in components as

$$
  \begin{aligned}
  [v_i, v_j, v_k]_\rho
  & =\;
  \rho(D(v_i, v_j), v_k)
  \\
  & =\;
  \rho( D_{i j}{}^a t_a , v_k )
  \\
  & =
  \rho_{a}{}^m{}_k
  \underset{
    =
    g_{l j} k^{a b} \, \rho_b{}^l{}_i
  }{
    \underbrace{D_{i j}{}^a}
  }
  v_m
  \\
  & =
  \rho_{a}{}^m{}_k   g_{l j} k^{a b} \, \rho_b{}^l{}_i     v_m
  \\
  & =
  \big( \rho_{a}{}^m{}_k \rho^a{}_j{}_i \big)  v_m
  \end{aligned}
  \,.
$$

=--

+-- {: .num_prop #M2Brane3AlgebrasEquivalentToMetricLieRepresentation}
###### Proposition
**([[M2-brane 3-algebras]] are equivalent to [[metric Lie representations]])**

Given an orthogonal representation $\mathfrak{g} \otimes V \overset{\rho}{\to} V$ of a metric Lie algebra as in Def. \ref{ModulesOfMetricLieAlgebras}, its induced trilinear map $[-,-,-]_\rho$ (Def. \ref{TrilinearMapInducedFromOrthogonalRepOfMetricLieAlgebra}) satisfies the axioms of an M2-brane 3-algebra (Def. \ref{AxiomsOnM2Brane3Algebra}).

Hence this assignment defines a [[function]] from [[isomorphism classes]] of orthogonal representations of metric Lie algebras to [[isomorphism classes]] of M2-brane 3-algebras

$$
  \left\{
    {
    {\text{orthogonal representations}}
    \atop
    {\text{metric Lie algebras}}
    }
  \right\}_{/\sim}
  \underoverset
  {}
  {
     \;\;\;\;
     \rho \;\mapsto\; [-,-,-]_\rho
     \;\;\;\;
  }{\longrightarrow}
  \left\{
    {
    {\text{M2-brane}}
    \atop
    {\text{3-algebras}}
    }
  \right\}_{/\sim}
  \,.
$$

Moreover, restricted to [[faithful representations]], this function is a [[bijection]]:

$$
  \left\{
    {
    {
    {\text{faithful}}
    \atop
    {\text{orthogonal representations}}
    }
    \atop
    {\text{metric Lie algebras}}
    }
  \right\}_{/\sim}
  \underoverset
  {\simeq}
  {
     \;\;\;\;
     \rho \;\mapsto\; [-,-,-]_\rho
     \;\;\;\;
  }{\longrightarrow}
  \left\{
    {
    {\text{M2-brane}}
    \atop
    {\text{3-algebras}}
    }
  \right\}_{/\sim}
  \,.
$$

=--

This is [dMFFMER 08, Prop. 10 and Theorem 11](#dMFFMER08).

### Relation to Lie algebra weight systems on chord diagrams
 {#RelationToLieAlgebraWeightSystemsOnChordDiagrams}

In [[Penrose notation]] ([[string diagram]]-notation), the trilinear bracket induced by a [[metric Lie representation]] according to Def. \ref{TrilinearMapInducedFromOrthogonalRepOfMetricLieAlgebra} looks as follows:

With Prop. \ref{M2Brane3AlgebrasEquivalentToMetricLieRepresentation} this shows that M2-brane 3-algebras are equivalently the datum that [[Lie algebra weight systems]] on ([[horizontal chord diagram|horizontal]]) [[chord diagrams]] assign to a single chord.

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

## References

### Appearance in M2-brane theory

The idea that some higher-arity version of the Lie bracket may be relevant for [[M2-M5 brane intersections]] originates with attempts of generalizing [[Nahm's equations]] for [[fuzzy funnels]] of [[D2-D4 brane intersections]] in

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

Motivated by this, the M2-brane 3-algebra appears in the construction of the [[BLG model]] for the [[worldvolume]] [[quantum field theory]] on 2 coincident [[M2-branes]] in

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

* {#BaggerLambert07} [[Jonathan Bagger]], [[Neil Lambert]], _Gauge Symmetry and Supersymmetry of Multiple M2-Branes_, Phys. Rev. D77, 065008 (2008). ([arXiv:0711.0955](http://arXiv.org/abs/0711.0955))

further highlighted as such in

* {#Gustavsson07} [[Andreas Gustavsson]], _Algebraic structures on parallel M2-branes_ Nucl. Phys. B811, 66-76 (2009) ([arXiv:0709.1260](https://arxiv.org/abs/0709.1260))

(whence the "BLG" of the [[BLG model]])

and further explored in

* [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], _Metric Lie 3-algebras in Bagger-Lambert theory_, JHEP 0808 : 045, 2008 ([arXiv:0806.3242](https://arxiv.org/abs/0806.3242))

From here on a myriad of references followed up. Review includes:

(...)

### Equivalence to metric Lie representations

The full generalized axioms on the M2-brane 3-algebra and first insights into their relation to [[Lie algebra representations]] of [[metric Lie algebras]] is due to

* {#CherkisSaemann08} [[Sergey Cherkis]], [[Christian Saemann]], Section 4 of: _Multiple M2-branes and Generalized 3-Lie algebras_, Phys. Rev. D78:066019, 2008 ([arXiv:0807.0808](https://arxiv.org/abs/0807.0808))

The full identification of M2-brane 3-algebras with dualizable [[Lie algebra representations]] over [[metric Lie algebras]] is due to

* {#dMFFMER08} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], [[Patricia Ritter]], _On the Lie-algebraic origin of metric 3-algebras_, Commun. Math. Phys. 290:871-902, 2009 ([arXiv:0809.1086](http://arxiv.org/abs/0809.1086))

reviewed in

* [[José Figueroa-O'Farrill]], slide 145 onwards in: _Triple systems and Lie superalgebras_ in _M2-branes, ADE and Lie superalgebras_, talk at IPMU 2009 ([pdf](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/Hongo.pdf), [[FigueroaOFarrillM2Branes02.pdf:file]])

further explored in

* [[José Figueroa-O'Farrill]], _Simplicity in the Faulkner construction_,  Journal of Physics A: Mathematical and Theoretical, Volume 42, Number 44 ([arXiv:0905.4900](https://arxiv.org/abs/0905.4900))

and putting to use the [[Faulkner construction]] that was previously introduced in

* {#Faulkner73} [[John Faulkner]], _On the geometry of inner ideals_, Journal of Algebra, Volume 26, Issue 1, July 1973, Pages 1-9 (doi:10.1016/0021-8693(73)90032-X)

See also:

* Sam Palmer, [[Christian Saemann]], section 2 of _M-brane Models from Non-Abelian Gerbes_, JHEP 1207:010, 2012 ([arXiv:1203.5757](http://arxiv.org/abs/1203.5757))

* {#SaemannRitter13} [[Patricia Ritter]], [[Christian Saemann]], section 2.5 of _Lie 2-algebra models_, JHEP 04 (2014) 066 ([arXiv:1308.4892](http://arxiv.org/abs/1308.4892))

* {#Saemann16} [[Christian Saemann]], appendix A of _Lectures on Higher Structures in M-Theory_ ([arXiv:1609.09815](https://arxiv.org/abs/1609.09815))

and

* {#Palmkvist09} [[Jakob Palmkvist]], _Three-algebras, triple systems and 3-graded Lie superalgebras_, J. Phys. A43:015205, 2010 ([arXiv:0905.2468](https://arxiv.org/abs/0905.2468))

[[!redirects M-brane 3-algebras]]

[[!redirects M2-brane 3-algebra]]
[[!redirects M2-brane 3-algebras]]

[[!redirects Faulkner construction]]
[[!redirects Faulkner constructions]]

[[!redirects BLG 3-algebra]]
[[!redirects BLG 3-algebras]]


M-category

# $\mathcal{M}$-categories
* table of contents
{: toc}

## Idea

An $\mathcal{M}$-category is a [[category]] with two classes of [[morphisms]]: *tight* and *loose*.

A [[categorification]] of the concept of $\mathcal{M}$-category to $2$-[[2-category|categories]] is the concept of $\mathcal{F}$-[[F-category|category]].  (There are other possible categorifications, such as locally $\mathcal{M}$-enriched $2$-categories, or locally $\mathcal{M}$-enriched $\mathcal{F}$-categories.)  In particular, every $\mathcal{M}$-category is an $\mathcal{F}$-category with only identity $2$-[[2-morphism|morphisms]], and an $\mathcal{F}$-category becomes an $\mathcal{M}$-category upon forgetting its nonidentity $2$-cells.

## Definitions {#def}

{#mono} Let $\mathcal{M}$ (or $Mono$) be the category whose [[objects]] are [[injections]] ([[monomorphisms]] in the [[category of sets]]) and whose [[morphisms]] are [[commutative squares]]. $\mathcal{M}$ is a [[reflective subcategory]] of the [[arrow category]] $Set^\to$ (the [[Sierpinski topos]]), where the reflector [[product-preserving functor|preserves products]]; as a result, $\mathcal{M}$ is [[complete category|complete]], [[cocomplete category|cocomplete]] [[cartesian closed category]]. (In fact, $\mathcal{M}$ is a Grothendieck [[quasitopos]], as discussed [here](free+cartesian+category#Subset): it is the category of [[separated presheaves]] for the [[double negation topology]] on $Set^\to$.)

