halving group

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## Idea ##

An [[abelian group]] where one could halve each element or divide each element by two.

## Definition ##

A **halving group** is an [[abelian group]] *G* with a function $(-)/2:G \to G$ called **halving** or **dividing by two** such that for all $g \in G$, $g/2 + g/2 = g$.

## Properties

* Just as every abelian group is a $\mathbb{Z}$-[[module]], every halving group is a $\mathbb{D}$-[[module]], where $\mathbb{D}$ are the [[dyadic rational numbers]].

* The only element of a halving group with [[order of a group|order]] $2$ is the additive unit $0$.

## See also

* [[dyadic rational numbers]]

* [[abelian group]]

## External links

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

[[!redirects halving groups]]


Hamilton operator > history

< [[Hamilton operator]]


Hamilton's equations

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## Idea

The [[equation of motion]] as expressed in [[Hamiltonian mechanics]].

Given a [[phase space]] represented by a [[symplectic manifold]] $(X,\omega)$, and given a [[Hamiltonian]] function $H \colon X \longrightarrow \mathbb{R}$ the solutions to the [[equations of motion]] are [[trajectories]] $\gamma \colon \mathbb{R} \longrightarrow X$ which satisfy

$$
  \mathbf{d} H(-) = \omega(\dot \gamma,-)
  \,,
$$

hence which are [[flow lines]] of the [[flow]] induced by the [[Hamiltonian vector field]] associated with $H$.

## Related concepts

* [[classical mechanics]]

  * [[prequantized Lagrangian correspondence]]

* [[differential equation]]

* [[equations of motion]]

  * [[Newton's laws of motion]]

  * Hamilton's equations, [[de Donder-Weyl-Hamilton equation]]

    * [[symplectic integrator]]

  * [[Einstein's equations]]

* [[Liouville's equation]]

## References

* Richard Fitzpatrick, _Newtonian Dynamics -- [Hamilton's equations](http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node91.html)_

[[!redirects Hamilton's equation]]

[[!redirects Hamilton equation]]
[[!redirects Hamilton equations]]

[[!redirects Hamilton's equations of motion]]


Hamilton-Jacobi equation

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## Idea

The _Hamilton-Jacobi equation_ is a first-order nonlinear [[partial differential equation]] that arises in [[variational calculus]] and which gives, in particular, a formulation  of the [[laws of motion]] in [[classical mechanics]].

It relates the principal function ([[action functional]] as a function of certain arguments) with the [[Hamiltonian]]. In one dimension it is of the form

$$
\frac{\partial S}{\partial t} + H\left(t, x, \frac{\partial S}{\partial x}\right) = 0
$$

## Related concepts

* [[Hamilton's equations]]

* [[prequantized Lagrangian correspondence]]

* [[Bohmian mechanics]]

## References

* [[eom]]: [Hamilton-Jacobi theory](http://www.encyclopediaofmath.org/index.php/Hamilton-Jacobi_theory)
* [wikipedia](http://en.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi_equation), [scholarpedia](http://www.scholarpedia.org/article/Hamilton-Jacobi_equation)
* V.I. Arnol'd, _Mathematical methods of classical mechanics_
* [[Shlomo Sternberg]], _Lectures in differential geometry_
* L. Vitagliano, _Hamilton-Jacobi diffieties_, [arxiv/1104.0162](http://arxiv.org/abs/1104.0162)

Specifically for [[quadratic Hamiltonians]]:

* {#RobbinSalamon93} [[Joel Robbin]],  [[Dietmar Salamon]], remark 3.4 in _Feynman path integrals on phase space and the metaplectic representation_ in [[Dietmar Salamon]] (ed.), _Symplectic Geometry_, LMS Lecture Note series 192 (1993) ([[RobbinSalamonMetaplectic.pdf:file]])

[[!redirects Hamilton-Jacobi equations]]

[[!redirects Hamilton-Jacobi theory]]

[[!redirects Hamilton-Jacobi action]]
[[!redirects Hamilton-Jacobi actions]]


Hamiltonian

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Disambiguation: there is an unrelated notion of a Hamilton or Hamiltonian operator also called [[nabla]] in [[vector analysis]].

