multiset

# Multisets
* table of contents
{: toc}

## Idea

A multiset is like a [[set]], just allowing that the elements have multiplicities. Thus the multiset $\{1,1,2\}$ differs from the multiset $\{1,2\}$, while $\{1,2,1\}$ is the same as $\{1,1,2\}$. Multisets are useful in [[combinatorics]]. See [wikipedia](http://en.wikipedia.org/wiki/Multiset).

## Definitions

While it is possible to take multisets as a fundamental concept in [[foundations]], it is more common to define them in terms of [[sets]] and [[functions]].

A **multiset** $\mathcal{X} = \langle X,\mu_X\rangle$,can be defined as a set $X$ (its __underlying set__) together with a function $\mu_X$ (giving each element its __multiplicity__) from $X$ to a class of nonzero [[cardinal numbers]].  A multiset is **locally finite** if multiplicity takes values in the [[natural numbers]].  Many authors take all multisets to be locally finite; that is the default in combinatorics.  The multiset is **finite** if it is locally finite and $X$ is a [[finite set]].  We can also define a multiset to be a function from the [[proper class]] of all objects to the class of all cardinal numbers, with the proviso that the objects whose multiplicity is nonzero form a set (the set $X$ above).

If we are only interested in multisets with elements drawn from a given set $U$ (as is common in combinatorics), then an alternative definition is very useful: a __multisubset__ of $U$ is a function $f\colon B \to U$, where two multisubsets $f\colon B \to U$ and $f'\colon B' \to U$ are considered _equal_ if there is a [[bijection]] $g\colon B \to B'$ that makes a commutative triangle.  In other words, a multisubset of $U$ is an isomorphism class in the [[slice category]] $Set/U$.  (Compare this to the structural definition of _[[subset]]_ of $U$ as an [[injective function]] to $U$.)

A multisubset of $U$ is locally finite if every [[fibre]] is finite; it is finite if additionally the [[image]] of $f$ (which corresponds to $A$ in the original definition) is finite.  A locally finite multisubset can also be described as a function from $U$ to the set of natural numbers; this is just the multiplicity function $\mu$ again, now with $U$ (rather than $X$) specified as the domain and allowing the value $0$ to be taken.

## Operations on multisets

The operations on cardinal numbers induce operations on multisets (or on multisubsets of any given set $U$).

In the following, let $\mathcal{X} = \langle X,\mu_X\rangle$ and $\mathcal{Y} = \langle Y,\mu_Y\rangle$ be multisets.

*  The [[cardinality]] of a multiset is given by

   $$|\mathcal{X}| = \sum_{e\in X} \mu_X(e).$$

*  The __[[intersection]]__ of multisets is the multiset whose cardinality is given by the [[infimum]] operation on cardinal numbers.

   $$\mathcal{X}\cap\mathcal{Y} = \langle X\cap Y, \min(\mu_X,\mu_Y)\rangle.$$

*  The __[[union]]__ of multisets is the multiset whose cardinality is given by the [[supremum]] operation on cardinal numbers.

   $$\mathcal{X}\cup\mathcal{Y} = \langle X\cup Y, \max(\mu_X,\mu_Y)\rangle.$$

*  The _[[set difference]]_ of multisets is the multiset given by

   $$ \mathcal{X} \backslash \mathcal{Y} = \langle \{ a \in X \cup Y \;|\; \mu_X(a) \gt \mu_Y(a) \}, \mu_X - \mu_Y \rangle .$$

*  The __sum__ of multisets is the multiset whose cardinality is given by addition of cardinal numbers; this has no analogue for ordinary sets.

   $$\mathcal{X}+\mathcal{Y} = \langle X\cup Y, \mu_X+\mu_Y\rangle.$$

*  The __product__ of multisets (turning them into a [[rig]]) is the multiset whose cardinality is given by the product of cardinal numbers

   $$\mathcal{X}\mathcal{Y} = \langle X\cap Y, \mu_X\mu_Y\rangle.$$

   Note that if $\mathcal{X}$ is a set, then $\mathcal{X}\mathcal{X} = \mathcal{X}.$

*  The [[inner product of multisets]] is given by

   $$\langle\mathcal{X},\mathcal{Y}\rangle = \sum_{e\in{X\cap Y}} \mu_X(e) \mu_Y(e).$$

   Note that the inner product corresponds to the cardinality of the product

   $$\langle\mathcal{X},\mathcal{Y}\rangle = |\mathcal{X}\mathcal{Y}|.$$

## Examples

...

* See also [[inner product of multisets]].

## References

* [Mathematics of Multisets](http://asyropoulos.eu/papers/PDF/MathMSet.pdf), Apostolos Syropoulos

* [Categorical Models of Multisets](http://asyropoulos.eu/papers/PDF/mset.pdf), Apostolos Syropoulos

* [An Overview of the Applications of
Multisets](http://www.emis.de/journals/NSJOM/Papers/37_2/NSJOM_37_2_073_092.pdf), D. Singh, A. M. Ibrahim, T. Yohanna and J. N. Singh

[[!redirects finite multiset]]
[[!redirects finite multisets]]

[[!redirects multiset]]
[[!redirects multisets]]

[[!redirects multisubset]]
[[!redirects multisubsets]]

[[!redirects bag]]
[[!redirects bags]]


multisimplicial set

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

#Content#
* table of contents
{:toc}

## Idea

Multisimplicial sets are the analogs of [[simplicial sets]] with [[simplices]] replaced by [[multisimplices]] (perhaps more appropriately called multiplexes).

## Definition
 {#Definition}

+-- {: .num_defn}
###### Definition

An **$n$-fold multisimplicial set**  is a [[presheaf]] on the $n$-fold [[multisimplex category]] $\Delta^n$, that is, a [[functor]] $X\colon(\Delta^n)^{op}\to Sets$, equivalently a [[multisimplicial object]] in the category [[Set]] of sets.

=--

## Relation to simplicial sets

The category of $n$-fold multisimplicial sets can be
equipped with a [[model structure]] that turns
it into a [[model category]] that is Quillen equivalent
to the standard [[Kan–Quillen model structure]] on [[simplicial sets]].

Cofibrations of multisimplicial sets are precisely [[monomorphisms]].
Weak equivalences are induced from [[simplicial sets]] by the diagonal functor.

The corresponding [[Quillen adjunction]]
is constructed as as a [[nerve-realization adjunction]]
for the functor
$$\Delta^n\to sSet$$
that sends a multisimplex $(m_1,\ldots,m_n)$ to $\Delta^{m_1}\times\cdots\times\Delta^{m_n}$.

The left adjoint is given by a [[left Kan extension]].
The right adjoint sends a [[simplicial set]] $X$ to its multisimplicial nerve,
which sends a multisimplex $(m_1,\ldots,m_n)$ to $sSet(\Delta^{m_1}\times\cdots\times\Delta^{m_n},X)$.

To see that this [[Quillen adjunction]] is a [[Quillen equivalence]],
one can either argue directly, by computing the derived [[unit]] and derived [[counit]] and showing that they are [[weak equivalences]],
or, much more elegantly, invoke a theorem by [[Grothendieck]] and [[Maltsiniotis]].

The latter theorem ([Maltsiniotis](#Maltsiniotis), Proposition 1.6.8)
states that a totally aspherical [[small category]] $C$ with a separating aspherical interval is a [[strict test category]],
and, therefore, the category of presheaves of sets on $C$
admits a [[model structure]] whose cofibrations are monomorphisms
and weak equivalences are induced by the [[category of elements]] functor
from the [[Thomason model structure]] on the category of [[small categories]].
Furthermore, the resulting model category of presheaves of sets on $C$
is a cartesian [[combinatorial model category]] that is Quillen equivalent
to the [[Kan–Quillen model structure]] on the category of [[simplicial sets]].

We now verify the conditions of the theorem:

* The category of multisimplices is nonempty.

* The categorical product of two representable multisimplicial sets is acyclic, i.e., weakly equivalent to the terminal object.

* The multisimplex $([1],[1],\ldots,[1])$ together with the two canonical inclusions of $([0],\ldots,[0])$ is a separating aspherical interval.

## Exterior product

An important operation on multisimplicial sets is the _exterior product_
$$Fun((\Delta^m)^{op},Set)\times Fun((\Delta^n)^{op},Set)\to Fun((\Delta^{m+n})^{op},Set)$$
defined as the [[left Kan extension]] of the tautological functor
$$\Delta^m\times\Delta^n\to\Delta^{m+n}.$$

The exterior product is a [[left Quillen bifunctor]]
whose [[left derived bifunctor]] model the cartesian
product in the [[∞-category]] of spaces.

The exterior product is useful when it is desirable
to have a product operation that does not require subdivision.

## Related concepts

* [[simplicial sets]]

* [[cubical sets]]

* [[Cisinski model category]]

* [[test category]], [[strict test category]]

## References

* [[Zouhair Tamsamani]], _On non-strict notions of $n$-category and $n$-groupoid via multisimplicial sets_ ([arXiv:alg-geom/9512006](http://arxiv.org/abs/alg-geom/9512006))

* [[Gerd Laures]] and [[James E. McClure]],
_Multiplicative properties of Quinn spectra_
([arXiv:0907.2367v2](http://arxiv.org/abs/0907.2367v2))

* Yifeng Liu and Weizhe Zheng,
_Gluing restricted nerves of $\infty$-categories_
([arXiv:1211.5294](http://arxiv.org/abs/1211.5294))

* {#Maltsiniotis} [[Georges Maltsiniotis]], _La théorie de l'homotopie de Grothendieck_, Astérisque 301 (2005).


multispan

[[!redirects multispans]]
[[!redirects multicospan]]
[[!redirects multicospans]]

#Idea#

* A _multi(co)span_ is supposed to be something that generalizes [[span]] (and [[cospan]]) both horizontally and vertically: it may have a number of legs different from 2, but more importantly it need not be a single roof but can be a more complex diagram.

* A multispan is supposed to be a model for a _hierarchical cell complex_ with

  * a single top "cell" $K(\top) \in C$ (an object in some ambient category $C$);

  * for each cell $K(a)$ a collection of morphisms $\{K(b_i) \to K(a)\}_i$ into it, to be thought of as pieces of boundary components of $K(a)$;

    * in the language of [[hyperstructures]] we would say that $K(a)$ is a _bond_ for the $K(b_i)$

  * and so on;

  * such that the entire resulting diagram commutes, expressing the fact how one boundary pieces may be part of different higher order boundary pieces.

Multi-cospans of [[cubical]] shape have been introduced and studied by Marco Grandis in his work on [[Cospans in Algebraic Topology]]. Grandis also formulates the idea that an [[FQFT|extended QFT]] should be a morphism with domain [[extended cobordisms]] modeled as multi-cospans.

#Examples#

Special cases of multispans of the above general kind are

* ordinary [[spans]] and [[cospans]]: these are obtained by restricting the domain poset to be of the form
$\wedge := a_1 \to \top \leftarrow a_2$.

* the double cospans of the form $\wedge^{\times 2}$ with $\wedge$ as above, described in

  * Jeffrey Morton, _Double Bicategories and Double Cospans_ ([arXiv](http://xxx.lanl.gov/abs/math.CT/0611930)),

 * Grandis' higher cubical cospans, which generalize the previous example: these are obtained by restricting the domain posets to be the cartesian powers
$\wedge^{\times n}$ (with $\wedge$ as above).

An attempt at a discussion of multispans in greater generality is at [[hyperstructure]].

#Related concepts#

* [[span trace]]

* [[co-span co-trace]]

* [[Cospans in Algebraic Topology]]


multisymplectic geometry

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

#Contents#
* table of contents
{:toc}

## Idea
  {#Idea}

### Traditional

Multisymplectic geometry is a generalization of [[symplectic geometry]] in the context of [[variational calculus]] and [[mechanical systems]] in which the [[symplectic form]] is generalized from a closed 2-form to a closed $n+1$-form, for $n \geq 1$ -- the [[n-plectic form]].

It is closely related to the [[de Donder-Weyl formalism]] of [[variational calculus]]. In the context of [[quantization]] it is meant to provide a refinement of [[geometric quantization]] which is well-adapted to $n$-dimensional [[quantum field theory]]. However, details of the multisymplectic quantization procedure remain under investigation.

### From the $n$POV

We comment a bit on how to, presumably, think of multisymplectic geometry
from the [[nPOV]], in the context of [[higher geometric quantization]]. Readers may want to skip ahead to traditional technical discussion at _[Extended phase space](#extendedphasespace)_.