A ([[locally small category|locally small]]) __$\mathcal{M}$-category__ is simply a [[category enriched]] over $\mathcal{M}$. In other words, where the hom-objects are pairs of sets $(T, L)$ where $T \subseteq L$.

In more detail, an __$\mathcal{M}$-category__ consists of the following data:

* a [[class]] of __[[objects]]__;
* for each pair $x, y$ of objects, a [[set]] $L(x,y)$ of __loose [[morphisms]]__;
* for each pair $x, y$ of objects, a [[subset]] $T(x,y)$ of $L(x,y)$ of __tight morphisms__;
* for each object $x$, a tight __[[identity morphism]]__ $id_x$; and
* for each triple $x, y, z$ of objects, a [[binary operation]] __[[composition]]__ $\circ_{x,y,z}$ from $L(y,z)$ and $L(x,y)$ to $L(x,z)$ such that:

  * $f \circ g$ is tight whenever $f, g$ are;
  * $\id_y \circ f = f = f \circ \id_x$ whenever this makes sense; and
  * $(f \circ g) \circ h = f \circ (g \circ h)$ whenever this makes sense.

Equivalently, an __$\mathcal{M}$-category__ is a [[category]] of objects and __loose__ morphisms together with a [[wide subcategory]] of __tight__ morphisms.

Note that the "underlying ordinary category" of an $\mathcal{M}$-category, in the usual sense that that phrase is used in [[enriched category]] theory, is its category of *tight* morphisms. This is because the underlying ordinary category is induced by a monoidal change-of-base functor $\hom(I, -) \colon \mathcal{M} \to Set$ represented by the monoidal unit $I = (1, 1)$, where we have

$$\hom_{\mathcal{M}}((1, 1), (T, L)) \cong T.$$

Nevertheless, it is frequently also useful to think of the category of loose morphisms as a sort of "underlying ordinary category" of an $\mathcal{M}$-category. This is effected by the monoidal change-of-base functor $\hom_{\mathcal{M}}((0, 1), -) \colon \mathcal{M} \to Set$, where we have

$$\hom_{\mathcal{M}}((0, 1), (T, L)) \cong L.$$

## Examples

*  Every $\dagger$-[[dagger-category|category]] is an $\mathcal{M}$-category in which the tight morphisms are the [[unitary isomorphisms]].  In particular, [[Hilbert spaces]] form an $\mathcal{M}$-category with [[unitary operators]] as tight morphisms.

*  A more interesting way to make Hilbert spaces into an $\mathcal{M}$-category [[Hilb]] uses (as in the $\dagger$-category $Hilb$) all [[bounded linear operators]] as loose morphisms but only [[short linear operators]] (those with norm at most $1$) as tight morphisms.  This gives the same tight isomorphisms as the $\dagger$-category $Hilb$ (but also has non-invertible tight morphisms).

*  Similarly, [[Ban]] (the category of [[Banach spaces]]) is an $\mathcal{M}$-category with all bounded linear operators as loose morphisms but only short linear operators as tight morphisms.  In [[functional analysis]], a loose isomorphism in $Ban$ is traditionally called an '[[norm isomorphism|isomorphism]]' while a tight isomorphism is called a '[[global isometry]]'.

*  We can make [[Met]] (the category of [[metric spaces]]) into an $\mathcal{M}$-category is several ways, with [[short maps]] contained in [[Lipschitz map]]s contained in [[uniformly continuous maps]] contained in [[continuous maps]]; we can also take Lipschitz maps contained in [[bounded maps]].  (For linear operators between Banach spaces, the continuous and bounded operators are the same and are already Lipschitz.)

*  Any [[strict category]] is an $\mathcal{M}$-category with [[equality|equalities]] as the tight morphisms.  (Thus the wide subcategory of tight morphisms is [[skeletal category|skeletal]].)  In particular, the [[category of sets]] (or any category) in [[material set theory]] is an $\mathcal{M}$-category.

*  A more interesting way to make [[material set theory]]\'s [[category of sets]] into an $\mathcal{M}$-category has all [[subset]] inclusions as tight morphisms.  Again the tight isomorphisms are simply the equalities.

*  Given any category $C$ and object $a$ of $C$, the [[subobjects]] of $a$ form an $\mathcal{M}$-category whose category of loose morphisms is the [[full subcategory]] of $C$ on the subobjects of $a$ and whose category of tight morphisms is the [[subobject poset]] of $a$.

*  Similarly, [[quotient objects]] form an $\mathcal{M}$-category.

*  The __[[core]]__ of every category $C$ is an $\mathcal{M}$-category whose loose morphisms are the morphisms of $C$ and whose tight morphisms are the [[isomorphisms]] of $C$.

*  $\mathcal{M}$ itself is an $\mathcal{M}$-category whose objects are sets equipped with a specified subset, whose tight morphisms are functions which map the subset into each other, and whose loose morphisms are arbitrary functions.

* The [[Kleisli category]] for any [[monad]], whose tight morphisms are those induced from the underlying category by postcomposition with the unit.

*  Given any [[faithful functor]] $F\colon C \to D$, we may make $C$ into an $\mathcal{M}$-category whose tight morphisms are the original morphisms of $C$ and whose loose morphisms from $x$ to $y$ are the $D$-morphisms from $F(x)$ to $F(y)$.  This includes all of the examples above; up to [[equivalence of categories|equivalence]], this includes all examples.

* Let $T$ be a strict 2-monad on a strict 2-category. Then the strict $T$-algebras form an $\mathcal{M}$-category $T \mathrm{Alg}$ where tight morphisms are the strict algebra morphisms and the loose morphisms are the pseudo algebra morphsims. This is just a decategorified version of the $\mathcal{F}$-category of $T$-algebras, which also includes the 2-cells between algebras.

* More generally, any strict $\mathcal{F}$-category can be made into an $\mathcal{M}$-category by forgetting the non-identity 2-cells.

## Applications

* Any $\mathcal{M}$-category whose tight morphisms form a [[preorder]] can be made into a [[strict category]] in a canonical way: declare two objects to be equal if they are *tightly* isomorphic.  This is an unusual sort of strict category in that its "equality predicate" on objects may not be literal equality (even in a foundational system where the latter makes sense).  Many strict categories that arise in practice underlie $\mathcal{M}$-categories, such as the category of sets in material set theory.

  Note that two *equivalent* $\mathcal{M}$-categories with posetal tight categories (in the usual sense of equivalence for enriched categories) have *isomorphic* underlying strict categories (in the appropriate sense, i.e. making use of the stipulated equality predicate on objects to define "isomorphism").  In this way, some examples which may seem on the surface to be [[evil]], by referring to an isomorphism of categories, can alternatively be described non-evilly by recognizing the presence of a neglected $\mathcal{M}$-enrichment.

* For instance, let $G = Gal(E/F)$ be the [[Galois group]] of a finite [[Galois extension]] $E/F$.  Then there is an $\mathcal{M}$-category whose objects are intermediate [[fields]] $F\subset K\subset E$, whose loose maps are arbitrary field homomorphisms that fix $F$ pointwise, and whose tight maps are those which commute with the inclusions into $E$.  There is also an $\mathcal{M}$-category whose objects are [[orbits]] $G/H$, whose loose maps are arbitrary maps of $G$-sets, and whose tight maps are those which commute with the quotient maps from $G$.  The fundamental theorem of classical [[Galois theory]] says that these two $\mathcal{M}$-categories are equivalent *as $\mathcal{M}$-categories*.

  This is a stronger statement than saying that their underlying strict categories, as above, are isomorphic, which is yet stronger than saying that their underlying categories of loose maps are equivalent.  See [this post](http://article.gmane.org/gmane.science.mathematics.categories/5877) ([archive](http://web.archive.org/web/20150326125356/http://article.gmane.org/gmane.science.mathematics.categories/5877)) by [[Peter May]].

## Related Concepts

* [[relative category]]

* [[F-category]]

## References

$\mathcal{M}$-categories are mentioned as $Subset$-categories (thinking of $\mathcal{M}$ as the category of [[subset]] inclusions) in

*  [[John Power]] (2002), Premonoidal categories as categories with algebraic structure, Theoretical Computer Science 278, [pdf](http://www.inf.ed.ac.uk/publications/online/0413.pdf).

We discussed them in the [[nlabmeta:n-Forum]] as part of a discussion of the category of [[Banach spaces]]:

*  [[Mark Meckes]] et al (2011), Banach space, n-Forum, ([discussion link](https://nforum.ncatlab.org/discussion/3289/banach-space/)).

[[!redirects M-category]]
[[!redirects M-categories]]
[[!redirects Subset-category]]
[[!redirects Subset-categories]]


m-cofibrant space

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###Context###
#### Topology
+--{: .hide}
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#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
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=--

#Contents#
* automatic table of contents goes here
{:toc}

## Definition

An **m-cofibrant space** is a [[topological space]] that is [[homotopy equivalence|homotopy equivalent]] to a [[CW complex]].