## Definition

Given a [[Poisson manifold]] $(X, \{-,-\})$ and a [[vector field]] $v \in \Gamma(T X) $, a **Hamiltonian** for $v$ is a [[smooth function]] $h_v \in C^\infty(X)$ such that $\{h_v,-\}$ is the [[derivation]] corresponding to $v$.

Conversely, one says that $v$ is the [[Hamiltonian vector field]] of $h_v$.

## In mechanical systems

Given a [[classical mechanical system]] evolving in time, there is a [[symplectic manifold]] (or at least [[Poisson manifold]]) equipped with the [[vector field]] that generates time evolution. Its Hamiltonian is often called _the_ Hamiltonian. This is the concept that Hamilton originally considered and which hence gives the name to the general situaiton.

### In classical mechanics

The simplest, so-called "natural", Hamiltonian (function) of a [[dynamical system]] is the sum of the kinetic and potential energy:
\[ H = T + V. \]

Knowing only $H$ as a function on [[phase space]] (so as a function of [[position]] $q^i$ and [[momentum]] $p_i$), we can derive other quantities as functions on phase space.  In particular, we have:
*  [[velocity]], $v^i = \partial{H}/\partial{p_i}$,
*  [[force]], $f_i = -\partial{H}/\partial{q^i}$.

Setting $v^i = \mathrm{d}q^i/\mathrm{d}t$ and $f_i = \mathrm{d}p_i/\mathrm{d}t$, we derive the [[equations of motion]] in [[Hamiltonian mechanics]].

### In quantum mechanics

The [[quantum mechanics]] of a point [[particle]] in the
_[[Schrödinger picture]]_ is encoded in a [[Hilbert space]] [[bundle]] $\mathcal{H} \to \mathbb{R}$ [[connection on a bundle|with connection]] $\nabla$ over the real line -- the _worldline_ -- of the particle.

For $t \in \mathbb{R}$ the fiber $\mathcal{H}_t$ is the **space of [[quantum state]]s** of the system, at given parameter time $t$. Since this bundle is necessarily trivializable, we imagine fixing a trivialization $\mathcal{H} \simeq \mathcal{H}_0 \times \mathbb{R}$. Then the flat connection on the bundle is canonically a [[differential form|1-form]] on $\mathbb{R}$ with values in [[linear operator]]s on $\mathbf{H}$.

$$
  A = H \;d t \in \Omega^1(\mathbb{R}, End(\mathcal{H}))
  \,.
$$

The component $H \in End(\mathcal{H})$ of this canonical 1-form is the **Hamilton(ian) operator** (or the *quantum Hamiltonian*) of the system.

Its [[parallel transport]] is the **time evolution** of quantum states. If $H$ is constant as a function on $\mathbb{R}$, this parallel transport assigns to the path $\gamma$ from $t_1$ to $t_2$ in $\mathbb{R}$ the map

$$
  U :
  (t_1 \stackrel{\gamma}{\to} t_2)
   \mapsto
  (\mathcal{H}_{t_1}
  \stackrel{exp\left(-\frac{i}{\hbar}H (t_2-t_1)\right)}{\to} \mathcal{H}_{t_2})
  \,.
$$

If instead $H$ does depend on $t$ -- called the case of _time-dependent quantum mechanics_ -- then the full formula for parallel transport applies, which is given by the [[path-ordered exponential]]

$$
  U :
  (t_1 \stackrel{\gamma}{\to} t_2) \mapsto
  (\mathcal{H}_{t_1} \stackrel{P exp \left(-\frac{i}{\hbar}\int_{t_1}^{t_2}H  d t\right)}{\to} \mathcal{H}_{t_2})
  \,.
$$

In the physics literature this path-ordered exponential is known as the **[[Dyson formula]]** .

#### Physical meaning and relation to unitary transformations

The [[eigenvalue]]s of the Hamiltonian operator for a closed quantum system are exactly the energy eigenvalues of that system.  Thus the Hamiltonian is interpreted as being an "energy" operator.  Conservation of energy occurs when the Hamiltonian is time-independent.