+-- {: .standout}

Multisymplectic geometry is (or should be) to [[symplectic geometry]] as [extended quantum field theory](http://ncatlab.org/nlab/show/FQFT#Extended) is to non-extended [[quantum field theory]]:

in the multisymplectic **extended phase space** of an $n$-dimensional [[classical field theory|field theory]] a state is not just a point, but an $n$-dimensional subspace.

=--

See also [[n-plectic geometry]].

**Multisymplectic geometry** is a generalization of [[symplectic geometry]] to cases where the symplectic 2-form is generalized to a higher degree [[differential form]].

In as far as [[symplectic geometry]] encodes [[Hamiltonian mechanics]], multisymplectic geometry may be regarded as resolving the symplectic geometry of the [[Hamiltonian mechanics]] of [[classical field theory]]: the kinematics of an $n$-dimensional field theory may be encoded in an degree $(n+1)$ symplectic form.

In this application to physics, multisymplectic geometry is also known as the **covariant symplectic approach** to field theory (e.g. [section 2 here](http://arxiv.org/PS_cache/physics/pdf/9801/9801019v2.pdf#page=18)).

The idea is that under a suitable fiber integration multisymplectic geometry becomes ordinary symplectic form on the ordinary phase space of the theory, similar to, and in fact as a special case of, how for instance a [[line bundle]] on a [[loop space]] with a 2-form [[Chern class]] may arise by [[transgression]] from a [[bundle gerbe]] down on the original space, with a 3-form class.

By effectively undoing this implicit transgression in the ordinary [[Hamiltonian mechanics]] of [[classical field theory]], multisymplectic geometry provides a general framework for a geometric, covariant formulation of [[classical field theory]], where _covariant formulation_ means that spacelike and timelike directions on a given space-time be treated on equal footing.

## Extended phase spaces in covariant field theory
 {#extendedphasespace}

We discuss here the refinement in multisymplectic geometry of the [[covariant phase spaces]] of [[classical field theory]]/[[prequantum field theory]] from ([[presymplectic manifold|pre]]-)[[symplectic manifolds]] of initial value data in a [[Cauchy surface]] to multisymplectic manifolds of local initial value data.

Recall that an ordinary [[phase space]] of a [[physical system]] is a [[symplectic manifold]] whose points correspond to the _[[states]]_ of the system. The **extended phase space** of an $n$-dimensional quantum field theory is a multisymplectic space whose points correspond to pairs consisting of

* a point in the field theory's parameter space -- an "event";

* a state of the theory "at that event".

So **extended phase spaces _localizes_ the information about states** : a point in here encodes not just the entire state of the system, but remembers explicitly what that state is like over any point in parameter space.

### Covariant configuration bundle

Consider [[classical field theory]] over a **parameter space** $\Sigma$. From the point of view of [[FQFT]] $\Sigma$ will be one fixed [[cobordism]] on which we want to understand the (classical) field theory.

We assume that a **[[field (physics)|field configuration]]** on $\Sigma$ is a [[section]]  $\phi : \Sigma \to E$ of some prescribed [[bundle]] $E \to \Sigma$: the _[[field bundle]]_.

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

For instance an $n$-dimensional [[sigma-model]] [[quantum field theory]] is one whose field configurations on $\Sigma$ are given by maps

$$
  \phi : \Sigma \longrightarrow X
$$

into some prescribed **[[target space]]** $X$. This is the case where $E = \Sigma \times X$ is a **trivial [[bundle]]**.

=--

+-- {: .num_remark}
###### Remark

Beware of the standard source of confusion here when correlating this formalism with actual physics: the physical [[spacetime]] that we inhabit may be given either by $\Sigma$ or by $X$:

* in the description of the [[quantum mechanics]] of objects propagating _in_ our physical [[spacetime]], subject to [[forces]] exerted by fixed [[background gauge fields]] (such as electrons propagating in our particle accelerator, subject to the electromagnetic field in the accelerator tube), physical spacetime is identified with target space $X$, while $\Sigma$ is the **worldvolume** of the object that propagates through $X$. The _field configurations_ on $\Sigma$ are really the maps $\Sigma \to X$ that determine how the object sits in spacetime.

* in quantum mechanics of fields on spacetime, such as the quantized electromagnetic field in a laser, it is $\Sigma$ which represents physical spacetime, and $X$ is some abstract space, for instance a smooth version of the [[classifying space]] $\mathcal{B}U(1)$, so that a field configuration $\Sigma \to X$ encodes a [[line bundle]] [[connection on a bundle|with connection]] that encodes a configuration of the [[electromagnetic field]].

=--

+-- {: .num_defn}
###### Definition

The **configuration space** of the system is the space of all field configurations, hence the space $\Gamma_\Sigma(E)$ of [[sections]] of the [[bundle]] $E$.

=--

In the [[sigma-model]] example this is some incarnation of the
[[mapping space]] $[\Sigma,X]$.

+-- {: .num_remark}
###### Remark

Beware that in low dimensions one often distinguishes between the space of _configurations_ $\Sigma \to X$ and that of _trajectories_ or _histories_ $\Sigma \times \mathbb{R} \to X$. This comes from the case $\Sigma = *$ where for a particle propagating on $X$ the maps $[*,X] \simeq X$ are the possible configurations of the particle at a given parameter times, while maps $[* \times \mathbb{R}, X] = [\mathbb{R}, X]$ are the trajectories. But for the higher dimensional and [[FQFT|extended]] field theories under discussion here, this distinction becomes a bit obsolete and trajectories become just a special case of configurations.

=--

+-- {: .num_remark}
###### Remark

In the **non-covariant** approach one would try to consider the a [[cotangent bundle]] of the configuration space $\Gamma(E)$ as _phase space_ . Contrary to that, in the **covariant approach** one considers the much smaller space $E$ instead. This is then called the **covariant configuration space** or **covariant configuration bundle**.

=--

+-- {: .num_defn}
###### Definition

Write $J^1 E \to \Sigma$ for the first order [[jet bundle]] of the configuration space bundle $E \to \Sigma$. Its [[fiber]] over $s \in \Sigma$ are equivalence classses of [[germs]] of [[sections]] at $x$, where two germs are identified if their first derivatives coincide.

=--

### Covariant phase space

+-- {: .num_defn #DualFirstJetBundle}
###### Definition

Given a [[vector bundle]] $E \to \Sigma$ over a [[smooth manifold]] of [[dimension]] $dim(\Sigma)  = n+1$, the **affine dual first jet bundle**
(or often just **dual first jet bundle** for short)  $(J_1 E)^\ast \to \Sigma$ is the [[vector bundle]] whose [[fiber]] at $e \in E_s$ is the set of [[affine maps]]

$$
  J_e^1 E \longrightarrow \Lambda_s^{n+1} \Sigma
$$

from the first [[jet bundle|jets]] at $e$ to the degree-$(n+1)$ [[differential forms]] at $s$ on $\Sigma$.

=--

+-- {: .num_defn #ExtendedCovariantPhaseSpace}
###### Definition

Given a [[spacetime]]/[[worldvolume]] $\Sigma$ and a [[field bundle]] $E \to \Sigma$, the **extended covariant phase space** is the multisymplectic manifold

* whose underlying manifold is the dual first jet bundle, def. \ref{DualFirstJetBundle}, of the field bundle

  $$
    (J^1 E)^* \to E
    \,,
  $$

* equipped with the canonical degree-$(n+2)$ [[differential form]]

  $$
    \omega = d \alpha
    \,,
  $$

  where $\alpha$ is the canonical $(n+1)$-form

=--

Given $\pi \colon E \to \Sigma$, with $\mathrm{dim} \Sigma =n+1$, the dual jet bundle $(J^1 E)^*$ is isomorphic to a particular vector sub-bundle of the $n+1$-form bundle $\Lambda^{n+1}T^{*}E$.
To see this, first consider the following

+-- {: .num_defn #nHorizontalForms}
###### Definition

Given a point $y \in E$, a tangent vector $v \in T_{y} E$
is said to be **[[vertical vector field|vertical]]** if $d \pi(v) = 0$.   Define

$$
  \Lambda^{n+1}_{1}T^{\ast}E
  \hookrightarrow
  \Lambda^{n+1} T^{\ast} E
$$

to be the subbundle of the $n+1$-form bundle whose fiber at $y \in E$
consists of all $\beta \in \Lambda^{n+1} T^{*}_{y} E$ such that
$$
  \iota_{v_1}\iota_{v_2} \beta =0
$$
for all [[vertical vector field|vertical vectors]] $v_1,v_2 \in T_{y}E$.
Sections of $\Lambda^{n+1}_{1}T^{*}E$ are called **$n$-horizontal** $\mathbf{n+1}$**-forms**.

=--

In words, an $n$-horizontal $(n+1)$-form is one which has at most one "leg" not along $\Sigma$. This is made very explicit in the proof of the following proposition.

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

Let $E \to \Sigma$ be a [[vector bundle]] over a [[smooth manifold]] $\Sigma$ of [[dimension]] $dim \Sigma = (n+1)$ and assume that  $\Sigma$ is [[orientation|orientable]], then there is an [[isomorphism]]

$$
  (J^1 E)^{\ast} \simeq \Lambda^{n+1}_{1} T^{\ast}E
$$

of [[vector bundles]] over $\Sigma$ between the dual first jet bundle of $E$, def. \ref{DualFirstJetBundle}, and the bundle of $n$-horizontal $(n+1)$-forms on $E$, def. \ref{nHorizontalForms}.

=--

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

It suffices to work locally with respect to a [[good open cover]], so we reduce the statement to the special case of the [[sigma model]] i.e. the trivial bundle  $E = \Sigma \times X$ over $\Sigma$. By the assumoption that $\Sigma$ admits an [[orientation]] we may pick a [[volume form]] $\vol \in \Gamma(\Lambda^{n+1}T^\ast \Sigma)$.

Let $q^1, \dots, q^{n+1}$ be local [[coordinates]] on $\Sigma$ and let
$u^1, \dots , u^d$ be local coordinates on $X$.  Then $\Lambda_1^{n+1} T^* E$ has a local [[basis]] of sections given by $(n+1)$-forms of two types: first, the wedge product of all $n+1$ [[cotangent vectors]] of type $\mathbf{d}q^i$:

$$
  \vol = \mathbf{d}q^1 \wedge \cdots \wedge \mathbf{d}q^{n+1}
$$

and second, wedge products of $n$ cotangent vectors of type $\mathbf{d}q^i$ and a single one of type $\mathbf{d}u^a$:

$$
  \mathbf{d}q^1 \wedge \cdots \wedge
   \mathbf{d} q^{i-1}
   \wedge
   \mathbf{d}q^{i+1}
   \wedge \cdots \wedge \mathbf{d}q^{n+1} \wedge \mathbf{d}u^a
  \,.
$$

If $y = (p,u) \in \Sigma \times X$, this basis gives an isomorphism

$$
  \Lambda^{n+1}_1 T^*_y E
   \;\; \simeq\;\;
  \left(\Lambda^{n+1} T^*_p \Sigma \right)
    \; \oplus \;
  \left(\Lambda^{n} T^*_p \Sigma \otimes T^*_u X \right)
  \,.
$$

The [[volume form]] on $\Sigma$ also determines isomorphisms

$$
  \mathbb{R} \overset{\simeq}{\longrightarrow}
  \Lambda^{n+1} T^*_p \Sigma
$$

$$
                 c \mapsto  c \, \vol_p
$$

and

$$
  T_p \Sigma   \overset{\simeq}{\longrightarrow}  \Lambda^{n} T^*_p \Sigma
$$

$$
         v      \mapsto     \iota_v \vol_p
  \,.
$$

We thus have obtained an isomorphism

$$
  \Lambda^{n+1}_1 T^*_y E \;\; \cong \;\; \mathbb{R} \; \oplus \;
T_p \Sigma \otimes T^*_u X
  \,.
$$

On the other hand, the trivialization $E = \Sigma \times X$ gives
an isomorphism of [[affine spaces]]

$$
  J^1_y E \; \; \cong \; \; T^*_p \Sigma \otimes T_u X
$$

which has the side-effect of exhibiting on $J^1_y E$ the structure of a  [[vector space]]. Since we've identified

$\Lambda^{n+1} T^*_p \Sigma$ with $\mathbb{R}$, an affine map from $J^1_y E$ to $\Lambda^{n+1} T^*_p \Sigma$ is just
an element of $T_x \Sigma \otimes T^*_u X$ plus a constant.
So, we obtain
$$   (J^1_y E)^* \; \; \cong \; \; \mathbb{R} \; \oplus \;
T_p \Sigma \otimes T^*_u X .$$
This gives a specific vector bundle isomorphism
$(J^1 E)^* \cong \Lambda_1^{n+1} T^* E$, as desired.