## Idea

The importance of m-cofibrant spaces comes from the fact any [[weak homotopy equivalence]] between m-cofibrant spaces is a homotopy equivalence. It is a classical theorem -- the [[Whitehead theorem]] -- that this is true for CW complexes, and it is easy to see that it is a property preserved under homotopy equivalence.  In fact, it is easy to see that m-cofibrant spaces are the largest subcategory $C$ of $Top$ which includes all CW complexes and such that any weak homotopy equivalence between objects of $C$ is a homotopy equivalence.

Moreover, since any space is weakly equivalent to a CW complex, any space is _a fortiori_ weakly equivalent to an m-cofibrant one.  It follows that the [[homotopy category]] of topological spaces in which the weak homotopy equivalences are inverted is equivalent to the homotopy category of m-cofibrant spaces in which homotopic maps are identified.

Therefore, m-cofibrant spaces are a natural place to do [[homotopy theory]] if one is interested in spaces up to weak homotopy type, but doesn't want to have to worry about the distinction between weak homotopy equivalence and homotopy equivalence.  For instance, when higher category theorists say that an $\infty$-[[infinity-groupoid|groupoid]] "is" a topological space, they generally mean an m-cofibrant space, so that $\infty$-functors, natural transformations, and so on can be identified with continuous maps, homotopies, etc.

Milnor, in particular, has argued that the category of m-cofibrant spaces is a convenient category of spaces for algebraic topology.  It is not [[cartesian closed category|cartesian closed]], but Milnor proved that if $Y$ is m-cofibrant and $X$ is [[compact Hausdorff space|compact Hausdorff]], then $Y^X$ is also m-cofibrant.  Hence many desirable constructions are still available, in particular finite and infinite [[loop space]]s.

More recently it has been discovered that there is a [[model category|model structure]] on [[Top]], called the _mixed model structure_, in which the weak equivalences are the weak homotopy equivalences, the cofibrant objects are precisely the m-cofibrant spaces, and the fibrations are the Hurewicz fibrations.  This is where the name "m-cofibrant" comes from; "m-" is for the "mixed" model structure, as contrasted with the "q-" or Quillen model structure and the "h-" or Hurewicz/Strøm model structure (see [[model structure on topological spaces]]).

[[!redirects m-cofibrant spaces]]


M-complete category

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### Context
#### Category theory
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#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

* table of contents
{:toc}

## Idea

A [[category]] is $M$-complete or $E$-cocomplete if has certain [[limits]] or [[colimits]] of [[morphisms]] in a given [[class]] of [[diagrams]] $M$ or $E$.

Not to be confused with an [[M-category]].

## Definitions

Let $C$ be a [[category]] and let $M$ be a class of [[monomorphisms]] in $C$.  (Often, $M$ will be the right class in an [[orthogonal factorization system]].)  We say that $C$ is **$M$-complete** if it admits all (even [[large category|large]]) [[intersections]] of $M$-[[subobjects]].  This means that it admits all (even large) [[wide pullbacks]] of families of $M$-morphisms, and such pullbacks are again in $M$.  (If $M$ is the right class of an OFS, then any intersection of $M$-morphisms which exists is automatically in $M$.)

If $M$ is the class of all monomorphisms, we may say **mono-complete** for $M$-complete.

Dually, if $E$ is a class of [[epimorphisms]], we say $C$ is **$E$-cocomplete** if it admits all [[cointersection]]s of $E$-morphisms, and **epi-cocomplete** if $E$ is the class of all epimorphisms.

## Examples

* If $C$ is $M$-[[well-powered category|well-powered]], then no large limits are required in the definition of $M$-completeness.  Therefore, if $C$ is well-powered and [[complete category|complete]], it is $M$-complete whenever $M$ is the right class in an OFS.  Dually, if $C$ is well-copowered and cocomplete, it is $E$-cocomplete whenever $E$ is the left class in an OFS.

* For similar reasons, the category [[FinSet]] is mono-complete and epi-cocomplete---although it is not complete or cocomplete, it is [[finitely complete category|finitely complete]] and [[finitely cocomplete category|cocomplete]], and its subobject lattices and quotient lattices are likewise [[essentially finite category|essentially finite]].

* If $C$ is a [[topological concrete category]] over a category $D$ which is mono-complete or epi-complete, then $C$ is also mono-complete or epi-complete.  For the faithful forgetful functor $U\colon C\to D$ preserves and reflects monos and epis, and so the [[weak structure|initial]] $C$-structure on an [[intersection]] of underlying monos in $D$ gives an intersection in $C$ and the [[strong structure|final]] $C$-structure on a [[cointersection]] of underlying epis in $D$ gives a cointersection in $C$.

## Construction of factorization systems

$M$-completeness is useful for constructing [[orthogonal factorization systems]].  The following is Lemma 3.1 in [CHK](#CHK).

+-- {: .num_theorem #ConstructingOFS}
###### Theorem
Let $M$ be a class of maps in a category $C$, and assume that

1. $M$ consists of monomorphisms,
2. $M$ is closed under [[composition]],
3. all [[pullbacks]] of $M$-morphisms exist in $C$ and are again in $M$, and
4. $C$ is $M$-complete.

Then there is an orthogonal factorization system $(E,M)$, with $E = {}^\perp M$.
=--
+-- {: .proof}
###### Proof
Given $f\colon A\to B$, let $m$ be the intersection of all $M$-morphisms $n\colon X \to B$ through which $f$ factors.  Then by the universal property of this intersection, we have $f = m e$ for some $e$; thus it suffices to show $e\in E$.  Suppose given a commutative square
$$\array{ A & \overset{g}{\to} & Z \\
  ^e \downarrow & & \downarrow^p \\
  Y & \underset{h}{\to} & W} $$
with $p\in M$.  By pulling $p$ back to $Y$ (since pullbacks of $M$-morphisms exist), we may assume that $Y=W$ and $h$ is the identity.  But now the composite $m p$ is an $M$-morphism through which $f$ factors, so by definition, $m$ factors through it.  Thus $p$ is an isomorphism and so the lifting problem can be solved.
=--

In fact, it is easy to see that the same proof constructs a [[factorization structure for sinks]].

Note that if $M$ is already part of a [[prefactorization system]], then any composite, pullback, or intersection of $M$-morphisms which exists is automatically also in $M$, since $M = E^\perp$.

+-- {: .num_cor}
###### Corollary
Let $(E,M)$ be a [[prefactorization system]] on a category $C$, and assume that

1. $M$ consists of monomorphisms,
2. All pullbacks of $M$-morphisms exist in $C$, and
3. $C$ is $M$-complete.

Then $(E,M)$ is an orthogonal factorization system.
=--

The following is a slight generalization of Theorem 3.3 of [CHK](#CHK).  There it is stated only for the case $M=$ [[strong monomorphisms]], in which case a finitely complete and $M$-complete category is called **finitely well-complete**.

+-- {: .num_theorem #OFSFromAdjunction}
###### Theorem
Let $S : A \rightleftarrows C : T$ be an [[adjunction]], and assume that $A$ is finitely complete and $M$-complete for some OFS $(E,M)$, where $M$ consists of monomorphisms and contains the split monics.  Define $E_S$ to be the class of maps inverted by $S$, and $M_S = (E_S)^\perp$; then $(E_S,M_S)$ is an OFS on $A$.
=--
+-- {: .proof}
###### Proof
First of all, since $M_S$ belongs to a prefactorization system, it is closed under composites, pullbacks, and any intersections which exist.  Therefore, if we define $M' \coloneqq M \cap M_S$, then $M'$ satisfies the hypotheses of Theorem \ref{ConstructingOFS}, and so we have an OFS $(E',M')$.

Moreover, it is useful to notice that $E_S={}^\perp T(\hom(C))$: this is an easy consequence of the fact that if $S\dashv T$, then $S a\perp b\iff a\perp T b$, since $f\perp T u\iff S f\perp u$ for each $u\in\hom(C)$, so that $S f $ is an isomorphism.

Now suppose given $f\colon A\to B$; we want to construct an $(E_S,M_S)$-factorization.  Let $v$ be the pullback of $T S f$ along the [[unit of an adjunction|unit]] $\eta_B \colon B \to T S B$.  The naturality square for $\eta$ at $f$ shows that $f$ factors through $v$, say $f = v w$.
$$\begin{array}{ccccc}
A & \\
 &\overset{w}\searrow\\
 && P &\overset{u}\to& T S A \\
 && {}^v\downarrow  && \downarrow^{T S f}\\
 && B &\underset{\eta_B}\to& T S B
\end{array}
$$
Since $T S f$ is evidently in $M_S=({}^\perp T(\hom(C)))^\perp\supseteq T(\hom (C))$, so is $v$; thus it suffices to find an $(E_S,M_S)$-factorization of $w$.