Transformations and evolutions in standard quantum mechanics are represented via [[unitary operator]]s where a time evolving unitary is related to the Hamiltonian $H$ via

$U(0,t) = $exp$\left(-\frac{i}{\hbar}H t\right),$

provided the Hamiltonian is time-independent.

## Related concepts

* [[quadratic Hamiltonian]]

[[!include Hamiltonian and Lagrangian -- table]]

* [[Hamilton's equations]]

* [[quantum operator]]

* [[Hamiltonian action]]

* [[Hamiltonian form]]

* [[propagator]]

[[!include geometric quantization extensions - table]]

## References

Named after [[William Rowan Hamilton]].

[[!redirects Hamiltonian]]
[[!redirects hamiltonian]]
[[!redirects Hamiltonians]]
[[!redirects hamiltonians]]
[[!redirects Hamiltonian function]]
[[!redirects hamiltonian function]]
[[!redirects Hamiltonian functions]]
[[!redirects hamiltonian functions]]
[[!redirects Hamiltonian operator]]
[[!redirects hamiltonian operator]]
[[!redirects Hamiltonian operators]]
[[!redirects hamiltonian operators]]
[[!redirects Hamilton operator]]
[[!redirects hamilton operator]]
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[[!redirects hamilton operators]]


Hamiltonian action

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## Definition

### Integrated

Given a [[symplectic manifold]] $(X,\omega)$, there is the [[group]] of [[Hamiltonian symplectomorphisms]] $HamSympl(X,\omega)$ [[group action|acting]] on $X$. If $(X,\omega)$ is [[prequantum line bundle|prequantizable]] this lifts to the group of [[quantomorphisms]], both of them covering the [[diffeomorphisms]] of $X$:

[[quantomorphisms]] $\to$ [[Hamiltonian symplectomorphisms]] $\to$ [[diffeomorphisms]] .

A _Hamiltonian action_ of a [[Lie group]] $G$ on $(X,\omega)$ is an [[action]] by [[quantomorphisms]], hence a [[Lie group]] [[homomorphism]] $\hat \phi \colon G \to Quant(X, \omega)$

$$
  \array{
     && Quant(X, \omega)
     \\
     & {}^{\mathllap{\hat \phi}}\nearrow & \downarrow
     \\
     G &\stackrel{\phi}{\to}& HamSympl(X, \omega)
     \\
     & {}_{\mathllap{}}\searrow & \downarrow
     \\
     && Diff(X)
  }
  \,.
$$

See [Brylinski, prop. 2.4.10](#Brylinski).

### Differentially

In the literature this is usually discussed at the [[infinitesimal object|infinitesimal]] level, hence for the corresponding  [[Lie algebras]]:

[[smooth functions]]+[[Poisson bracket]] $\to$ [[Hamiltonian vector fields]] $\to$ [[vector fields]]

Now an (infinitesimal) **Hamiltonian action** is a Lie algebra [[homomorphism]] $\mu \colon \mathfrak{g} \to (C^\infty(X), \{-,-\})$ to the [[Poisson bracket]]-algebra:

$$
  \array{
     && (C^\infty(X),\{-,-\})
     \\
     & {}^{\mathllap{\mu}}\nearrow & \downarrow
     \\
     \mathfrak{g} &\stackrel{}{\to}& HamVect(X, \omega)
     \\
     & {}_{\mathllap{}}\searrow & \downarrow
     \\
     && Vect(X)
  }
  \,.
$$

[[dual linear map|Dualizing]], the homomorphism $\mu$ is equivalently a linear map

$$
  \tilde \mu \colon X \to \mathfrak{g}^\ast
$$

which is a [[homomorphism]] of [[Poisson manifolds]], called the **[[moment map]]** of the (infinitesimal) Hamiltonian $G$-action.