=--

+-- {: .num_remark #CanonicalFormInGoodCoordinates}
###### Remark

In practice it is better to use the [[pullback of differential forms|pulled back]] [[volume form]] $\pi^* \vol$ as a substitute for the coordinate-dependent $n+1$-form $\mathbf{d}q^1 \wedge \cdots \wedge \mathbf{d}q^{n+1}$ on $E$.  This gives another [[basis]] of
sections of $\Lambda_1^{n+1} T^* E$, whose elements we write suggestively

$$
  dQ \coloneqq \pi^* \vol
$$

and

$$
  dQ_i^a
  \coloneqq
  \left(\pi^* \iota_{\partial/\partial q^i} \vol\right)
   \wedge
   \mathbf{d}u^a
  \,.
$$

Corresponding to this [[basis]] then there are
local [[coordinates]] $P$ and $P^i_a$ on $\Lambda_1^{n+1} T^* E$, which combined with the coordinates $q^i$ and $u^a$ pulled back from $E$ give a local coordinate system on $\Lambda_1^{n+1} T^* E$.

In these [[coordinates]] the canonical $n+1$-form on $(J^1 E)^* \cong \Lambda_1^{n+1} T^* E$ is:

$$
  \alpha = P \wedge dQ + P^i_a \wedge dQ_i^a,
$$

and the $n+2$ [[multisymplectic form]] is

$$
  \omega = \mathbf{d}P \wedge dQ + \mathbf{d}P^i_a \wedge dQ_i^a.
$$

=--

### De Donder-Weyl-Hamilton field equations
 {#DonderWeylHamiltonFieldEquations}

We discuss the [[Euler-Lagrange equations of motion]] of a [[local field theory]] expressed in multisymplectic geoemtry via [[de Donder-Weyl formalism]].

+-- {: .num_defn #LocalLagrangian}
###### Definition

Given a [[field bundle]] $E \to \Sigma$ as above, a (first order) [[local Lagrangian]] is a [[smooth function]]

$$
  \mathbf{L} \;\colon\; J^1 E \longrightarrow \wedge^n T^\ast \Sigma
$$

on the first [[jet bundle]] of $E$ with values in [[densities]]/[[volume forms]]. Equivalently this is a degree $(n,0)$-form on the jet bundle, in terms of [[variational bicomplex]] grading.

=--

+-- {: .num_defn #LegendreTransform}
###### Definition

Given a [[local Lagrangian]], def. \ref{LocalLagrangian},
its **local [[Legendre transform]]** is the [[smooth function]]

$$
  \mathbb{F}\mathbf{L} \;\colon\;
  J^1 E \longrightarrow (J^1 E)^\ast
$$

from first [[jets]] to the affine dual jet bundle, def. \ref{DualFirstJetBundle}, which sends $\mathbf{L}$
to its first-order [[Taylor series]].

=--

This definition was proposed in ([Forger-Romero 04, section 2.5](#ForgerRomero04)).

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

In terms of the local [[coordinates]] of remark \ref{CanonicalFormInGoodCoordinates} the Legendre transform of def. \ref{LegendreTransform} is the function with [[coordinates]]

$$
  P^i_a = \frac{\partial \mathbf{L}}{\partial q^a_{, i}}
$$

and

$$
  P = \mathbf{L} - \frac{\partial \mathbf{L}}{\partial q^a_{,i}}q^a_{,i}
  \,.
$$

=--

([Forger-Romero 04, section 2.5 (41)](#ForgerRomero04)).

+-- {: .num_remark}
###### Remark

The second term in prop. \ref{LegendreTransformInLocalCoordinates} is what is traditionally called the Legendre transform in multisymplectic geometry/[[de Donder-Weyl formalism]]. Def. \ref{LegendreTransform} may be regarded as explaining the conceptual role of this expression, in particular in view of the following proposition.

=--

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

Given a [[local Lagrangian]] $\mathbf{L}$, the [[pullback of differential forms|pullback]] $\omega_{\mathbf{L}}$ of the canonical [[n-plectic form|pre-n-plectic form]] $\omega$, def. \ref{ExtendedCovariantPhaseSpace}, along the [[Legendre transform]] $\mathbb{F}\mathbf{L}$, def. \ref{LegendreTransform}, to the first [[jet bundle]] is the sum of the [[Euler-Lagrange equation]] $EL_{\mathbf{L}}$ and the canonical symplectic form $\mathbf{d}_v \theta_{\mathbf{L}}$ from [[covariant phase space]] formalism:

$$
  \begin{aligned}
    \omega_{\mathbf{L}}
    & \coloneqq
    \mathbb{F}\mathbf{L}^\ast \omega
    \\
    & =
    EL_{\mathbf{L}} + \mathbf{d}_v \theta_{\mathbf{L}}
  \end{aligned}
  \,.
$$

It follows that

1. $(\iota_{v_n} \cdots \iota_{v_1})  \omega_{\mathbf{L}} = 0$ is the [[Euler-Lagrange equation of motion]] in [[de Donder-Weyl formalism|de Donder-Weyl-Hamilton]]-form;

1. for any [[Cauchy surface]] $\Sigma_{n-1}$, the [[transgression]]
   $\omega_\Sigma \coloneqq \int_{\Sigma_{n-1}}\omega_{\mathbf{L}}$ is the canonical pre-symplectic form on [[phase space]] (as discussed there).

=--

This statement is essentially the content of ([Forger-Romero 04, equation (54) and theorem 1](#ForgerRomero04)). In the above form in terms of [[variational bicomplex]] notions this statement has been amplified by [[Igor Khavkine]].

### Examples

#### Free field theory
 {#FreeFieldTheory}

We write out the multisymplectic geometry corresponding to a [[free field theory]].

Let $\Sigma = (\mathbb{R}^{d-1;1}, \eta)$ be [[Minkowski spacetime]].
Write the canonical [[coordinates]]

$$
  \sigma^i \;\colon\; \Sigma \longrightarrow \mathbb{R}
  \,.
$$

Let $(X,g)$ be a [[Riemannian manifold]]. For simplicity of notation we assume that $X \simeq \mathbb{R}^k$ is a [[vector space]], too. Write its canonical coordinates as

$$
  \phi^a \;\colon\; X \longrightarrow \mathbb{R}
  \,.
$$

Let $X \times \Sigma \to \Sigma$ be the [[field bundle]].
Its first [[jet bundle]] then has canonical coordinates

$$
  \{ \sigma^i \}, \{\phi^a\}, \{\phi^a_{,i}\} \;\colon\; j_\infty^1(\Sigma \times X) \longrightarrow X
  \,.
$$

+-- {: .num_defn}
###### Definition

The [[local Lagrangian]] for [[free field theory]] with this [[field bundle]] is

$$
  L
   \coloneqq
  \left(
     \frac{1}{2} g^{i j} \eta_{a b} \phi^a_{,i} \phi^a_{,j}
  \right)
  \wedge
  \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d
  \,.
$$

=--

The [[canonical momenta]] are

$$
  \begin{aligned}
    p_a^i \wedge \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d
      & \coloneqq
    \frac{\partial}{\partial u^a_i} L
    \\
      & =
    \left(
       g^{i j} \eta_{a b} \phi^a_{,j}
    \right)
    \wedge
    \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d
  \end{aligned}
$$

So the boundary term $\theta$ in [[variational calculus]],
(see [this remark](covariant+phase+space#CanonicalThetaDensityInLocalCoordinates) at _[[covariant phase space]]_ ) is

$$
  \begin{aligned}
    \mathbf{d}u^a
     \wedge
    \iota_{\partial_i}
    \left(
      \frac{\partial}{\partial \phi^a_{,i}}L
    \right)
     & =
    p^i_a \wedge (\iota_{\partial_{\sigma^i}} vol) \wedge \mathbf{d}u^a
    \\
    & =  p^i_a \wedge dq_i^a
    \,,
  \end{aligned}
$$

where in the last line we adopted the notation of remark \ref{CanonicalFormInGoodCoordinates}.

This shows that the canonical [[multisymplectic form]]
is the "covariant symplectic potential current density"
which is induced by the free field Lagrangian.

See also ([Forger Romero 04, section 3.2](#ForgerRomero04)).

#### Bosonic particle propagating on a manifold

> scratch

Ordinary point [[particle]] [[mechanics]] on a [[manifold]] $X$ involves [[trajectories]] $\mathbb{R} \to X$ in $X$, with parameter space $\Sigma = \mathbb{R}$ the [[real line]], thought of as the abstract "[[worldline]]" of the particle.

* parameter space: $\Sigma = \mathbb{R}$, the **worldline**;

* target space: $X$, some [[manifold]] -- **spacetime**;

* configuration bundle: $(E \to \mathbb{R}) = (\mathbb{R}\times X \to \mathbb{R})$;

* jet bundle: $J^1 E = \mathbb{R} \times T X$ .

for $U \subset E = \mathbb{R} \times X$ a local patch with coordinate functions $\{t, q^i\}$, there are canonically induced coordinates on $J^1 E$ written $\{t,q^i, v^i\}$.

Here a collection of vaues $(q^i_0)$ is a **position** of the particle and $(v^i_0)$ is a **velocity** of the particle. Notice that in this covariant approach these are not positions and velocities "at a given time". Rather, a point in $J^1 E$ specified a parameter time and a corresponding position and velocity.

....

Let $U \to X$ be a local patch of $X$ with canonical coordinates $\{x^i\}$.

The canonical 2-form on the extended phase space in this case is traditionally locally written as

$$
  \omega|_U = \mathbf{d} \alpha|_U = \mathbf{d}( p_i \wedge \mathbf{d} q^i + H \wedge d t )
  \,.
$$

... blah-blah-blah...

#### Electromagnetism

* parameter space: $\Sigma$ is spacetime;

A field configuration of the [[electromagnetic field]] is a [[line  bundle]] [[connection on a bundle|with connection]] on $\Sigma$. If we assume the corresponding bundle to be trivial, then this is just a [[differential form|1-form]] on $\Sigma$. So in this simplified case we can take

* configuration bundle: $T^* \Sigma$ is the [[cotangent bundle]] of $\Sigma$.

#### Bosonic string propagating on a manifold

* parameter space: $\Sigma$ some 2-dimensional manifold -- the **worldsheet** -- for instance $\Sigma = S^1 \times \mathbb{R}$ models
a closed string propagating without interaction.

* target space. $X$ spacetime;

* covariant configuration bundle $E = \Sigma \times X$.

We will work out the covariant Hamiltonian formalism (also known as the **[[de Donder-Weyl formalism]]**) for this example in detail. We follow here the exposition found in ([Hélein 02](#Helein02)).

For simplicity we will only consider the case where $\Sigma$ is the
cylinder $\mathbb{R}\times S^1$ and $X$ is $d$-dimensional Minkowski spacetime, $\mathbb{R}^{1,d-1}$.
A solution of the classical bosonic string is then a map $\phi : \Sigma \to X$ which is a critical point of the area subject to certain boundary conditions.

Equivalently, by exploiting symmetries
in the variational problem, one can describe solutions $\phi$ by equipping $\mathbb{R} \times S^{1}$ with its standard Minkowski metric and then solving the
$1+1$ dimensional field theory specified by the Lagrangian density
$$
\mathcal{L}=\frac{1}{2} g^{ij}\eta_{ab} \frac{\partial
  \phi^{a}}{\partial q^i}\frac{\partial \phi^{b}}{\partial q^j}.
$$

Here $q^i$ $(i = 0,1)$ are standard coordinates on $\mathbb{R} \times S^1$ and
$g=\mathrm{diag}(1,-1)$ is the Minkowski metric on $\mathbb{R} \times S^1$, while
$\phi^a$ are the coordinates of the map $\phi$ in $\mathbb{R}^{1,d-1}$ and $\eta =
\mathrm{diag}(1,-1,\cdots,-1)$ is the Minkowski metric on $\mathbb{R}^{1,d-1}$. The corresponding Euler-Lagrange equation is just a version of the
wave equation:
$$
g^{ij}\partial_{i} \partial_{j} \phi^a =0.
$$

The space $E=\Sigma \times X$ can be thought of as a trivial bundle over $\Sigma$, and the graph of a function $\phi : \Sigma \to X$ is a smooth section of $E$.  We write the coordinates of a
point $(q,u)\in E$ as $\left(q^i,u^a \right)$.
Let $J^1 E \to E$ be the first jet bundle of $E$.
We may regard $J^1 E$ as a vector bundle whose
fiber over $(q,u)\in E$ is $T^*_q \Sigma \otimes T_u X$.
The Lagrangian density for
the string can be defined as a smooth function on $J^1 E$:
$$
\mathcal{L}=\frac{1}{2} g^{ij}\eta_{ab}u^{a}_{i}u^{b}_{j},
$$
which depends in this example only on the fiber coordinates $u^a_{i}$.