Let $w = n g$ be the $(E',M')$-factorization of $w$.  Since $M' \subseteq M_S$, it suffices to show that $g\in E_S$.  Note also that since $w$ is a first factor of the unit $\eta_A$, by passing to adjuncts we find that $S w$ is [[split monic]]: in the former diagram we have $u w=\eta_A$, so that the adjunct $\epsilon_{S A} \cdot S u\cdot  S n\cdot  S g=1$, hence also $S g$ is a split monic.  But $T S g$ is then also split monic, hence belongs to $M$ and thus also to $M'$ (since it obviously belong to $M_S=({}^\perp T(\hom(C)))^\perp\supseteq T(\hom (C))$).  Therefore, since $g\in E'$, the naturality square for $\eta$ at $g$ contains a lift: there is an $\alpha\colon X\to T S A$ such that in the diagram
$$\begin{array}{ccc}
A & \overset{\eta_A}\to & T S A \\
{}^g\downarrow && \downarrow^{T S g} \\
X &\overset{\eta_X}\to& T S X
\end{array}$$
$\alpha\cdot g=\eta_A$ and $T S g \cdot \alpha=\eta_X$. Passing to adjuncts again, we find that $S g$ is also [[split epic]], since we can consider the diagram
$$\begin{array}{ccccc}
S A & \overset{S\eta_A}\longrightarrow & S T S A &\overset{\epsilon_{S A}}\longrightarrow & S A\\
{}^{S g}\downarrow && \phantom{aaa}\downarrow_{S T S g} &&\downarrow^{S g}\\
S X &\underset{S\eta_X}\longrightarrow & S T S X &\underset{\epsilon_{S X}}\longrightarrow & S X
\end{array}$$
and the commutativity
$$
S g \cdot \epsilon_{S A} \cdot S\alpha = \epsilon_{S X} \cdot  S T S g \cdot S\alpha = \epsilon_{S X}\cdot S\eta_X = 1
$$
Hence $S g$ is an isomorphism; thus $g\in E_S$ as desired.
=--

This is useful in the construction of [[reflective factorization systems]].

## Related concepts

* [[complete category]], [[cocomplete category]]

* [[bicomplete category]]

* [[locally bounded category]]

* [[ideal completion]]

## References

* {#CHK} Cassidy and Hébert and [[Max Kelly|Kelly]]. _Reflective subcategories, localizations, and factorization systems_.  *J. Austral. Math Soc. (Series A)* 38 (1985), 287--329

[[!redirects M-complete category]]
[[!redirects M-complete categories]]
[[!redirects M-limit]]
[[!redirects M-limits]]

[[!redirects mono-complete category]]
[[!redirects mono-complete categories]]

[[!redirects E-cocomplete category]]
[[!redirects E-cocomplete categories]]
[[!redirects E-colimit]]
[[!redirects E-colimits]]

[[!redirects epi-cocomplete category]]
[[!redirects epi-cocomplete categories]]

[[!redirects finitely well-complete category]]
[[!redirects finitely well-complete categories]]


M-cotopos

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[[!include category theory - contents]]
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#### Topos Theory
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[[!include topos theory - contents]]
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# Contents
* table of contents
{:toc}

## Idea

A [[cotopos]] with all [[M-types]]

## See also

* [[M-type]]

* [[cotopos]]


M-matrix

## Definition

An $M$-matrix is a real matrix with non-positive off-diagonal entries and positive principal minors.

## Literature

Treatment of [[Coxeter group]]s of finite rank via M-matrices

* Robert B. Howlett, _Coxeter groups and M-matrices_, Bull. London Math. Soc. __14__:2 (1982) 137-141 [doi](https://doi.org/10.1112/blms/14.2.137)

A generalization of M-matrices in the setting of proper cones

* Ronald J. Stem, Michael Tsatsomeros, _Extended M-matrices and subtangentiality_, [pdf](https://www.math.wsu.edu/faculty/tsat/files/extmmatr.pdf)

[[!redirects M-matrices]]


M-string

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#### String theory
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[[!include string theory - contents]]
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#### Physics
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[[!include physicscontents]]
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#Contents#
* table of contents
{:toc}

## Idea

The [[brane intersection]] of an [[M2-brane]] with an [[M5-brane]] (i.e. a [[self-dual string]] in the M5's [[worldvolume]] [[D=6 N=(2,0) SCFT]] is called an _M-string_ if the _other_ end of the [[M2-brane]] intersects another, parallel, [[M5-brane]].

> graphics grabbed from [HLV 14](E-string#HLV14)

In contrast, if the other end of the M2 intersects an [[MO9-plane]], then the former intersection is an _[[E-string]]_.

## Properties

### M-string elliptic genus

See at _[[M-string elliptic genus]]_.

## Related concepts

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

## References

### General

* [[Babak Haghighat]], [[Amer Iqbal]], [[Can Kozcaz]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _M-Strings_, Commun. Math. Phys. 334, 779–842 (2015) ([arXiv:1305.6322](https://arxiv.org/abs/1305.6322), [doi:10.1007/s00220-014-2139-1](https://doi.org/10.1007/s00220-014-2139-1))

* [[Babak Haghighat]], [[Can Kozcaz]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _On orbifolds of M-Strings_, Physical Review D 89.4 (2014): 046003 ([arXiv:1310.1185](https://arxiv.org/abs/1310.1185))

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

[[!redirects M-strings]]


M-theory

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

## Idea

There are various hints (originally observed in [Witten 95](#Witten95)) that all [[perturbation theory|perturbative]] [[superstring theories]] ([[type II string theory|type II]] (A and B), [[type I string theory|type I]], [[heterotic string theory|heterotic]] ($SO(32)$ and $E_8 \times E_8$)) have a joint [[coupling constant|strong coupling]] [[non-perturbative quantum field theory|non-perturbative]] limit whose low energy [[effective field theory]] description is [[11-dimensional supergravity]] and which reduces to the various string theories by [[Kaluza-Klein compactification]] on an [[orientifold]] torus bundle, followed by various [[duality in string theory|string dualities]]. Since the string itself is thought to arise from a [[membrane]]/[[M2-brane]] in 11-dimensions after [[double dimensional reduction]] this hypothetical theory has been called "M-theory" short for "membrane theory"; e.g. in [Horava-Witten 95](#HoravaWitten95):

> As it has been proposed that the eleven-dimensional theory is a supermembrane theory but there are some reasons to doubt that interpretation, we will non-committally call it the _M-theory_, leaving to the future the relation of M to membranes.

The "reasons to doubt" that interpretation is that the [[M2-brane]] does not support a [[perturbation theory]] the way that the [[superstring]] does. This is part of the reason why the actual nature of "M-theory" remains mysterious. On the other hand, later it was argued that there is a regularization of the M2-brane worldvolume theory, which makes it becomes the [[BFSS matrix model]] ([Nicolai-Helling 98](#NicolaiHelling98), [Dasgupta-Nicolai-Plefka 02](#DasguptaNicolaiPlefka02)). In reaction to these developments it was suggested that "M-theory" could be read as "matrix theory".

> {#MagicMysteryQuote} Later, the membranes were interpreted in terms of matrices. Purely by chance, the word "matrix" also starts with "m", so for a while I would say that the M stands for magic, mystery, or matrix. ([Witten 2014, last paragraph](#Witten14))

The defining characteristic of M-theory is that it exhibits [[duality between M-theory and type IIA string theory|duality with type IIA string theory]] in the following way:

\[
  \label{MTheoryDiagram}
  \array{
      M-Theory(?) &\stackrel{low\;energy\;limit}{\to}& 11d Supergravity
      \\
      {}^{\mathllap{ \array{ small \\ coupling \\ limit}}}
      \Bigg\downarrow
      &&
      \Bigg\downarrow
      {}^{\mathrlap{ \array{KK\;reduction \\ on S^1 }}}
      \\
      type IIA string theory
        &\stackrel{low\;energy\;effective\;QFT}{\to}&
      10d Supergravity
  }
\]

(see also e.g. ([Obers-Pioline 98, p. 12](#ObersPioline98))). The unknown top left corner here has optimistically been given a name, and that is "M-theory". But even the rough global structure of the top left corner has remained elusive.

> We still have no fundamental formulation of "[[M-theory]]" - the hypothetical theory of which [[11-dimensional supergravity]] and the five [[string theories]] are all special limiting cases. Work on formulating the fundamental principles underlying M-theory has noticeably waned. $[...]$. If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of M-theory is surely going to lead to new and novel mathematics. Regrettably, it is a problem the community seems to have put aside - temporarily. But, ultimately, Physical Mathematics must return to this grand issue.

> ([Moore 14, section 12](Physical+Mathematics+and+the+Future#Moore14))

## Hints for M-theory
 {#Hints}

The available evidence that there is something like M-theory consists of various facets in the bottom left and the top right entry of the above diagram (eq:MTheoryDiagram) that seem to have a common origin in the top left corner.