**Warning** The lift from $\phi$ to $\hat \phi$ above, hence from the _existence_ of [[Hamiltonians]] to an actual _choice_ of Hamiltonians is in general not unique. In the literature the difference between $\hat \phi$ and $\phi$ (or of their Lie theoretic analogs) is not always made clear.

## Properties

### Characterization

By [Atiyah & Bott](#AtiyahBott), the action of a Lie algebra on a symplectic manifold is Hamiltonian if and only if the symplectic form has a ([[basic differential form|basic]], [[closed differential form|closed]]) extension to [[equivariant de Rham cohomology]].

## Related concepts

* [[moment map]]

* [[symplectic quotient]]

* [[symplectic reduction]]

* [[Guillemin-Sternberg geometric quantization conjecture]]

* [[projective representation|projective]] Hamiltonian action: [[classical anomaly]]

## References

A comprehensive account:

* [[Viktor Ginzburg]], [[Victor Guillemin]], [[Yael Karshon]], around section 2.1 of: _Moment maps, cobordisms and Hamiltonian group actions_ [[pdf](http://kolxo3.tiera.ru/M_Mathematics/MD_Geometry%20and%20topology/MDdg_Differential%20geometry/Ginzburg.pdf)]

The perspective on Hamiltonian actions in terms of maps to extensions, infinitesimally and integrally, is made explicit in

* {#Brylinski} [[Jean-Luc Brylinski]], Prop. 2.4.10 of: _Loop spaces, characteristic classes and geometric quantization_, Birkhäuser (1993)

The characterization in  [[equivariant cohomology]] is due to

* {#AtiyahBott} [[Michael Atiyah]], [[Raoul Bott]], _The moment map and equivariant cohomology_, Topology **23** 1 (1984) [[pdf](https://www.math.sunysb.edu/~mmovshev/MAT570Spring2008/BOOKS/atiyahbott_moment.pdf)]

Generalization to Hamiltonian actions by a [[Lie algebroid]] (instead of just a [[Lie algebra]]) is discussed in

* {#Bos} Rogier Bos, _Geometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids_ [[arXiv:math.SG/0604027](http://arxiv.org/abs/math.SG/0604027)]

[[!redirects hamiltonian action]]

[[!redirects Hamiltonian actions]]
[[!redirects hamiltonian actions]]

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[[!redirects Hamiltonian G-actions]]


Hamiltonian and Lagrangian -- table

| [[Hamiltonian]] | $\leftarrow$ [[Legendre transform]] $\rightarrow$ | [[Lagrangian]] |
|--|--|--|
| [[Lagrangian correspondence]] | [[prequantization]] | [[prequantized Lagrangian correspondence]] |


Hamiltonian dynamics on Lie groups

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## Idea

Many important special cases of [[classical mechanics]] involve physical systems whose [[configuration space]] is a [[Lie group]], for instance [[rigid body dynamics]] but also (for infinite-dimensional Lie groups) [[fluid dynamics]].

All these systems have special properties, notably they are formally [[integrable system]]s.

## Details

Let $G$ be a [[Lie group]]. Write $\mathfrak{g}$ for its [[Lie algebra]].

Choose a [[Riemannian metric]]

$$
  \langle -,-\rangle \in Sym^2_{C^\infty(G)} \Gamma(T G)
$$

on $G$ which is [[left invariant]].

On the [[tangent bundle]] $T G$ this induces the [[Hamiltonian]]

$$
  H : (v \in T G) \mapsto \frac{1}{2}\langle v,v\rangle
  \,.
$$

The [[Euler-Arnold equation]] is the associated differential equation governing the velocity of geodesic flow.

## Examples

* For $G = SO(n)$ the [[special orthogonal group]] the above yields is [[rigid body dynamics]]. The left invariant metric $\langle -,-\rangle$ is the [[moment of inertia]] of the body.

* For $G$ the infinite-dimensional Lie group of [[volume]]-preserving [[diffeomorphism]] of a [[compact space|compact]] [[smooth manifold]], the above yields is Euler's equations of [[hydrodynamics]].