From the Lagrangian $\mathcal{L} : J^{1}E \to \mathbb{R}$, the **[[de
Donder-Weyl Hamiltonian]]** $\mathcal{H} : T \Sigma \otimes T^*X \to \mathbb{R}$ can be constructed via a Legendre transform.   It is given as follows:
$$
\mathcal{H}= p^{i}_{a}u^{a}_{i}- \mathcal{L} =\frac{1}{2} \eta^{ab}g_{ij}p_{a}^{i}p_{b}^{j},
$$
where $u^a_{i}$ are defined implicitly by
$p_a^{i}=\partial \mathcal{L} / \partial u^{a}_{i}$,
and $p_a^{i}$ are coordinates on the
fiber $T^{*}_{u}X \otimes T_{q}\Sigma$.
Note that $\mathcal{H}$ differs from the standard (non-covariant) Hamiltonian density for a field theory:
$$
p^{0}_{a}u^{a}_{0} - \mathcal{L}.
$$

Let $\phi$ be a section of $E$ and let $\pi$ be a smooth section of $T \Sigma \otimes T^*X$ restricted to $\phi(\Sigma)$ with fiber coordinates $\pi_{a}^{i}$.  It is then straightforward to show
that $\phi$ is a solution of the Euler-Lagrange equations
if and only if $\phi$ and $\pi$ satisfy the following system of equations:
$$
\frac{\partial \pi^{i}_{a}}{\partial q^i} =
  -  \left.\frac{\partial \mathcal{H}}{\partial
  u^{a}} \right \vert_{u=\phi,p=\pi}
$$
$$
\frac{\partial \phi^{a}}{\partial q^i} = \left.\frac{\partial \mathcal{H}}{\partial
  p^i_{a}} \right \vert_{u=\phi,p=\pi}.
$$

This system of equations is a generalization of Hamilton's equations for the point particle.

As explained above, the covariant phase space for the bosonic string is the dual jet bundle
$(J^1 E)^*$, and this space is equipped with a canonical 2-form $\alpha$
whose exterior derivative $\omega = d \alpha$ is a [[multisymplectic form|multisymplectic 3-form]].
Using the isomorphism
$$         (J^1 E)^* \cong  T \Sigma \otimes T^*X \times \mathbb{R} ,$$
a point in $(J^1 E)^{*}$ gets coordinates $(q^i,u^a,p^{i}_{a},e)$.
In terms of these coordinates,
$$
\alpha= e dq^{0} \wedge dq^{1} +
\left(p_{a}^{0} du^{a} \wedge dq^{1} -
p_{a}^{1} du^{a} \wedge dq^{0} \right) .
$$
The multisymplectic structure on $(J^1 E)^*$ is thus
$$
\omega =
de \wedge dq^0 \wedge dq^{1} +
\left(dp_a^0 \wedge du^a \wedge dq^{1} -
dp_a^1 \wedge du^a \wedge dq^{0} \right) .
$$
So, the variable $e$ may be considered as canonically conjugate to the area form $dq^{0} \wedge dq^{1}$.

As before, let $\phi$ be a section of $E$ and let $\pi$ be a smooth section of $T \Sigma \otimes T^*X$ restricted to $\phi(\Sigma)$.  Consider the submanifold $S \subset (J^1 E)^*$ with coordinates:
$$
(q^i,\phi^{a}(q^j),\pi_{a}^{i}(q^j),-\mathcal{H}).
$$
Note that $S$ is constructed from $\phi$, $\pi$ and
from the constraint $e + \mathcal{H}=0$. This constraint is analogous to the one that is used in finding constant energy solutions in the extended phase space approach to classical mechanics.  At each point in
$S$, a tangent bivector $v=v_{0} \wedge v_{1}$ can be defined as
$$
v_{0} =\frac{\partial}{\partial q^{0}} +
\frac{\partial \phi^a}{\partial q^{0}}\frac{\partial}{\partial u^a} +
\frac{\partial \pi_{a}^{i}}{\partial q^{0}}
\frac{\partial}{\partial p_{a}^{i}}
$$
$$
v_{1} = \frac{\partial}{\partial q^{1}} +
\frac{\partial \phi^a}{\partial q^{1}}
\frac{\partial}{\partial u^a} +
\frac{\partial \pi_{a}^{i}}{\partial q^{1}}
\frac{\partial}{\partial p_{a}^{i}}.
$$

Explicit computation reveals that the submanifold $S$ is
generated by solutions to Hamilton's equations if and only if
$$
\omega(v_{0},v_{1},\cdot)=0.
$$

## Hamiltonian $n$-dimensional flow

* [[Hamiltonian n-vector field]]

## Relation to $n$-symplectic manifolds

There is also the notion of

* [[n-symplectic manifold]]s.

Which is different, but related...

....blah-blah....

## Survey of developments in the field {#SurveyDevelopments}

There is in this sense a covariant form of the [[Legendre transformation]] which associates to every field variable as many conjugated momenta -- the multimomenta -- as there are space-time dimensions. These, together with the field variables, those of $n$-dimensional space-time, and an extra variable, the energy variable, span the multiphase space [1]. For a recent exposition on the differential geometry of this construction, see [2]. Multiphase space, together with a closed, nondegenerate differential $(n+1)$-form, the [[multisymplectic form]], is an example of a multisymplectic manifold [3].

Among the achievements of the multisymplectic approach is a geometric formulation of the relation of infinitesimal symmetries and covariantly conserved quantities ([[Noether's theorem]]), see [4] for a recent review, and [5,6] for a clarification of the improvement techniques ("Belinfante-Rosenfeld formula") of the energy-momentum tensor in classical field theory.

Multisymplectic geometry also provides convenient sets of variational integrators for the numerical study of partial differential equations [7].

Since in multisymplectic geometry, the symplectic 2-form of classical [[Hamiltonian mechanics]] is replaced by a closed [[differential form]] of higher tensor degree, multivector fields and differential forms have their natural appearance. (See [8] for an interpretation of multivector fields as describing solutions to field equations infinitesimally.) Multivector fields form a [[graded Lie algebra]] with the [[Schouten bracket]] (see [9] for a review and unified viewpoint). Using the multisymplectic $(n+1)$-form, one can construct a new bracket for the differential forms, the Poisson forms [10], generalizing a well-known formula for the Poisson brackets related to a symplectic 2-form.

A remarkable fact is that in order to establish a Jacobi identity, the multisymplectic form has to have a potential, a condition that is not seen in symplectic geometry. Further, the admissible differential forms, the Poisson forms, are subject to the rather strong restrictions on their dependence on the multimomentum variables [11]. In particular, $(n-1)$-forms in this context can be shown to arise exactly from the covariantly conserved currents of Noether symmetries [11], which allows their pairing with spacelike hypersurfaces to yield conserved charges in a geometric way.

Not much is known about the interpretation of Poisson forms of form degree between zero and n-1. However, as $(n-1)$-forms describe vector fields and hence collections of lines [2, 10], and as (certain) functions describe n-vector fields and hence collections of bundle sections [8], it seems natural to speculate that the intermediate forms may be useful for the [[brane]]s of [[string theory]].

The Hamiltonian, infinite dimensional formulation of [[classical field theory]] requires the choice of a spacelike hypersurface ("Cauchy surface") [12] which manifestly breaks the general covariance of the theory at hand. For $(n-1)$-forms, the above mentioned new bracket reduces to the Peierls-deWitt bracket after integration over the spacelike hypersurface [13]. With the choice of a hypersurface, a constraint analysis [14] _à la_ Dirac [15,16] can be performed [17]. Again, the necessary breaking of general covariance raises the need for an alternative formulation of all this [18]; first attempts have been made to carry out a [[Marsden-Weinstein reduction]] [19] for multisymplectic manifolds with symmetries [20]. However, not very much is known about how to quantize a multisymplectic geometry, see [21] for an approach using a path integral.

This discussion so far concerns field theories of first order, i.e. where the Lagrangian depends on the fields and their first derivatives. Higher order theories can be reduced to first order ones for the price of introducing auxiliary fields. A direct treatment would involve higher order jet bundles [22]. A definition of the covariant Legendre transform and the multiphase space has been given for this case [3].

## References

### On classical multisymplectic geometry

A comprehensive source on covariant field theory with plenty of further references is

* Mark J. Gotay, James Isenberg, [[Jerrold Marsden]], Richard Montgomery, _Momentum maps and classical relativistic fields. Part I: covariant field theory_ ([arXiv:physics/9801019](http://arxiv.org/abs/physics/9801019))

Much of the material in the [section on covariant field theory](#extendedphasespace) is based on this.

Other discussions include

* {#Helein02} [[Frédéric Hélein]], _Hamiltonian formalisms for multidimensional calculus of variations and perturbation theory_ ([arXiv:math-ph/0212036](http://arxiv.org/abs/math-ph/0212036))

* [[Narciso Román-Roy]], _Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories, SIGMA 5 (2009), 100 ([journal](http://www.emis.de/journals/SIGMA/2009/100/), [arXiv:math-ph/0506022](http://arxiv.org/abs/math-ph/0506022))

The relation to [[covariant phase space]] methods is discussed in

* [[Frédéric Hélein]], _Multisymplectic formalism and the covariant phase_ ([arXiv:1106.2086](http://arxiv.org/abs/1106.2086))
  {#Helein}

A discussion of [[Hamiltonian n-forms]] as [[observables]] is in

* [[Frédéric Hélein]], _The notion of observable in the covariant Hamiltonian formalism for the calculus of variations with several variables_ ([arXiv:math-ph/0401047](http://arxiv.org/abs/math-ph/0401047))

Other texts include

* [[Carlo Rovelli]], _Covariant hamiltonian formalism for field theory: Hamilton-Jacobi equation on the space $\mathcal{G}$_ ([arXiv:gr-qc/0207043](http://arxiv.org/abs/gr-qc/0207043))

Much of the above [survey of recent developments](#SurveyDevelopments) and of the following list of references is reproduced from the web-page

* [[Cornelius Mertzlufft-Paufler]], _[Multisymplectic geometry](http://www.mertzlufft-paufler.de/multisymplectic_geometry.html)_

References mentioned above are

* [1] J. Kijowski, W. Szczyrba, _A Canonical Structure For Classical Field Theories_ . Commun. Math. Phys. 46 (1976) 183.

* [2] M. J. Gotay, J. Isenberg, J. E. Marsden: _Momentum maps and classical relativistic fields. I: Covariant field theory_ [arXiv:physics/9801019v2](http://arxiv.org/abs/physics/9801019v2).

* [3] M. J. Gotay, _A multisymplectic framework for classical field theory and the calculus of variations. I: Covariant Hamiltonian formalism_
In M. Francaviglia (ed.), _Mechanics, analysis and geometry: 200 years after Lagrange_ Amsterdam etc.: North-Holland (1991), 203--235.

* [4] M. de Leon, D. Martin de Diego, A. Santamaria-Merino, _Symmetries in Classical Field Theory_ [arXiv:math-ph/0404013](http://arxiv.org/abs/math-ph/0404013).

* [5] M. J. Gotay, J. E. Marsden: _Stress-energy-momentum tensors and the Belinfante--Rosenfeld formula_ Contemp. Math. vol. 132, AMS, Providence, 1992, 367--392.

* [6] M. Forger, H. Römer, _Currents and the energy-momentum tensor in classical field theory: a fresh look at an old problem_ Ann. Phys. (N.Y.) 309 (2004) 306--389. [arXiv:hep-th/0307199](http://arxiv.org/abs/hep-th/0307199).

* [7] A. Lew, J. E. Marsden, M. Ortiz, M. West, _An overview of variational integrators_
In L. P. Franca (ed.), Finite Element Methods: 70's and Beyond. Barcelona (2003).

* [8] C. Paufler, H. Römer, _The Geometry of Hamiltonian $n$-vector fields in Multisymplectic Field Theory_ J. Geom. Phys. 44, No.1(2002), 52--69. [arXiv:math-ph/0102008](http://arxiv.org/abs/math-ph/0102008).

* [9] Y. Kosmann-Schwarzbach: Derived brackets. Lett. Math. Phys. 69 (2004) 61--87 [arXiv:math.DG/0312524](http://arxiv.org/abs/math.DG/0312524).