### Membranes

Notably, from the [[black brane]]-solution structure in [[11-dimensional supergravity]] and from the [[brane scan]] one finds that it contains a 2-[[brane]], called the _[[M2-brane]]_, and to the extent that one has this under control one can show that under "double dimensional reduction" this becomes the [[string]]. However, it is clear that this cannot quite give a definition of the top left corner by [[perturbation theory]] as the [[superstring]] [[sigma-model]] does for the bottom left corner. At the bottom of it,  this is simply because, by the very nature of the conjecture, the top left corner is supposed to be given by a non-perturbative strong-coupling limit of the bottom left corner. But one may also see that the evident guess for a would-be membrane analog of the [[string perturbation series]] fails

> {#WittenOnProblemsOfMembraneTheory2014} [[Mike Duff]], [[Paul Townsend]], and other physicists working on [[supermembranes]] had spent a couple of years in the mid-1980s saying that there should be a theory of fundamental membranes analogous to the theory of fundamental strings. That wasn't convincing for a large number of reasons. For one thing, a three-manifold doesn't have an [[Euler characteristic]], so there isn't a topological expansion as there is in string theory. Moreover, in three dimensions there is no [[conformal invariance]] to help us make sense of membrane theory; membrane theory is [[renormalization|nonrenormalizable]] just like [[general relativity]].

> ([[Edward Witten]] in interview by [[Hirosi Ooguri]], Notices
Amer. Math. Soc, May 2015 p491 ([pdf](http://www.ams.org/notices/201505/rnoti-p491.pdf)))

This issue is the very root of the abbreviation "M-theory":

> As it has been proposed that the eleven-dimensional theory is a supermembrane theory but there are some reasons to doubt that interpretation, we will non-committally call it the _M-theory_, leaving to the future the relation of M to membranes. ([Horava-Witten 95](#HoravaWitten95))

> {#Witten14LastParagraph} M-theory was meant as a temporary name pending a better understanding. Some colleagues thought that the theory should be understood as a membrane theory. Though I was skeptical, I decided to keep the letter "m" from "membrane" and call the theory M--theory, with time to tell whether the M stands for magic, mystery, or membrane. Later, the membranes were interpreted in terms of matrices. Purely by chance, the word "matrix" also starts with "m", so for a while I would say that the M stands for magic, mystery, or matrix. ([Witten 14, last paragraph](#Witten14))

### Strongly coupled type IIA strings and D$0$-branes
 {#StronglyCoupledTypeIIAAndD0Branes}

There is a bunch of consistency checks on the statement that the [[KK-compactification]] of [[11-dimensional supergravity]] on a circle gives the [[non-perturbative quantum field theory|strong coupling]] refinement of [[type IIA string theory]]. See at _[[duality between M-theory and type IIA string theory]]_.

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

Discussion of the relation of [[gauge enhancement]] of M-theory at [[ADE singularities]] and the corresponding coincident [[D-brane]] geometries in type IIA string theory is in ([Sen 97](#Sen97)).

More on the decomposition of the [[nLab:supergravity C-field]] in 11d to the [[RR-fields]] and the NS-fields in type IIA is in ([Mathai-Sati 03, section 4](#MathaiSati03)).

For survey of how the components maps see also the table at _[Relation to F-theory](#RelationToFTheory)_.

### U-duality

Another hint comes from the fact that the [[U-duality]]-structure of [[supergravity]] theories forms a clear pattern in those dimensions where one understands it well, giving rise to a description of higher dimensional supergravity theories by [[exceptional generalized geometry]]. Now, this pattern, as a mathematical pattern, can be continued to the case that would correspond to the top left corner above, by passing to [[exceptional generalized geometry]] over _hyperbolic_ [[Kac-Moody Lie algebras]] such as first [[E10]] and then, ultimately [[E11]]. The references there show that these are huge algebraic structures inside which people incrementally find all kinds of relations that are naturally identified with various aspects of M-theory. This leads to the conjecture that M-theory somehow _is_ $E_{11}$ in some way. But it all remains rather mysterious at the moment.

### Relation to F-theory
 {#RelationToFTheory}

The [[KK-compactification|compactification]] of M-theory on a torus yields [[type II string theory]] -- directly type IIA, and then type IIB after [[T-duality|T-dualizing]]. It turns out that the [[axio-dilaton]] of the resulting type II-B string theory is equivalently the [[complex structure]]-[[moduli|modulus]] of this [[elliptic fibration]] by the compactification torus. This gives a description of [[non-perturbative quantum field theory|non-perturbative]] aspects of type II which has come to be known as _[[F-theory]]_ (see e.g. [Johnson 97](#Johnson97)).

In slightly more detail, write, topologically, $T^2 = S^1_A\times S^1_B$ for the compactification torus of M-theory, where contracting the first $S^1_A$-factor means passing to type IIA. To obtain type IIB in noncompact 10 dimensions from M-theory,  also the second $S^1_B$ is to be compactified (since [[T-duality]] sends the radius $r_A$ of $S^1_A$ to the inverse radius $r_B = \ell_s^2 / R_A$ of $S^1_B$).
Therefore type IIB sugra in $d = 10$ is obtained from 11d sugra compactified on the [[torus]] $S^1_A \times S^1_B$. More generally, this torus may be taken to be an [[elliptic curve]] and this may vary over the 9d base space as an [[elliptic fibration]].

Applying T-duality to one of the compact direction yields a 10-dimensional theory which may now be thought of as encoded by a 12-dimensional elliptic fibration. This 12d elliptic fibration encoding a 10d type II supergravity [[vacuum]] is the input data that [[F-theory]] is concerned with.

A schematic depiction of this story is the following:

|  |  |  |
|--|--|--|
| [[11d supergravity|M-theory]] in $d = 11$ | | F-theory in $d = 12$ |
| $\downarrow$ [[Kaluza-Klein mechanism|KK-reduction]] along [[elliptic fibration]] | | $\downarrow$ [[axio-dilaton]] is modulus of [[elliptic fibration]] |
| [[type IIA string theory|IIA string theory]] in $d = 9$ | $\leftarrow$[[T-duality]]$\rightarrow$ | [[type IIB string theory|IIB string theory]] in $d = 10$ |

In the simple case where the elliptic fiber is indeed just $S^1_A \times S^1_B$, the [[imaginary part]] of its complex modulus is

$$
  Im(\tau) = \frac{R_A}{R_B}
  \,.
$$

By following through the above diagram, one finds how this determines the [[coupling constant]] in the [[type II string theory]]:

First, the [[KK-compactification]] of M-theory on $S^1_A$ yields a type IIA [[string coupling constant]]

$$
  g_{IIA} = \frac{R_A}{\ell_s}
  \,.
$$

Then the T-duality operation along $S^1_B$ divides this by $R_B$:

$$
  \begin{aligned}
    g_{IIB} & = g_{IIA} \frac{\ell_s}{R_B}
    \\
    & = \frac{R_A}{R_B}
    \\
    & = Im(\tau)
   \end{aligned}
   \,.
$$

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

### Cohomological properties

A derivation of [[D-brane charge]], [[RR-fields]] and other [[K-theory]] structure in [[type II superstring theory]] from M-theory was argued in ([FMW 00](#FMW00)).

See also at _[[cubical structure in M-theory]]_.

### Kaluza-Klein compactifications

[[!include KK-compactifications of M-theory -- table]]

### Via AdS/CFT

The [[AdS-CFT duality]] for the [[black brane|black]][[M5-brane]] may be turned around to deduce from the [[6d (2,0)-superconformal QFT]] on the [[M5-brane]] [[scattering amplitudes]] in the 11-dimensional [[bulk]]-spacetime, hence in putative M-theory. While the [[6d (2,0)-superconformal QFT]] is not completely known, [[conformal invariance]] and [[supersymmetry]] tightly constrains it ("[[conformal bootstrap]]") and does allow to extract results.

This approach to computing putative M-theory scattering amplitudes is due to  ([ChesterPerlmutter18](#ChesterPerlmutter18)).

### More

(...)

## The open problem of formulating M-theory
 {#TheOpenProblem}

The tight web of hints and plausiblity checks notwithstanding, an actual formulation of M-theory as an actual [[theory (physics)|theory]] remains an open problem.

This is not outrageous in itself: In [[mathematics]] there are good examples of cases where a collection of situations was or is suspected to be limiting cases of a single unified theory, without that theory itself having been or being known.

One example of this is the putative theory "absolute geometry over the [[field with one element]]". In this [[analogy]], the various [[perturbative string theories]] ([[heterotic string theory|HET]], [[type I string theory|I]], [[type IIA string theory|IIA]], [[type IIB string theory|IIB]] and their [[KK-compactifications]]) correspond to [[arithmetic geometry|arithmetic geometries]] over [[ground field|base]] [[prime field]] $\mathbb{F}_p$ for $p \geq 2$, and the would-be M-theory corresponds to a  theory of a "[[field with one element]]" that unifies all this, by describing it at a deeper level (literally: a deeper base).

On the other hand, parts of the physics-minded literature tends to forget or downplay the [[conjecture|conjectural]] nature of many assumptions or leaps of faiths that are being made when it comes to discussion of [[D-brane]]/[[M-brane]] dynamics and generally of [[non-perturbative effects]] in string theory.