* [10] M. Forger, C. Paufler, H. Römer: _The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory_ Rev. Math. Phys. 15 (2003) 705 [arXiv:math-ph/0202043](http://arxiv.org/abs/math-ph/0202043).

* [10] M. Forger, C. Paufler, H. Römer, _Hamiltonian Multivector Fields and Poisson Forms in Multisymplectic Field Theory_ [arXiv:math-ph/0407057](http://arxiv.org/abs/math-ph/0407057).

* [11] M. J. Gotay, _A multisymplectic framework for classical field theory and the calculus of variations. II: Space + time decomposition_ Differ. Geom. Appl. 1(4) (1991), 375--390.

* [12] M. O. Salles, _Campos Hamiltonianos e Colchete de Poisson na Teoria Geométrica dos Campos_ , PhD thesis, IME-USP, June 2004.

* [13] M. J. Gotay, J. M. Nester, _Generalized constraint algorithm and special presymplectic manifolds_
In G. E. Kaiser, J. E. Marsden, Geometric methods in mathematical physics, Proc. NSF-CBMS Conf., Lowell/Mass. 1979, Berlin: Springer-Verlag, Lect. Notes Math. 775 (1980) 78--80.

* [14] P. A. M. Dirac, _Lectures on Quantum Mechanic_ Belfer Graduate School of Science, Yeshiva University, N.Y., 1964.

* [15] [[Marc Henneaux]], [[Claudio Teitelboim]], _[[Quantization of Gauge Systems]]_ Princeton University Press, 1992.

* [16] M. J. Gotay, J. Isenberg, J. E. Marsden, R. Montgomery, _Momentum Maps and Classical Relativistic Fields II: Canonical Analysis of Field Theories_ (2004) [arXiv:math-ph/0411032](http://arxiv.org/abs/math-ph/0411032).

* [17] N. P. Landsman, _Against the Wheeler-DeWitt equation_ Class. Quan. Grav. 12 (1995) L119-L124. [arXiv:gr-qc/9510033](http://arxiv.org/abs/gr-qc/9510033).

* [18] J. E. Marsden, A. Weinstein _Reduction of symplectic manifolds with symmetry_ Rept. Math. Phys. 5 (1974) 121--130.

* [19] F. Munteanu, A. M. Rey, M. Salgado, _The Günther's formalism in classical field theory: momentum map and reduction_ J. Math. Phys. 45, No. 5 (2004) 1730--1750.

* [20] D. Bashkirov, G. Sardanashvily, _Covariant Hamiltonian Field Theory. Path Integral Quantization_
[arXiv:hep-th/0402057](http://arxiv.org/abs/hep-th/0402057).

* [21] D. J. Saunders, _The Geometry of Jet Bundles_ Lond. Math. Soc. Lect. Notes Ser. 142, Cambr. Univ. Pr., Cambridge, 1989.

A [[higher category theory|higher categorial]] interpretation of [[2-plectic geometry]] and connection to $L_\infty$-algebras is given in

* [[John Baez]], [[Chris Rogers]], _Categorified symplectic geometry and the string Lie 2-algebra_. [arXiv:0901.4721](http://arxiv.org/abs/0901.4721).

* [[John Baez]], Alexander E. Hoffnung, [[Chris Rogers]], _Categorified symplectic geometry and the classical string_ ([arXiv:0808.0246](http://arxiv.org/abs/0808.0246/))

* [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis (2011) ([arXiv:1106.4068](http://arxiv.org/abs/1106.4068)); _$L_\infty$-algebras from multisymplectic geometry_, Lett. Math. Phys., 100(1):29–50 (2012); _2-plectic geometry, Courant algebroids, and categorified prequantization_, J. Symplectic Geom. 11(1):53–91 (2013)

* Antonio Michele Miti, [[Marco Zambon]], _Observables on multisymplectic manifolds and higher Courant algebroids_, [arXiv:2209.05836](https://arxiv.org/abs/2209.05836)

Higher order [[moment maps]] are considered in

* Thomas Bruun Madsen, Andrew Swann, _Closed forms and multi-moment maps_ ([arXiv:1110.6541](http://arxiv.org/abs/1110.6541))

* Martin Callies, [[Yaël Frégier]], [[Chris Rogers]], [[Marco Zambon]], _Homotopy moment maps_, Advances in Mathematics __303__ (2016) 954--1043 ([arXiv:1304.2051](https://arxiv.org/abs/1304.2051))

A [[higher differential geometry]]-generalization of [[contact geometry]] in line with multisymplectic geometry/[[n-plectic geometry]] is discussed in

* Luca Vitagliano, _$L_\infty$ algebras from multicontact geometry_, Diff. Geom. Appl. __39__ (2015) 147-165 [arXiv:1311.2751](http://arxiv.org/abs/1311.2751) [doi](https://doi.org/10.1016/j.difgeo.2015.01.006)

### Relation to covariant phase space formalism

On the relation of multisymplectic formalism to [[covariant phase space]] and [[variational bicomplex]] methods:

* {#ForgerRomero04} [[Michael Forger]], [[Sandro Romero]], *Covariant Poisson brackets in geometric field theory*, Commun. Math. Phys. **256** (2005) 375-410 [[arXiv:math-ph/0408008](http://arxiv.org/abs/math-ph/0408008), [doi:10.1007/s00220-005-1287-8](https://doi.org/10.1007/s00220-005-1287-8)]

* [[Narciso Román-Roy]], _Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories_, SIGMA 5 (2009), 100, 25 pages ([arXiv:math-ph/0506022](http://arxiv.org/abs/math-ph/0506022))

On multisymplectic [[boundary field theory]]:

* [[José Zapata]], _Multisymplectic effective General Boundary Field Theory_ (2013) [[pdf](http://relativity.phys.lsu.edu/ilqgs/zapata090313.pdf), [[Zapata-MultisymplecticBoundary.pdf:file]]]

### On quantization of multisymplectic geometry {#RefsonQuantization}

The following articles discuss the [[quantization]] procedure for multisymplectic geometry, generalizing [[geometric quantization]] of [[symplectic geometry]].

* [[Igor V. Kanatchikov]], _De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory_, Rept. Math. Phys. **43** (1999) 157-170 [[arXiv:hep-th/9810165](http://arxiv.org/abs/hep-th/9810165), <a href="https://doi.org/10.1016/S0034-4877(99)80024-X">doi:10.1016/S0034-4877(99)80024-X</a>]

* [[Igor V. Kanatchikov]] *Geometric (pre)quantization in the polysymplectic approach to field theory*, in *Differential Geometry and Its Applications*, Proc. Conf., Opava (August 2001) [[arXiv:hep-th/0112263](https://arxiv.org/abs/hep-th/0112263), [pdf](http://conferences.math.slu.cz/8icdga/PDF/309-322.pdf)]

Kanatchikov's "algebra of observables" is what he calls a "higher-order Gerstenhaber algebra". (The "bracket" in this structure fails to satisfy Leibniz's rule as a derivation of the product.) The relationship between it and the [[Lie superalgebra]] of observables constructed by Forger, Paufler, and Roemer is discussed in this paper:

* [[Michael Forger]], C. Paufler, and H. Roemer, _The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory_ , ([arXiv:math-ph/0202043](http://arxiv.org/abs/math-ph/0202043v1))

and ([Forger-Romero 04](#ForgerRomero04)) above.

Kanatchikov's formalism was used by S.P. Hrabak to propose a multisymplectic refinement of [[BV-BRST formalism]]. See there for more details.

[[!redirects covariant field theory]]

[[!redirects extended phase space]]
[[!redirects extended phase spaces]]

[[!redirects multisymplectic manifold]]
[[!redirects multisymplectic manifolds]]


multivalued group

[[!redirects n-valued group]]

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of *$n$-valued groups* is an analogue of [[groups]] where the multiplication takes values in the $n$-th "symmetric power" of the [[underlying]] set. 2-valued [[formal groups]] appeared first in the study of [[characteristic classes]] by [Buchstaber & Novikov 1971](#BuchstaberNovikov71).

## Definition

Let $n$ be a [[positive number|positive]] [[integer]]. Denote by $Sym^n(G)$ the $n$-th symmetric power of a set $G$, or equivalently the set of [[multisets]] of $n$ not necessarily different elements $[g_1,\ldots,g_n]$ in $G$.

An *$n$-valued group* is given by a [[set]] $G$ together with a function ("multiplication")
$$
  \ast
    \;\colon\;
  G \times G \longrightarrow Sym^n(G)
  \,,
$$

satisfying:

* ([[associativity]]) The following $n^2$-element
multisets are equal:

  $\big[ a\ast (b\ast c)_1, a\ast (b\ast c)_2,\ldots, a\ast
(b\ast c)_n\big]$ and $\big[(a\ast b)_1\ast c,\ldots,(a\ast b)_n\ast c\big]$,

* ([[unitality]]) there is a [[neutral element]] $\mathrm{e}$ such that for all $x\in G$, $\mathrm{e} \ast x = x \ast \mathrm{e} = [x,x,\ldots,x]$,

* ([[inverses]]) For every $g\in G$ there exists $g^{-1}\in G$ such that $\mathrm{e} \in g^{-1}\ast g$ and $\mathrm{e} \in g\ast g^{-1}$.

## Literature

Origin of the notion in multivalued [[formal groups]] appearing in [[complex oriented cohomology theory]]:

* {#BuchstaberNovikov71} В. М. Бухштабер, С. П. Новиков, Формальные группы, степенные системы и операторы Адамса, Мат.
Сборник (Н.С.) 84 (1971) 81--118;

  English translation:

  [[Victor M. Buchstaber]], [[Sergei P. Novikov]], _Formal groups, power systems and Adams operators_, Math. USSR-Sb. **13** (1971) 80-116 [[doi:10.1070/SM1971v013n01ABEH001030](https://iopscience.iop.org/article/10.1070/SM1971v013n01ABEH001030)]

* [[Victor M. Buchstaber]], Two-valued formal groups. Some applications to cobordisms, Uspehi Mat. Nauk **26** 3 (1971),  (159) 195–196 [MR 0461533]

* [[Victor Buchstaber]], §5 in: _Cobordisms in problems of algebraic topology_,  J Math Sci **7** (1977) 629–653 [[doi:10.1007/BF01084983](https://doi.org/10.1007/BF01084983)]

The notion has in fact been sporadically studied much earlier, e.g. in the context of hypergroups.

* J. Delsarte, _Hypergroupes et opérateurs de permutation et de transmutation_, La théorie des équations aux dérivées partielles. Nancy, 9-15 avril 1956, pp. 29--45
Colloq. Internat. CNRS, LXXI [International Colloquia of the CNRS] Centre National de la Recherche Scientifique, Paris, 1956 [MR0116151](https://mathscinet.ams.org/mathscinet/relay-station?mr=MR0116151)

Some newer articles

* D. Coulette, F. Déglise, J Fasel, J. Hornbostel, _Formal ternary laws and Buchstaber’s 2-groups_, Manuscripta Math. (2023) [doi](https://doi.org/10.1007/s00229-023-01512-4)

* В. М. Бухштабер, Функциональные уравнения ассоциированные с теоремами сложения для эллиптических функций и двузначные алгебраические группы, Успехи Мат. Наук 45:3 (1990) 185--186 (transl.: V. M. Bukhstaber, _Functional equations that are associated with addition theorems for elliptic functions and two-valued algebraic groups_, Russian Math. Surveys 45:3 (1990) 213--215.)

In 2010. V. Dragović has observed that the integrability of Kovalevskaia top is related to associativity of certain $n$-valued group. A related later developments on [[integrable systems]] are touched upon in:

* [[Victor M. Buchstaber]], Vladimir Dragović, _Two-valued groups, Kummer varieties, and integrable billiards_, Arnold Math J. 4, 27--57 (2018) [[doi:10.1007/s40598-018-0085-2](https://doi.org/10.1007/s40598-018-0085-2)]

[[!redirects multivalued groups]]

[[!redirects n-valued group]]
[[!redirects n-valued groups]]

[[!redirects 2-valued group]]
[[!redirects 2-valued groups]]

category: algebra


multivalued partial function

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

\tableofcontents

## Definition

In [[dependent type theory]], given types $A$ and $B$, a **multivalued partial function** or **partial multivalued function** from $A$ to $B$ is a type family $x:A \vdash P(x)$ with a function

$$f:\left(\sum_{x:A} P(x)\right) \to B$$

or equivalently by [[currying]],

$$f:\prod_{x:A} P(x) \to B$$

or equivalently

$$x:A, p:P(x) \vdash f(x, p):B$$

## Properties

In dependent type theory, [[correspondences]] are [[type families]] $x:A, y:B \vdash R(x, y)$, and [[spans]] are types $C$ with functions $x:C \vdash g_A(x):A$ and $x:C \vdash g_B(x):B$. Correspondences and spans are interdefinable with multivalued partial functions:

* From every span one could get a multivalued partial function by defining the type family $x:A \vdash P(x)$ as $P(x) \coloneqq \sum_{y:C} g(y) =_A x$ and the family of elements $x:A, p:P(x) \vdash f(x, p):B$ as $f(x, p) \coloneqq h(\pi_1(p))$.

* From every multivalued partial function one could get a span by defining the type $C$ as $C \coloneqq \sum_{x:A} P(x)$ and the family of elements $x:C \vdash g(x):A$ as $g(x) \coloneqq \pi_1(x)$.

* From every multivalued partial function one could get a correspondence by defining the type family $x:A, y:B \vdash R(x, y)$ as $R(x, y) \coloneqq \sum_{p:P(x)} f(x, p) =_B y$.

* From every correspondence one could get a multivalued partial function by defining the type family $x:A \vdash P(x)$ as $P(x) \coloneqq \sum_{y:B} R(x, y)$, and the family of elements $x:A, p:P(x) \vdash h(x, p):B$ as $h(x, p) \coloneqq \pi_1(p)$

* From every span one could get a correspondence by defining the type family $x:A, y:B \vdash R(x, y)$ as $R(x, y) \coloneqq \sum_{z:C} (g(z) =_A x) \times (h(z) =_B y)$.

* From every correspondence one could get a span by defining the type $C$ as $C \coloneqq \sum_{x:A} \sum_{y:B} R(x, y)$, the family of elements $z:C \vdash g(z):A$ as $g(z) \coloneqq \pi_1(z)$, and the function $z:C \vdash h(z):B$ as $h(z) \coloneqq \pi_1(\pi_2(z))$

## Examples

* A multivalued partial function for which there is an element $p(x):P(x)$ for all $x:A$ is a [[multivalued function]].

* A multivalued partial function for which the dependent type $P(x)$ is a [[mere proposition]] for all $x:A$ is a [[partial function]].

* A multivalued partial function for which the dependent type $P(x)$ is a [[contractible type]] for all $x:A$ is a [[total function]].

## Dependent multivalued partial functions

In the same way that there is a notion of [[dependent function]] in addition to (non-dependent) function in dependent type theory, there is also a notion of dependent multivalued partial functions in addition to non-dependent multivalued partial functions.

In [[dependent type theory]], given a type $A$ and a type family $x:A \vdash B(x)$, a **dependent multivalued partial function** or **dependent partial multivalued function** from $A$ to $B$ is a type family $x:A \vdash P(x)$ with a dependent function

$$f:\prod_{x:A} P(x) \to B(x)$$

or equivalently

$$x:A, p:P(x) \vdash f(x, p):B(x)$$

## See also

* [[correspondence]], [[correspondence type]]
* [[multivalued function]]
* [[partial function]]

[[!redirects multivalued partial function]]
[[!redirects multivalued partial functions]]
[[!redirects partial multivalued function]]
[[!redirects partial multivalued functions]]


multivector field

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Definition

For $X$ a [[smooth manifold]] and $T X$ its [[tangent bundle]] a **multivector field** on $X$ is an element of the [[exterior algebra]] bundle $\wedge^\bullet_{C^\infty(X)}(\Gamma(T X))$ of skew-symmetric [[tensor power]]s of [[section]]s of $T X$.

* In degree $0$ these are simply the [[smooth function]]s on $X$.

* In degree $1$ these are simply the [[tangent vector field]]s on $X$.

* In degree $p$ these are sometimes called the __$p$-vector fields__ on $X$.

## Properties

### Hochschild cohomology

In suitable contexts, multivector fields on $X$ can be identified with the [[Hochschild cohomology]] $HH^\bullet(C(X), C(X))$ of the algebra of functions on $X$.

### Schouten bracket

There is a canonical bilinear pairing on multivector fields called the [[Schouten bracket]].

### Isomorphisms with de Rham complex

Let $X$ be a [[smooth manifold]] of [[dimension]] $d$, which is equipped with an [[orientation]] exhibited by a [[differential form]]
$\omega \in \Omega^d(X)$.

Then contraction with $\omega$ induces for all $0 \leq n \leq d$ an [[isomorphism]] of [[vector space]]s

$$
  \omega(-) :
  T^n_{poly}(X)
    \stackrel{\simeq}{\to}
  \Omega^{(d-n)}(X)
  \,.
$$

The transport of the [[de Rham complex|de Rham differential]] along these isomorphism equips $T^\bullet_{poly}$ with the structure of a [[chain complex]]

$$
  \array{
    \Omega^{n}(X) &\stackrel{d_{dR}}{\to}& \Omega^{n+1}(X)
    \\
    \downarrow^{\mathrlap{\simeq}}
      &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    T^{d-n}_{poly} &\stackrel{div_\omega}{\to}&
    T^{d-n-1}_{poly}
  }
  \,,
$$

The operation $div_\omega$ is a [[derivation]] of the [[Schouten bracket]] and makes multivectorfields into a [[BV-algebra]].

A more general discussion of this phenomenon in ([Cattaneo–Fiorenza–Longoni](#CattaneoFiorenzaLongoni)). Even more generally, see [[Poincaré duality]] for [[Hochschild cohomology]].

## Related concepts

* [[bivector]]

## References

The isomorphisms between the de Rham complex and the complex of polyvector field is reviewed for instance on p. 3 of

* [[Thomas Willwacher]], [[Damien Calaque]], _Formality of cyclic cochains_ ([arXiv:0806.4095](http://arxiv.org/abs/0806.4095))
{#WillwacherCalaque}

and in section 2 of

* [[Alberto Cattaneo]], [[Domenico Fiorenza]], Riccardo Longoni, _On the Hochschild-Kostant-Rosenberg map for graded manifolds_, [arXiv:math/0503380](https://arxiv.org/abs/math/0503380)
{#CattaneoFiorenzaLongoni}

and on p. 6 of

* [[Claude Roger]], _Gerstenhaber and Batalin-Vilkovisky algebras_, Archivum mathematicum, Volume 45 (2009), No. 4 ([pdf](http://www.emis.de/journals/AM/09-4/roger.pdf))

[[!redirects multivector field]]
[[!redirects multivector fields]]
[[!redirects multi-vector field]]
[[!redirects multi-vector fields]]
[[!redirects multi vector field]]
[[!redirects multi vector fields]]

[[!redirects multivector]]
[[!redirects multivectors]]

[[!redirects multi-vector]]
[[!redirects multi-vectors]]

[[!redirects tangent p-vector field]]
[[!redirects tangent p-vector fields]]

[[!redirects p-vector field]]
[[!redirects p-vector fields]]

[[!redirects polyvector field]]
[[!redirects polyvector fields]]
[[!redirects poly-vector field]]
[[!redirects poly-vector fields]]
[[!redirects poly vector field]]
[[!redirects poly vector fields]]


multiverse

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Cosmology
+-- {: .hide}
[[!include cosmology -- contents]]
=--
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]], the term "multiverse" refers to certain picture of [[cosmology]]. The intended meaning tends to differ between authors and/or remain vague (see [below](#General)) but broadly what goes with it is the idea that, just as the configuration of [[matter]] and [[energy]] changes from place to place within our [[observable universe]], clearly, it could be that _beyond_ the [[observable universe]] (for immediate or not so immediate notions of "beyond") eventually even the laws of [[physics]] might change from place to place, possibly altogether, but at least as concerns the values of [[coupling constants]] or even [[field (physics)|field]]-content.

The idea remains hypothetical, naturally, and might ultimately be empty as an idea about [[physics]] as opposed to [[philosophy]], but it is arguably in historical continuation with a sequence of famously confirmed insights about [[cosmology]], which led from an archaic picture of a cosmos centered all around the world directly experienced by humans (the continent a disk inside an ocean hung in the center of the heavenly spheres), via the [Copernican revolution](https://en.wikipedia.org/wiki/Copernican_Revolution), then the confirmation of distant [[galaxies]] etc., to the realization that the [[observable universe]] is immense and varied and our physical place and whereabouts inside it just a random result of chance, subject only to the tautological condition that it allows us to be here at all -- known as the "anthropic principle" in these discussions.

Accordingly, one motivation for the idea of the multiverse is as an explanation for apparent '[[naturalness|fine-tuning]]' in the "constants of nature", in particular concerning the presence of complex life. The thought here is that were the constants/laws of nature to be ever so slightly different from what they are, then the universe would not have been to able to evolve to support complex, intelligent life forms (see [SEP: Fine-tuning](https://plato.stanford.edu/archives/fall2017/entries/fine-tuning/)). But an account which proposes that a vast number of universes exist governed by laws spanning some full range of possibilities removes the surprise that we should live in such a special universe, since any form of life as ourselves could only occur within such a universe.

Another motivation of contemplating a multiverse, even in the face of its ontological elusiveness, comes from attempts of formulating fundamental [[theory (physics)|theories of physics]] such as [[quantum gravity]] or [[theories of everything]], since the motivation and success of such theories depends to some extent on what one thinks they can or should explain or predict, and how much room for chance and chaos there may remain in a truly fundamental theory of physics.

For example, if a fundamental [[theory of everything]] were to _predict_, say, the [[generations of fundamental particles]] and the values of the [[Yukawa couplings]] in the [[standard model of particle physics]], then these would be fixed -- by pure [[logic]], ultimately -- to be what they are, and would be the same strictly "everywhere", thus making obsolete the idea of a multiverse.

Part of the interest in the idea of the multiverse derives from the feeling that this scenario of the [[standard model of particle physics]], with its particular [[field (physics)|field content]] and [[coupling constants]], being fixed by pure logic, is no more likely than [[Johannes Kepler]]'s attempt to derive the distances in the solar system from the [[Platonic solids]] (the old _[[Music of the Spheres]]_), which today we understand is a confusion of what is fundamental law and what is [[random variable|random]] initial conditions.

In this sense, the idea of the multiverse is a form of [[empiricism]] and in contrast to [[theoretical aestheticism]] such as _[[universal exceptionalism]]_.

### General
 {#General}

On the very largest scales observable in astrophysical [[experiment]], the cosmos is well described by an [[FRW model]] with [[cosmological constant]] $\Lambda$, with plenty of [[dark matter]] and with primordial [[cosmic inflation]] (called the $\Lambda$-CDM "[[standard model of cosmology|concordance model of cosmology]]"). A [[horizon]] in there and everything on smaller scales is the _[[observable universe]]_.

There has never been a reason to assume that beyond this _cosmic horizon_ visible to us today, the cosmos would not extend further. In fact in the simple [[FRW models]] with positive cosmological constant, the [[spacetime]] [[manifold]] is not a [[closed manifold]] and extends indefinitely beyond our observable horizon. This is directly analogous to the ancient mariner who would stand on the deck of his ship and see a few miles of ocean around him. That was the world within his horizon, but there was no telling what lay beyond.

Therefore it is important to distinguish the _[[observable universe]]_ from the _universe_ as such. Unfortunately, intellectual laziness tended to ignore this distinction and at some point some people who said "universe" to mean just the stretch of our cosmic horizon felt the need to have a new term for whatever may lie beyond. For better or worse, that new term has become wide-spread these days and is "multiverse".

### In view of cosmic inflation

The main scientific observation that goes with this is the observation that [[cosmic inflation]] -- for which there is by now excellent and ever increasing [[experiment|experimental]] evidence that indeed it happened -- does, at least by the simple method which is currently used to model it, naturally predict a large ambient space in which local regions perpetually undergo inflationary expansion -- [[eternal inflation]].

Hence what is called _multiverse_ these days might more sanely have been called _universe with multiple observable regions_ or the like, but the term has seen an immense increase in popularity and is apparently here to stay.

### In view of moduli fields

Another reason why the old speculation (going back at least to [[Giordano Bruno]]) that the [[observable universe]] is but one of many variants in a larger ambient cosmos received more empathetic attention in some corners of fundamental [[physics]] at the beginning of the 21st century is that it was argued to lead to a plausible picture of the apparent lack of the fundamental constants of nature (such as notably the [[cosmological constant]] but also the [[Yukawa couplings]] etc.) to follow any discernible law (see also the "[[hierarchy problem]]").