The following is a collection of quotes from authors that highlight the open problem:

### General
 {#OpenProblemGeneral}

{#Witten1988Quote} [Witten 1988](Edward+Witten#DavisBrown88) (before the notion of M-theory was formulated):

> String theory at its finest is, or should be, a new branch of geometry. $[\ldots]$ I would consider trying to elucidate this proper generalization of geometry as the central problem of physics, certainly the central problem of string theory. $[\ldots]$ What should have happened, by rights, is that the  correct mathematical structures should have been developed in the twenty-first or twenty-second century, and then finally physicists should have invented string theory as a physical theory that is made possible by these structures. $[\ldots]$ we are paying the price for the fact that we didn't come by this thing in the usual way.

\linebreak

[Duff 1996, totality of Section 6](#Duff96) (right after the idea of M-theory was formulated):

> The overriding problem in superunification in the coming years will be to take the Mystery out of M-theory, while keeping the Magic and the Membranes.

\linebreak

[Howe-Lambert-West 97, p. 2](DpDp2-brane+bound+state#HoweLambertWest97):

> the still rather mysterious M theory governs many aspects of the lower dimensional string theories.  What little is known of M theory is powerful enough to lead us to new phenomena in string theory and indeed new string theories.  In particular the M theory fivebrane is strongly believed hold a new kind of self-dual string theory on its worldvolume. This new and somewhat elusive theory has also appeared in other contexts such as the compactification of type IIB string theory on K3 , M(atrix) theory on $\mathbb{T}^5$ and the S-duality of $N = 4$, $D = 4$ super-Yang-Mills. Thus one may hope that a greater understanding of this self-dual string may lead directly to a greater understanding of duality, string theory and M theory.

\linebreak

[Duff 98a, last paragraph (p. 6)](#Duff98):

> Despite all these successes, physicists are glimpsing only small corners of M-theory; the big picture is still lacking.

> $[...]$

> Indeed future historians may judge the late 20th century as a time when theorists were like  children  playing  on  the  seashore, diverting themselves with the smoother pebbles or prettier shells of superstrings while the great ocean of M-theory lay undiscovered before them.

\linebreak

[Duff 98b, p. 6](#Duff98b):

> we are only just beginning to scratch the surface of the ultimate meaning of M-theory, and for the time being therefore, M stands for Magic and Mystery too.

\linebreak

[Nicolai-Helling 98, p. 2](BFSS+matrix+model#NicolaiHelling98):

> Despite the recent excitement, however, we do not think that [[BFSS matrix model|M(atrix) theory]] and the [[M2-brane|d= 11 supermembrane]] in their present incarnation are already the final answer in the search for [[M-theory|M-Theory]], even though they probably are important pieces of the puzzle. There are still too many ingredients missing that we would expect the final theory to possess. For one thing, we would expect a true theory of quantum gravity to exhibit certain pregeometrical features corresponding to a “dissolution” of space-time and the emergence of some kind of non-commutative geometry at short distances; although the matrix model does achieve that to some extent by replacing commuting coordinates by non-commuting matrices, it seems to us that a still more radical departure from conventional ideas about space and time may be required in order to arrive at a truly background independent formulation (the matrix model “lives” in nine _flat_ transverse dimensions only). Furthermore, there should exist some huge and so far completely hidden symmetries generalizing not only the duality symmetries of extended supergravity and string theory, but also the principles underlying general relativity.

\linebreak

[Duff 99, p.330](#Duff99):

> future historians may judge the period 1984-95 as a time when theorists were like boys playing by the sea shore, and diverting themselves with the smoother pebbles or prettier shells of perturbative ten-dimensiorial superstrings while the great ocean of non-perturbative eleven-dimensional M-theory lay all undiscovered before them.

\linebreak

{#CGNT04} [Cederwall-Gran-Nilsson-Tsimpis 04](higher+curvature+correction#CGNT04):

> Our understanding of M-theory is still very limited, mainly due to the lack of powerful methods to probe it at the microscopic level. One approach to encoding information about M-theory is through its low energy [[effective field theory]]. $[...]$ The ultimate goal is to be able to derive the [[higher curvature corrections|higher derivative corrections]], e.g. by means of a microscopic version of M-theory. Since this is not yet possible, our aim here is instead to solve the [[superspace]] [[Bianchi identities]] in order to obtain the most general form such correction terms can take restricted only by [[supersymmetry]] and [[Lorentz group|Lorentz invariance]] in eleven dimensions. To what extent such an approach can capture main features of M-theory is an interesting question to which we have no answer at this point.

\linebreak

{#Moore14} [[Physical Mathematics and the Future|Moore 14, pages 43-44]]:

>  We still have no fundamental formulation of "[[M-theory]]" - the hypothetical theory of which [[11-dimensional supergravity]] and the five [[string theories]] are all special limiting cases. Work on formulating the fundamental principles underlying M-theory has noticeably waned.

> {#AGoodStartWasGivenByTheMatrixTheory} A good start was given by the [[BFSS matrix model|Matrix theory]] approach of [[Tom Banks|Banks]], [[Willy Fischler|Fischler]], [[Stephen Shenker|Shenker]] and [[Leonard Susskind|Susskind]]. We have every reason to expect that this theory produces the correct [[scattering amplitudes]] of modes in the [[11-dimensional supergravity]] multiplet in 11-dimensional [[Minkowski space]] - even at energies sufficiently large that [[black holes]] should be created. (This latter phenomenon has never been explicitly demonstrated). But [[BFSS matrix model|Matrix theory]] is only a beginning and does not give us the whole picture of [[M-theory]]. The program ran into increasing technical difficulties when more complicated [[KK-compactification|compactifications]] were investigated. (For example, compactification on a six-dimensional torus is not very well understood at all. $[...]$). Moreover, to my mind, as it has thus far been practiced it has an important flaw: It has not led to much significant new mathematics.

\linebreak

[Chester-Perlmutter 18, p. 2](conformal+bootstrap#ChesterPerlmutter18):

> given our utter lack of a complete description of [[M-theory]], the [[bulk]] is  not terribly useful for determining finite aspects of the [[AdS/CFT|dual CFT]]. However, we can turn this problem around using the modern perspective of the [[conformal bootstrap]], which gives an a priori independent formulation of the (local sector of the) CFT. This provides an independent tool for constructing M-theory at the non-perturbative level, a philosophy that we will substantiate in this work.

\linebreak

{#Witten19} [Witten 19, USN Interview](Edward+Witten#WhatIsMissing):

> I actually believe that string/M-theory is on the right track toward a deeper explanation.  But at a very fundamental level it’s not well understood. And I’m not even confident that we have a good concept of what sort of thing is  missing or where to find it. The reason I am not is that in hindsight it is clear the view given in the 1980s of what is missing was too narrow. Instead of discovering what we thought was missing, we broadened the picture in the 90s, in unexpected directions.

\linebreak

{#Duff19} [Duff 19, USN interview](Michael+Duff#WeDontKnowWhatItIs):

> The problem we face is that we have a patchwork understanding of M-theory, like a quilt. We understand this corner and that corner, but what's lacking is the overarching big picture. So directly or indirectly, my research hopes to explain what M-theory really is. We don't know what it is.

> In a certain sense, and this is not a popular statement, I think it's premature to be asking: "What are the empirical consequences", because it's not yet in a mature enough state, where we can sensibly make falsifiable prediction.

\linebreak

{#Duff20} [[Duff interview at M-Theory-Mathematics 2020]]:

> (06:59) But the matrix model itself was not all of M-theory; it was a corner of M-theory, and it told us certain interesting things, but there were interesting things about M-theory that it didn't tell us.

> (07:13) I think we are still looking, in fact, for what M-theory really is.

> (07:19) We have a patchwork picture. We understand various corners. But the overarching big picture of M-theory is still waiting to be discovered, in my view.

> (08:12) M-theory in 1995 was very promising, and it's taught us a lot about the fundamental interactions; but the final theory is still not with us.

> (12:46) I wouldn't like to predict what the ultimate picture of M-theory will be; I imagine it will be something quite different from what we can imagine now.

> (16:36) That's why I think M-theory is not yet in a mature enough stage for us to make falsifiable predictions.

> (16:44) We don't understand the theory sufficiently well yet to do so.

\linebreak

{#WittenInterviewByZierler} [Witten, AIP interview 2022](https://www.aip.org/history-programs/niels-bohr-library/oral-histories/46968):

> String theory and M-theory have always been different. From the beginning, they were discovered by people who discovered formulas or bits and pieces of the theory without understanding what’s behind it at a more fundamental level.

> And what we understand now, even today, is extremely fragmentary, and I'm sure very superficial compared to what the real theory is. That’s the problem with the claim that supposedly I invented M-theory. It would make at least as much sense to say that M-theory hasn’t been invented yet. And you could also claim it had been invented before by other people.

\linebreak

### Non-abelian DBI-action
 {#OpenProblemNonabelianDBI}

A key ingredient of [[M-theory]] is supposedly the physics of [[intersecting branes]] with [[gauge enhancement]] in their [[worldvolume]] [[super Yang-Mills theory]]/[[DBI field theory]]. But the actual derivation or even formulation of the expected [[non-abelian DBI action]] remains open:

\linebreak

[Taylor-VanRaamsdonk 99, p. 1](Dirac-Born-Infeld+action#TaylorVanRaamsdonk99):

> For a system of multiple D$p$-branes, the world-volume action is much less well understood than for a single brane. The extension of $[$  the [[super Yang-Mills theory|super Yang-Mills action]] $]$ to a full super-symmetric or [[kappa-symmetry|κ-symmetric]] [[nonabelian Born-Infeld action]] is not known.