At the same time there are [[theory (physics)|theories]] of fundamental physics where such apparent "constants" of nature are in fact just the values of "[[moduli fields]]", hence of dynamical [[field (physics)|fields]], whose value however is approximately constant over the [[observable universe]] ("[[moduli stabilization]]"). Generically [[Kaluza-Klein mechanism|Kaluza-Klein theories]] have this property, and in particular [[string theory]] does. This has the remarkable consequence that in such theories the "constants of nature" may dynamically move through their [[moduli space]] -- hence through the "landscape" of possible sets of laws of nature -- just like any other field evolves dynamically throughout [[spacetime]]. (See also at [[string theory FAQ]], _[What does it mean that string theory has a "landscape" of solutions?](string+theory+FAQ#WhatDoesItMeanToSayStringTheoryHasALandscapeOfSolutions)_.)

When the [[landscape of string theory vacua]] was (finally) realized to, apparently, have no reason to be "small", hence when it was realized that indeed the theory alone did not constrain its [[moduli fields]] but allowed all kinds of values for them, then it was argued (first in ([Schellekens 98](#Schellekens98))) that combining this apparent embarrassment of a surplus of theoretically admissible "kinds of universes" with the apparent embarrassment of a surplus of [[cosmic inflation|cosmically inflating]] [[observable universe|observable universes]] together yields a plausible picture: each inflating "bubble" in [[eternal inflation]] randomly has its [[moduli fields]] determined by the [[vacuum fluctuation]] that gives rise to the inflationary process, and hence literally the [[moduli space]] of "constants/laws of nature" becomes inhabited by a kind of [[stochastic process]]. The particular value of the [[cosmological constant]] or the [[Yukawa couplings]] which we observe would be just as random (or just as "anthropically" constrained) as, say, the number, nature and distance of the planets in the [[solar system]].

Whatever the status of this story, it is to a large part responsible for the renewed interest in the idea of the "multiverse".

One must beware here that, while [[Kaluza-Klein theory]]/[[string theory]] constitute concrete theoretical frameworks realizing the concept of dynamically evolving "constants of nature", the [[landscape of string theory vacua]] is poorly understood in detail, even by the standards of non-rigorous physics-style arguments; and, worse, [[string theory]] itself is fully understood only [[perturbative quantum field theory|perturbatively]], while its full microscopic  [[non-perturbative effect|non-perturbative completion]] (working title: "[[M-theory]]", "[[F-theory]]"), despite a large and tight network of hints, remains an open problem, for the time being. But it is exactly [[non-perturbative effects]] that may be expected to be relevant for true understanding of the [[landscape of string theory vacua]], by possibly ruling out regions which look consistent only in perturbation theory. Hence for all that is known, [[universal exceptionalism]] instead of a multiverse remains a possibility in [[string theory]]. Only improved mathematical understanding of the theory will eventually be able to tell.

## Critique

From ([Ellis 11](#Ellis11)):

> Similar claims [about a multiverse] have been made since antiquity by many cultures. What is new is the assertion that the multiverse is a scientific theory, with all that implies about being mathematically rigorous and experimentally testable. I am skeptical about this claim. I do not believe the existence of those other universes has been proved—or ever could be. Proponents of the multiverse, as well as greatly enlarging our conception of physical reality, are implicitly redefining what is meant by 'science.'—pg 39

> For a cosmologist, the basic problem with all multiverse proposals is the presence of a cosmic visual horizon. The horizon is the limit to how far away we can see, because signals traveling toward us at the speed of light (which is finite) have not had time since the beginning of the universe to reach us from farther out. All the parallel universes lie outside our horizon and remain beyond our capacity to see, now or ever, no matter how technology evolves. In fact, they are too far away to have had any influence on our universe whatsoever. That is why none of the claims made by multiverse enthusiasts can be directly substantiated. —pg 40-41

> A remarkable fact about our universe is that physical constants have just the right values needed to allow for complex structures, including living things. Steven Weinberg, Martin Rees, Leonard Susskind and others contend that an exotic multiverse provides a tidy explanation for this apparent coincidence: if all possible values occur in a large enough collection of universes, then viable ones for life will surely be found somewhere. This reasoning has been applied, in particular, to explanation the density of the dark energy that is speeding up the expansion of the universe today. I agree that the multiverse is a possible valid explanation for the value of this density; arguably, it is the only scientifically based option we have right now. But we have no hope of testing it observationally. —pg 42

> All in all, the case for the multiverse is inconclusive. The basic reason is the extreme flexibility of the proposal: it is more a concept than well-defined theory. Most proposals involve a patchwork of different ideas rather than a coherent whole. The basic mechanism for eternal inflation does not itself cause physics to be different in each domain in a multiverse; for that, it needs to be coupled to another speculative theory. Although they can be fitted together, there is nothing inevitable about it. … Nothing is wrong with scientifically based philosophical speculation, which is what multiverse proposals are. But we should name it for what it is. —pg 43

> As skeptical as I am, I think the contemplation of the multiverse is an excellent opportunity to reflect on the nature of science and on the ultimate nature of existence: why we are here… In looking at this concept, we need an open mind, though not too open. It is a delicate path to tread. Parallel universes may or may not exist; the case is unproved. We are going to have to live with that uncertainty. Nothing is wrong with scientifically based philosophical speculation, which is what multiverse proposals are. But we should name it for what it is.

## Related concepts
 {#RelatedConcepts}

* Vaguely related is the [[possible worlds semantics]] in [[modal logic]] for the formalization of [[metaphysics|metaphysical]] [[necessity]] and [[possibility]]. For instance ([Kripke 80](#Kripke80)) is much concerned with the logic of "[[possible worlds semantics|possible worlds]]".

* Not really related but arguably somewhat analogous is, in the [[foundations of mathematics]], the [[set-theoretic multiverse]] of different [[set theories]], and similarly the [[2-category]] [[Topos]] of [[toposes]], hence of [[constructive mathematics|constructive]] [[foundations of mathematics]] (see at _[[topos theory]]_ for more on this).

## References

Apparently the first mentioning of the idea that a multiverse due to [[eternal inflation]] together with the [[landscape of string theory vacua]] provides a plausible story for the nature of "constants of nature" is due to

* {#Schellekens98} [[Bert Schellekens]], _Naar een waardig slot_, inauguration speech at University of Nijmegen, September 1998, ISBN 90-9012073-4

reproduced as

* {#Schellekens06} [[Bert Schellekens]], _The Landscape "avant la lettre"_ ([arXiv:physics/0604134](http://arxiv.org/abs/physics/0604134))

More recent account of how the relevant community pictures the situation includes

* {#Bousso11} [[Raphael Bousso]], _The State of the Multiverse: The String Landscape, the Cosmological Constant, and the Arrow of Time_, 2011 ([pdf](http://www.ctc.cam.ac.uk/stephen70/talks/swh70_bousso.pdf))

* [[Brian Greene]], _Is our universe the only universe?_, talk at TED2012 ([web](http://www.ted.com/talks/brian_greene_why_is_our_universe_fine_tuned_for_life))

* [[Andrei Linde]], _A brief history of the multiverse_ ([arXiv:1512.01203](https://arxiv.org/abs/1512.01203))

Further and popular expositions include

* {#Ellis11} [[George Ellis]], _Does the multiverse really exist?_, Scientific American, August 2011  ([web](http://www.scientificamerican.com/article/does-the-multiverse-really-exist/)

For more references see at _[[eternal cosmic inflation]]_.

Of course a bunch of links are also here:

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

though the discussion there seems to show an undue preference of some points of view.

"Possible worlds" in the context of [[logic]] are discussed in [[modal logic]], notably in

* {#Kripke80} [[Saul Kripke]], _[[Naming and Necessity]]_ (1980)


Mumford class

There is a very beautiful story on several families of stable classes of the compactification $\mathcal{M}_{g,n}$ of the moduli space of algebraic genus $g$ curves with $n$ marked points, playing major role in geometry (especially intersection theory), Gromov-Witten theory, etc. They are sometimes referred by standard notation and sometimes by their names; one has families of Mumford-Miller classes and of Morita classes.

Kontsevich associates to any cyclic/symplectic $A_\infty$-algebra certain partition function which is a inhomogeneous class in [[graph homology]].
Igusa and Mondello have substantiated Kontsevich's claim that the Mumford-Miller-Morita classes are induced from certain family of $A_\infty$-algebras;

Related entries [[Witten conjecture]], [[mapping class group]].

* [[David Mumford]], _Towards an enumerative geometry of the moduli space of curves_, Arithmetic and geometry, Vol. II, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328, [MR85j:14046](http://www.ams.org/mathscinet-getitem?mr=85j:14046)
* E. Y. Miller, _The homology of the mapping class group_, J. Differential Geom. __24__ (1986), no. 1, 1–14.
* John Harer, _The cohomology of the moduli space of curves_, Lec. Notes in Math. 1337, p. 138–221. Springer, Berlin, 1988.
* Ib Madsen, Michael Weiss, _The stable moduli space of Riemann surfaces: [[Mumford's conjecture]]_,  Ann. of Math. (2) __165__ (2007), no. 3, 843--941, [MR2009b:14051](http://www.ams.org/mathscinet-getitem?mr=2009b:14051), [doi](http://dx.doi.org/10.4007/annals.2007.165.843), [math.AT/0212321](http://arxiv.org/abs/math.AT/0212321)
* Shigeyuki Morita, _Characteristic classes of surface bundles_, Invent. Math. __90__ (1987),  no. 3, 551--577, [doi](http://dx.doi.org/10.1007/BF01389178), [MR89e:57022](http://www.ams.org/mathscinet-getitem?mr=89e:57022)
* Sh. Morita, _Introduction to mapping class groups of surfaces and related groups_, in: Handbook of Teichmüller theory (A. Papadopoulos, editor), vol. I, EMS Publishing House, Zürich, 2007, 353–386.
* C. Faber, R. Pandharipande, _Logarithmic series and Hodge integrals in the tautological ring_, With an appendix by Don Zagier. Dedicated to William Fulton on the occasion of his 60th birthday.  Michigan Math. J. __48__ (2000), 215--252, [MR2002e:14041](http://www.ams.org/mathscinet-getitem?mr=2002e:14041), [doi](http://dx.doi.org/10.1307/mmj/1030132716)
* Carel Faber, _Hodge integrals, tautological classes and Gromov-Witten theory_, Proc. Workshop _Algebraic Geometry and Integrable Systems related to String Theory_ (Kyoto, 2000).  Sūrikaisekikenkyūsho Kōkyūroku __1232__ (2001), 78--87, [MR1905884](http://www.ams.org/mathscinet-getitem?mr=1905884)
* K. Igusa, _Graph cohomology and Kontsevich cycles_, Topology __43__ (2004), n. 6, p. 1469-1510, [MR2005d:57028](http://www.ams.org/mathscinet-getitem?mr=2005d:57028), [doi](http://dx.doi.org/10.1016/j.top.2004.03.004)
* Gabriele Mondello, _Riemann surfaces, ribbon graphs and combinatorial classes_, in: Handbook of Teichmüller theory. Vol. II,  151--215, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, 2009; draft with index: [pdf](http://www.mat.uniroma1.it/~mondello/me/papers/ober-definitive.pdf), arxiv version [math.AG/0705.1792](http://arxiv.org/abs/0705.1792), [MR2010f:32012](http://www.ams.org/mathscinet-getitem?mr=2010f:32012)
* G. Mondello, _Combinatorial classes on $\mathcal{M}_{g,n}$ are tautological_, Int. Math. Res. Not. __44__ (2004), 2329-–2390, [MR2005g:14056](http://www.ams.org/mathscinet-getitem?mr=2005g:14056), [doi](http://dx.doi.org/10.1155/S1073792804131462), [math.AG/0303207](http://arxiv.org/abs/math/0303207)

[[!redirects Mumford classes]]
[[!redirects Mumford's class]]
[[!redirects Mumford's classes]]
[[!redirects Mumford-Morita-Miller classes]]


Mumford's conjecture

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[conjecture]] in ([Mumford 83](#Mumford83)) concerned the [[ordinary cohomology]] with [[rational numbers]] [[coefficients]] of the [[classifying space]] of the stable [[mapping class group]], which is essentially the stable [[orbifold cohomology]] of the [[moduli stack of curves]] over the [[complex numbers]].

The conjecture was proven in ([Madsen-Weiss 02](#MadsenWeiss02)).

Exposition and review is in ([Madsen 07](#Madsen07)).

## References

The conjecture is due to

* {#Mumford83} [[David Mumford]], _Towards an enumerative geometry of the moduli space of curves_, In Arithmetic and geometry, Vol. II, Progr. Math. 36, Birkhäuser, Boston, MA, 1983, 271–328.