> $[$the proposal by [Tseytlin 97](Dirac-Born-Infeld+action#Tseytlin97)$]$ has not yet been derived from any more fundamental principles

\linebreak

[Schwarz 01, Section 2](Dirac-Born-Infeld+action#Schwarz01):

> The explicit construction of such an action is a difficult problem that has been studied extensively (starting with [17]), but is not yet completely settled.

\linebreak

[Chemissany 04, p. 5 (7 of 84)](Dirac-Born-Infeld+action#Chemissany04)

> Several attempts to generalize the Born-Infeld action describing one D-brane to non-Abelian action describing a stack of them have been made. The proper (perhaps closed) form of it is however not known up to date.

### M5-Brane anomaly cancellation
 {#OpenProblemM5BraneAnomalyCancellation}

Another key ingredient of M-theory is the [[M5-brane]]. The argument for [[anomaly cancellation]] has a convoluted history (see [there](M5-brane#AnomalyCancellation)).

[Harvey 05, p. 46](#Harvey05):

> $[$...$]$ the solution is not so clear. $[$The established procedure$]$ will not work for the M5-brane. $[$...$]$ something new is required. What this something new is, is not a priori obvious. $[$...$]$ $[$This is$]$ a daunting task. To my knowledge no serious attempts have been made
to study the problem.

> $[$The proposal of [FHMM 98](M5-brane#FreedHarveyMinasianMoore98)$]$ probably should not be viewed as a final understanding of the
problem. One would eventually hope for a microscopic formulation of M-theory which makes some of the manipulations $[$proposed in [FHMM 98](M5-brane#FreedHarveyMinasianMoore98)$]$ appear more natural.

\linebreak

### Coincident M5-branes
 {#OpenProblemCoincidentM5Branes}

Formulating the [[D=6 N=(2,0) SCFT]] expected on [[coincident brane|coincident]] [[M5-branes]] (and thus the [[gauge enhancement]] on [[coincident brane|coincident]] [[D4-branes]] under [[duality between M-theory and type IIA string theory]]) remains an open problem:

[Moore 12, p. 77](6dN20SCFT#Moore12):

> The IR dynamics of $N$ coincident M5 branes $[...]$ remains largely mysterious.

[Lambert 12, p. 49](BLG+model#Lambert12):

> The M5-brane theory remains an important open problem.

[Hu 13, p. 1](6dN20SCFT#Hu13):

> The low energy effective theory on multiple M2 branes has been well understood in recent years.  However, the low energy effective theory on multiple M5 branes is still an open problem.

[Lambert 19, Section 3.6](6dN20SCFT#Lambert19):

> 3.6  Open problems and wishes

> Let me close this discussion of M5-branes with some open problems and wish list of results:

> i) Provide a field definition/construction of the (2,0) theory $[$...$]$

> ii) Find the mathematical structures that best capture aspects of the (2,0) theory $[$...$]$

\linebreak

## Related entries

* [[large 1/N limit]]

* [[supergravity C-field]]

  * [[shifted C-field flux quantization]]

  * [[C-field tadpole cancellation]]

* [[F-theory]]

* [[topological M-theory]], [[bosonic M-theory]]

* [[M2-brane]], [[M5-brane]]

* [[supergravity Lie 3-algebra]], [[supergravity Lie 6-algebra]], [[M-theory super Lie algebra]]

* [[Horava-Witten theory]], [[M9-brane]]

* [[Diaconescu-Moore-Witten anomaly]]

* [[BFSS matrix model]], [[IKKT matrix model]], [[membrane matrix model]]

## References
 {#References}

### General

First indications for M-theory came from the observation that the [[double dimensional reduction]] of the [[supermembrane]] [[Green-Schwarz sigma-model]] on [[11d supergravity]] [[target spacetimes]] (now called the [[M2-brane]] yields the [[type IIA superstring]]:

* {#DuffHoweInamiStelle87} [[Michael Duff]], [[Paul Howe]], T. Inami, [[Kellogg Stelle]], _Superstrings in $D=10$ from Supermembranes in $D=11$_, Phys. Lett. B **191** (1987) 70 [<a href="https://doi.org/10.1016/0370-2693(87)91323-2">doi:10.1016/0370-2693(87)91323-2</a>] and in: [[Michael Duff]] (ed.) _[[The World in Eleven Dimensions]]_ 205-206 (1987) [[spire](http://inspirehep.net/record/245249)]

and the first statement of what came to be known as the [[duality between type IIA string theory and M-theory]] is due to:

* [[Paul K. Townsend]], *The eleven-dimensional supermembrane revisited*, Phys. Lett. B **350** (1995) 184-187 [[arXiv:hep-th/9501068](https://arxiv.org/abs/hep-th/9501068), <a href="https://doi.org/10.1016/0370-2693(95)00397-4">doi:10.1016/0370-2693(95)00397-4</a>]

A comprehensive collection of early articles is in

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

For some time though the success of [[string theory]] in 10-dimensions caused resistence to the idea of a theory of [[membranes]] in 11-dimensions, an account is in ([Duff 99](#Duff99)) and in brevity on the first pages of

* [[Mike Duff]], _M-history without the M_ ([arXiv:1501.04098](http://arxiv.org/abs/1501.04098))

The article that convinced the community of M-theory was

* {#Witten95} [[Edward Witten]], esp. §2.2, §2.3 in: _[[String Theory Dynamics In Various Dimensions]]_, Nucl. Phys. B **443** (1995) 85-126 [[arXiv:hep-th/9503124](http://arxiv.org/abs/hep-th/9503124), <a href="https://doi.org/10.1016/0550-3213(95)00158-O">doi:10.1016/0550-3213(95)00158-O</a>]

A public talk announcing the conjecture that the [[non-perturbative field theory|strong-coupling limit]] of [[type IIA string theory]] is [[11-dimensional supergravity]] [[KK-compactification|KK-compactified]] on a circle is at 15:12 in

* {#Witten95Talk} [[Edward Witten]], talk at University of Southern California (1995) [video:[YT](https://youtu.be/1HYa4wxqe8Y)]

   > 19:33: "Ten years ago we had the embarrassment that there were five consistent string theories plus a close cousin, which was 11-dimensional supergravity." (19:40): "I promise you that by the end of the talk we have just one big theory."

The term "M-theory" originates in

* {#HoravaWitten95} [[Petr Hořava]], [[Edward Witten]], *Heterotic and Type I string dynamics from eleven dimensions*, Nucl. Phys. B **460** (1996) 506 [[arXiv:hep-th/9510209](http://arxiv.org/abs/hep-th/9510209), <a href="https://doi.org/10.1016/0550-3213(95)00621-4">doi:10.1016/0550-3213(95)00621-4</a>]

as a "non-committed" shorthand for "membrane theory"

> {#NonCommittal} As it has been proposed that the eleven-dimensional theory is a supermembrane theory but there are some reasons to doubt that interpretation, we will non-committally call it the M-theory, leaving to the future the relation of M to membranes. ([Hořava-Witten 95, p. 2](#HoravaWitten95))

and

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

which coined the association

> the eleven-dimensional "M-theory" (where M stands for magic, mystery, or membrane, according to taste) ([Witten 95, p. 1](#Witten95))

that later gained much publicity:

* {#Witten98} [[Edward Witten]], _Magic, Mystery, and Matrix_, Notices of the AMS, volume 45, number 9 (1998) ([pdf](http://www.ams.org/notices/199809/witten.pdf))

The argument that the regularized [[M2-brane]] [[worldvolume]] theory is the [[BFSS matrix model]] is discussed in

* {#NicolaiHelling98} [[Hermann Nicolai]], Robert Helling, _Supermembranes and M(atrix) Theory_,  Lectures given by H. Nicolai at the Trieste Spring School on Non-Perturbative Aspects of String Theory and Supersymmetric Gauge Theories, 23 - 31 March 1998 ([arXiv:hep-th/9809103](http://arxiv.org/abs/hep-th/9809103))

* {#DasguptaNicolaiPlefka02} Arundhati Dasgupta, [[Hermann Nicolai]], [[Jan Plefka]], _An Introduction to the Quantum Supermembrane_, Grav. Cosmol. **8** 1 (2002) Rev. Mex. Fis. **49S1**  (2003) 1-10 [[arXiv:hep-th/0201182](http://arxiv.org/abs/hep-th/0201182)]

Recollections include the last paragraph of

* {#Witten14} [[Edward Witten]], _Adventures in Physics and Math_, [Kyoto Prize lecture](http://www.kyotoprize.org/en/laureates/commemorative_lectures/)  2014 ([pdf](http://www.kyotoprize.org/wp/wp-content/uploads/2016/02/30kB_lct_EN.pdf), [[WittenKyotoPrizeLecture.pdf:file]])