The [[proof]] is due to

* {#MadsenWeiss02} [[Ib Madsen]], [[Michael Weiss]], _The stable moduli space of Riemann surfaces: [[Mumford's conjecture]]_,  Ann. of Math. (2) __165__ (2007), no. 3, 843--941, [MR2009b:14051](http://www.ams.org/mathscinet-getitem?mr=2009b:14051), [doi](http://dx.doi.org/10.4007/annals.2007.165.843)

See also

* Geoffrey Powell, _The Mumford conjecture (after Madsen and Weiss)_, Séminaire Bourbaki. Vol. 2004/2005.  Astérisque __307__ (2006), Exp. No. 944, viii, 247--282, [MR2009f:55018](http://www.ams.org/mathscinet-getitem?mr=2009f:55018)

* [[Michael Weiss]], _Cohomology of the stable [[mapping class group]]_,  Topology, geometry and quantum field theory,  379--404, London Math. Soc. Lecture Note Ser. __308__, Cambridge Univ. Press 2004, [MR2005e:57059](http://www.ams.org/mathscinet-getitem?mr=2005e:57059)

* [[Ib Madsen]], [[Ulrike Tillmann]], _The stable mapping class group and $Q(\mathbf{C} P^\infty_+)$_, Invent. Math. __145__ (2001), no. 3, 509--544. [MR2002h:55011](http://www.ams.org/mathscinet-getitem?mr=2002h:55011)

Review and exposition is in

* {#Madsen07} [[Ib Madsen]], _Moduli spaces from a topological viewpoint_, Proceedings of the International Congress of Mathematics, Madrid 2006 (2007) ([pdf](http://www.icm2006.org/proceedings/Vol_I/20.pdf))

[[!redirects Mumford conjecture]]


Muneto Nitta

* [webpage](http://topo-mat-sci.jp/en/members/profile_d01_nitta/)

## Selected writings

On [[Skyrmions]] and the [[complex Hopf fibration]]:

* {#GudnasonNitta20} [[Sven Bjarke Gudnason]], [[Muneto Nitta]], _Linking number of vortices as baryon number_ ([arXiv:2002.01762](https://arxiv.org/abs/2002.01762), [spire:1778698/](http://inspirehep.net/record/1778698/))

On [[anyons]] in [[topological superconductors]]:

* Yusuke Masaki, Takeshi Mizushima, [[Muneto Nitta]], *Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors* $[$[arXiv:2301.11614](https://arxiv.org/abs/2301.11614)$]$

category: people


muon

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _muon_ is one of the fundamental [[particles]]/[[matter]] [[field (physics)|fields]] in the [[standard model of particle physics]], one of the [[leptons]].

## Properties

* [[anomalous magnetic moment]]

## Related concepts

[[!include flavors of fundamental particles -- table]]

## References

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

[[!redirects muons]]


Murat Günaydin

[[!redirects Murat Gunaydin]]

* [webpage](http://www.phys.psu.edu/people/gxt)

## related $n$Lab entries

* [[4d supergravity]], [[magic supergravity]]

* [[4d supergravity Lie 2-algebra]]

* [[singleton representation]]

category: people


Muriel Livernet

* [personal page](https://webusers.imj-prg.fr/~muriel.livernet/), [publications](https://webusers.imj-prg.fr/~muriel.livernet/publications/index.html)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=56061)

## Selected writings

On [[recognition of relative loop spaces]]:

* {#HLS16} [[Eduardo Hoefel]], [[Muriel Livernet]], [[Jim Stasheff]], _$A_\infty$-actions and recognition of relative loop spaces_, Topology and its Applications **206** (2016) 126-147 [[arXiv:1312.7155](https://arxiv.org/abs/1312.7155), [doi:10.1016/j.topol.2016.03.023](https://doi.org/10.1016/j.topol.2016.03.023)]

On [[pre-Lie algebra]]s

* Frédéric Chapoton, Muriel Livernet, _Pre-Lie algebras and the rooted trees operad_, Int. Math. Res. Not. (2001) 395--408

category: people


Murray Gell-Mann

Murray Gell-Mann was an American physicist (1929-2019), best known for his work on the classification of elementary particles and their interactions.

* [Wikipedia entry](#http://de.wikipedia.org/wiki/Murray_Gell-Mann)

* Norman Dombey, _An Interview with Murray Gell-Mann on June 5 2001_ ([arXiv:2007.12615](https://arxiv.org/abs/2007.12615))

## Selected writings

Introducing the [[σ-meson]] to [[chiral perturbation theory]], together with the concept of the [[sigma-model]]:

* [[Murray Gell-Mann]], M. Lévy, _The axial vector current in beta decay_, Il Nuovo Cimento (1955-1965) volume 16, pages 705–726 (1960) ([doi:10.1007/BF02859738](https://doi.org/10.1007/BF02859738))

On [[quantum hadrodynamics]] (and mentioning the [[Gell-Mann principle]] of [[quantum compulsion]] in [[quantum  physics]]):

* {#GellMann56} [[Murray Gell-Mann]], *The interpretation of the new particles as displaced charge multiplets*,  Nuovo Cim **4** 2 (1956) 848–866 [[doi:10.1007/BF02748000](https://doi.org/10.1007/BF02748000)]

Introducing [[vector meson dominance]]:

* [[Murray Gell-Mann]], Fredrik Zachariasen, _Form Factors and Vector Mesons_, Phys. Rev. 124, 953 (1961) ([doi:10.1103/PhysRev.124.953](https://doi.org/10.1103/PhysRev.124.953))

The [[pole approximation]] for [[meson]] interaction:

* [[Murray Gell-Mann]], Fredrik Zachariasen, _Form Factors and Vector Mesons_, Phys. Rev. 124, 953 (1961) ([doi:10.1103/PhysRev.124.953](https://doi.org/10.1103/PhysRev.124.953))

Introducing the [[quark]] [[model (in theoretical physics)|model]]/[[quantum chromodynamics]]:

* {#GellMann64} [[Murray Gell-Mann]], _A Schematic Model of Baryons and Mesons_, Phys. Lett. 8 (1964) 214-215 ([spire:11880](http://inspirehep.net/record/11880), doi:10.1016/S0031-9163(64)92001-3)

On [[grand unified theories]], [[naturalness]], [[D=4 N=8 supergravity]], [[superstring theory]]:

* {#GellMann83} [[Murray Gell-Mann]], introductory talk at _[Shelter Island II](https://en.wikipedia.org/wiki/Shelter_Island_Conference)_, 1983 ([[Gell-Mann_ShelterIslandII_1983.pdf:file]])

  in: _Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics_. MIT Press. pp. 301–343. ISBN 0-262-10031-2.

* [[Murray Gell-Mann]], [[Pierre Ramond]], Richard Slansky, _Complex Spinors and Unified Theories_, Supergravity, [[Peter van Nieuwenhuizen]] and D.Z. Freedman, eds, North Holland Publishing Co, 1979, (reprinted as [arXiv:1306.4669](http://arxiv.org/abs/1306.4669))

A philosophical account of the nature of the universe, responding to some lines of [[Isaac Newton]], is in

* {#Gell-Mann96} [[Murray Gell-Mann]], _[[Nature conformable to herself]]: Some arguments for a unified theory of the universe_, Complexity 1(4): 9-12 (1996), ([doi:10.1002/cplx.6130010404](https://doi.org/10.1002/cplx.6130010404)).

## Related $n$Lab entries

* [[quark]], [[QCD]]

* [[renormalization group flow]], [[beta function]]

* [[naturalness]]

* [[grand unified theory]]

* [[D=4 N=8 supergravity]]

* [[gravitino]]

* [[superstring theory]]

* [[dark matter]]

category: people


Murray Gerstenhaber

* [webpage](http://www.math.upenn.edu/~mgersten/)

## Selected writings

* [[Murray Gerstenhaber]], [[Alexander Voronov]], _Homotopy G-algebras and moduli space operad_, Internat. Math. Research Notices (1995) 141-153 ([arXiv:hep-th/9409063](https://arxiv.org/abs/hep-th/9409063))

category: people


Music of the Spheres

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### Ancient idea

In ancient times, _Musica Universalis_ or _Music of the Spheres_ referred to a philosophical concept that regards proportions of orbits in the [[solar system]] as exhibiting a cosmic harmony ([Plato 380BC](#Plato380bc)).

[[Johannes Kepler]] proposed that this is governed by the shape of the [[platonic solid|Platonic solids]] ([Kepler 1596](#Kepler1596)). (See also at _[[multiverse]]_.)

### Modern idea
 {#ModernIdea}

More recently, it was suggested that the term refers to the [[stable homotopy groups of spheres]] $\pi_\bullet(\mathbb{S})$ and the absolute [[E-∞ geometry]] over [[Spec(S)]] as seen by the [[MU]]-[[Adams spectral sequence]].

> My initial inclination was to call this book [[The Music of the Spheres]], but I was dissuaded from doing so by my diligent publisher, who is ever mindful of the sensibilities of librarians. ([Ravenel 86, p. xv](#Ravenel86))

> For the last 50 years one of the basic problems in algebraic topology has been the determination of the homotopy groups of spheres ([Mahowald-Ravenel 87, Sec. 1](#MahowaldRavenel87))

> As is well known, it is our manifest destiny as 21st century algebraic topologists to compute homotopy groups of spheres. ([Wilson 13, p. 1](#Wilson13))

> One of the most fundamental problems in topology is to determine the set of homotopy classes of continuous based maps between spheres. ([Isaksen-Wang-Xu 20, Sec. 1](#IsaksenWangXu20))

Modern [[chromatic homotopy theory]] makes contact between this kind of music and [[physics]]:

1. the [[A-hat genus]] ([[partition function]] of the [[spinning particle]]) appears in the first chromatic level of the spheres;

1. the [[Witten genus]] ([[partition function]] of the [[spinning string]]) appears in the second chromatic level of the spheres.

Modern [[equivariant stable homotopy theory]] makes contact between this kind of music and Kepler's proposal:

* the [[equivariant sphere spectrum|equivariance]]  of the [[quaternionic Hopf fibration]] under the [[symmetry groups]] of the [[platonic solid|Platonic solids]] ([[icosahedral group]], [[octahedral group]], [[tetrahedral group]]) makes further [[equivariant stable homotopy groups]] of spheres appear, see at _[quaternionic Hopf fibration -- class in equivariant stable homotopy theory](quaternionic%20Hopf%20fibration#ClassInEquivariantStableHomotopyTheory)_.

> For whatever it's worth, all these items come together in the _[[schreiber:Equivariant cohomology of M2/M5-branes]]_

## Related concepts

* [[Cohomotopy]]

  * [[stable Cohomotopy]]

  * [[twisted Cohomotopy]]

  * [[equivariant Cohomotopy]]

  * [[differential Cohomotopy]]

## References

* Wikipedia, _[Musica universalis](https://en.wikipedia.org/wiki/Musica_universalis)_

* {#Plato380bc} [[Plato]], _[[Republic]]_

* {#Kepler1596} [[Johannes Kepler]], _[Mysterium Cosmographicum](https://en.wikipedia.org/wiki/Mysterium_Cosmographicum)_, 1595

* {#Ravenel86} [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_, 1986/2003

* {#MahowaldRavenel87} [[Mark Mahowald]], [[Doug Ravenel]], _Towards  a Global Understanding of the Homotopy Groups of Spheres_, in: [[Samuel Gitler]] (ed.): _The Lefschetz Centennial Conference: Proceedings on Algebraic Topology II_, Contemporary Mathematics volume 58, AMS 1987  ([pdf](http://www.math.rochester.edu/people/faculty/doug/mypapers/global.pdf), [[MahowaldRavenelHomotopyGroupsOfSpheres.pdf:file]], [ISBN:978-0-8218-5063-3](https://bookstore.ams.org/conm-58-2))

* {#Wilson13} [[Dylan Wilson]], _Spectral Sequences from Sequences of Spectra: Towards the Spectrum of the Category of Spectra_, lecture at [2013 Pre-Talbot Seminar](https://www.hiroleetanaka.com/pretalbot2013/), March 2013 ([[DylanWilsonOnANSS.pdf:file]])

* {#IsaksenWangXu20} [[Daniel Isaksen]], [[Guozhen Wang]], [[Zhouli Xu]], _Stable homotopy groups of spheres_, PNAS October 6, 2020 117 (40) 24757-2476 ([arXiv:2001.04247](https://arxiv.org/abs/2001.04247), [doi:10.1073/pnas.2012335117](https://doi.org/10.1073/pnas.2012335117))

[[!redirects The Music of the Spheres]]