The term became fully established with surveys including

* {#Duff96} [[Michael Duff]], _M-Theory (the Theory Formerly Known as Strings)_,  Int. J. Mod. Phys. A **11** (1996) 5623-5642 [[arXiv:hep-th/9608117](http://arxiv.org/abs/hep-th/9608117), [doi:10.1142/S0217751X96002583](https://doi.org/10.1142/S0217751X96002583)]

* {#Duff98} [[Michael Duff]], _The Theory Formerly Known as Strings_, Scientific American 1998 ([pdf](https://www.nikhef.nl/pub/services/biblio/bib_KR/sciam14395569.pdf), [[DuffFormerlyStrings98.pdf:file]])

* {#Duff98b} [[Michael Duff]], _A Layman's Guide to M-theory_, 	Abdus Salam Memorial Meeting, Trieste, Italy, 19 - 22 Nov 1997, pp.184-213 ([arXiv:hep-th/9805177](https://arxiv.org/abs/hep-th/9805177), [cds:355721](https://cds.cern.ch/record/355721))

Despite the magic and mystery, the relation to the original abbreviation for _membrane-theory_ was highlighted again for instance in

* [[Paul Townsend]], _M(embrane) theory on $T^0$_, Nucl. Phys. Proc. Suppl. 68:11-16, 1998 ([arXiv:hep-th/9708034](http://arxiv.org/abs/hep-th/9708034))

Early articles clarifying the relation to [[type II string theory]] now known as [[F-theory]]:

* {#Schwarz95} [[John Schwarz]], _The Power of M Theory_, Phys.Lett. B367 (1996) 97-103 ([arXiv:hep-th/9510086](http://arxiv.org/abs/hep-th/9510086))

* {#Johnson97} [[Clifford Johnson]], _From M-theory to F-theory, with Branes_, Nucl.Phys. B507 (1997) 227-244 ([arXiv:hep-th/9706155](http://arxiv.org/abs/hep-th/9706155))

The relation also to the [[heterotic string]] was understood in ([Horava-Witten 95](#HoravaWitten95))❛ see at _[[Horava-Witten theory]]_.

Early review:

* [[Paul Townsend]], _Four Lectures on M-theory_,  in: Proceedings of the ICTP summer school on High Energy Physics and Cosmology, Trieste, June 1996. ([arXiv:hep-th/9612121](http://arxiv.org/abs/hep-th/9612121))

* {#ObersPioline98} [[Niels A. Obers]], [[Boris Pioline]], *U-duality and M-Theory*, Phys. Rept. **318** (1999) 113-225 [[arXiv:hep-th/9809039](http://arxiv.org/abs/hep-th/9809039), <a href="https://doi.org/10.1016/S0370-1573(99)00004-6">doi:10.1016/S0370-1573(99)00004-6</a>]

* [[John Schwarz]], *Introduction to M Theory and AdS/CFT Duality*, in: *Quantum Aspects of Gauge Theories, Supersymmetry and Unification* Lecture Notes in Physics **525**, Springer (1999)  [[arXiv:hep-th/9812037](https://arxiv.org/abs/hep-th/9812037), [doi:10.1007/BFb0104239](https://doi.org/10.1007/BFb0104239)]

  >  (with an eye towards [[AdS/CFT duality]])

Surveys of the discussion of E-series [[Kac-Moody algebras]]/[[Kac-Moody groups]] in the context of M-theory include

* Sophie de Buyl, _Kac-Moody Algebras in M-theory_, PhD thesis ([pdf](http://theses.ulb.ac.be/ETD-db/collection/available/ULBetd-06072006-153117/unrestricted/kmalgebrasinmth.pdf))

* [[Paul Cook]], _Connections between Kac-Moody algebras and M-theory_ PhD thesis ([arXiv:0711.3498](http://arxiv.org/abs/0711.3498))

### Relation to AdS/CFT

Relation to [[AdS/CFT]] and [[conformal bootstrap]]:

* {#ChesterPerlmutter18} Shai M. Chester, [[Eric Perlmutter]], _M-Theory Reconstruction from $(2,0)$ CFT and the Chiral Algebra Conjecture_ ([arXiv:1805.00892](https://arxiv.org/abs/1805.00892))

### Cohomological considerations

Discussion of the [[cohomology|cohomological]] charge quantization in type II ([[RR-fields]] as cocycles in [[KR-theory]]) in relation to the M-theory [[supergravity C-field]] is in

* {#FMW00} D. Diaconescu, [[Gregory Moore]], [[Edward Witten]], _$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory_, Adv.Theor.Math.Phys.6:1031-1134,2003 ([arXiv:hep-th/0005090](http://arxiv.org/abs/hep-th/0005090)), summarised in _A Derivation of K-Theory from M-Theory_ ([arXiv:hep-th/0005091](http://arxiv.org/abs/hep-th/0005091))

See also

* {#GarciaUranga05} Inaki Garcia-Etxebarria, [[Angel Uranga]], _From F/M-theory to K-theory and back_, JHEP 0602:008,2006 ([arXiv:hep-th/0510073](https://arxiv.org/abs/hep-th/0510073))

For more on this perspective as 10d type II as a [[self-dual higher gauge theory]] in the boudnary of a kind of [[higher dimensional Chern-Simons theory|11-d Chern-Simons theory]] is in

* {#BelovMooreII} Dmitriy Belov, [[Greg Moore]], _Type II Actions from 11-Dimensional Chern-Simons Theories_ ([arXiv:hep-th/0611020](http://arxiv.org/abs/hep-th/0611020))

More complete discussion of the decomposition of the [[supergravity C-field]] as one passes from 11d to 10d is in

* {#MathaiSati03} [[Varghese Mathai]], [[Hisham Sati]], _Some Relations between Twisted K-theory and E8 Gauge Theory_, JHEP0403:016,2004 ([arXiv:hep-th/0312033](http://arxiv.org/abs/hep-th/0312033))

### Relation to D$0$-brane mechanics

Discussion of M-theory as arising from [[type II string theory]] via the effect of [[D0-branes]] is in

* [[Tom Banks]], W. Fischler, S.H. Shenker, [[Leonard Susskind]], _M Theory As A Matrix Model: A Conjecture_, Phys.Rev.D55:5112-5128,1997 ([arXiv:hep-th/9610043](http://arxiv.org/abs/hep-th/9610043))
 {#BanksFischlerShenkerSusskind97}

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

### More on the relation to type IIA string theory

* {#Sen97} [[Ashoke Sen]], _A Note on Enhanced Gauge Symmetries in M- and String Theory_, JHEP 9709:001,1997 ([arXiv:hep-th/9707123](http://arxiv.org/abs/hep-th/9707123))

### In terms of higher geometry

Discussion of phenomena of M-theory in [[higher geometry]] and [[generalized cohomology]] is in

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

See also the references at _[[exceptional generalized geometry]]_.

In fact, much of the broad structure of M-theory and its relation to the various [[string theory]] limits can be seen from the classification of exceptional [[super L-∞ algebras]] (such as the [[supergravity Lie 3-algebra]] and the [[supergravity Lie 6-algebra]]), as discussed in

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

Review:

* [[Branislav Jurčo]], [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]] (eds.)

  **Proceedings of the LMS/EPSRC Durham Symposium on

  *[[Higher Structures in M-Theory 2018]]*

  Fortschritte der Physik Special Issue

  Volume 67, Issue 8-9

  [doi:10.1002/prop.201870081](https://www.onlinelibrary.wiley.com/toc/15213978/2019/67/8-9)

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]:

  **[[schreiber:The rational higher structure of M-theory|The Rational Higher Structure of M‐theory]]**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue **67** 8-9

  [arXiv:1903.02834](https://arxiv.org/abs/1903.02834), [doi:10.1002/prop.201910017](https://doi.org/10.1002/prop.201910017)

{#ThisAnalysisReveals} This analysis reveals, in particular, that *[[rational homotopy theory|rationally]]* the [[supergravity C-field|M-theory C-field]] (and hence [[M-brane]]-charge) is [[flux quantization|flux quantized]] in (unstable/[[non-abelian cohomology|non-abelian]]) *[[Cohomotopy theory]]* in a way directly analogous to how the [[RR-field|string theory RR-field]] is [[charge quantization|flux quantized]] *[[K-theory classification of D-brane charge|in K-theory]]*; an observation first highlighted in:

* {#Sati13} [[Hisham Sati]], [Sec 2.5](https://arxiv.org/pdf/1310.1060.pdf#page=17) of: _Framed M-branes, corners, and topological invariants_, J. Math. Phys. **59** (2018), 062304 ([arXiv:1310.1060](http://arxiv.org/abs/1310.1060))

The natural hypothesis that, therefore, the [[supergravity C-field|M-theory C-field]] should find its proper mathematical definition as a [[cocycle]] in (not just [[rationalization|rationalized]] but) full-blown ([[twisted Cohomotopy|twisted]], [[equivariant Cohomotopy|equivariant]] & [[differential Cohomotopy|differential]]) [[non-abelian cohomology|non-abelian]] [[Cohomotopy theory]] is supported by some evidence, see at *[[Hypothesis H]]* for more.

### Activity

* Conference series _[[M-Theory and Mathematics]]_

[[!redirects non-perturbative string theory]]