## Related concepts

* [[Euler-Arnold equation]]

## References

The original influential article is

* [[Vladimir Arnol'd]], _Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits._ Ann. Inst. Fourier (Grenoble) 16 (1966) fasc. 1, 319–361. ([MathSciNet](http://www.ams.org/mathscinet-getitem?mr=202082))

A standard textbook reference is section 4.4 of

* [[Ralph Abraham]], [[Jerrold Marsden]], _[[Foundations of Mechanics]]_

[[!redirects Hamiltonian mechanics on Lie groups]]
[[!redirects dynamics on Lie groups]]


Hamiltonian flow

The [[flow]] of a [[Hamiltonian vector field]], or rather, that flow together with a specified [[Hamiltonian]]. A [[Hamiltonian action]].

[[!redirects Hamiltonian flows]]


Hamiltonian form

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## Definition

For $\omega \in \Omega^{n+1}(X)$ an [[n-plectic geometry]], and for $v \in \Gamma(X)$ a [[vector field]], a **Hamiltonian form** for $v$ is, if it exists, a [[differential form]] $h \in \Omega^n(X)$ such that

$$
  \iota_{v} \omega = \mathbf{d} h
  \,.
$$

For $n = 1$ this reduces to the notion of a [[Hamiltonian]] [[function]] on a [[symplectic manifold]].

If a Hamiltonian form for $v$ exists then $v$ is called a [[Hamiltonian vector field]].

The Hamiltonian forms are the local [[classical observables]]/[[prequantum observables]] in [[higher prequantum field theory]], often called _[[local currents]]_. They form the [[Poisson-bracket Lie n-algebra of local observables]].

## Related concepts

* [[Hamiltonian n-vector field]]

[[!include geometric quantization extensions - table]]

[[!redirects Hamiltonian forms]]

[[!redirects Hamiltonian n-form]]
[[!redirects Hamiltonian n-forms]]

[[!redirects Hamiltonian differential form]]
[[!redirects Hamiltonian differential forms]]


Hamiltonian mechanics

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## Idea

Hamiltonian mechanics is a formulation of [[mechanics]] in which the basic datum in a mechanical system is a function $H$, the [[Hamiltonian]] of the system, which gives the total energy in the system in terms of the [[position|positions]] and [[momentum|momenta]] of the objects in the system.

More abstractly, the Hamiltonian is a function on [[phase space]], a [[manifold]] whose coordinates are [[generalized position|generalised]] positions $q^i$ and momenta $p_i$.  (Compare this to [[Lagrangian mechanics]], in which the [[Lagrangian]] is a function on [[state space]], whose coordinates are generalised positions and [[velocity|velocities]].)  So to do Hamiltonian mechanics properly, you must 'mind your $p$s and $q$s' (blame [[John Baez]] for this pun).

To begin with, we often take phase space to be the [[cotangent bundle]] of [[configuration space]].  (Compare that state space is the [[tangent bundle]] of configuration space.)  This comes equipped with a natural $2$-[[differential form|form]]
$$ \omega = \sum_i \mathrm{d}p_i \wedge \mathrm{d}q^i ,$$
or simply $\omega = \mathrm{d}p_i \wedge \mathrm{d}q^i$ using the [[Einstein summation convention]].  This $2$-form is closed, in fact exact, since it is the [[exterior derivative|differential]] of the [[action functional|action form]] $\bar{\mathrm{d}}S = p_i \wedge \mathrm{d}q^i$, and therefore it is a [[symplectic form]].

However, it is also possible to take phase space to be *any* [[symplectic manifold]], or even any [[Poisson manifold]].  In any case, phase space itself gives only the [[kinematics]] (in a momentum-based rather than velocity-based sense); you need the Hamiltonian $H$ to get the [[dynamics]].

Hamiltonian mechanics was developed originally for [[classical mechanics]], but it is also the best known formulation of [[quantum mechanics]]; many students of [[physics]] (and even more so, students of [[chemistry]]) learn it only when they study the latter.  This sometimes leads to confusion about the essential differences between classical and quantum physics.

## Definition in terms of symplectic geometry

Hamiltonian mechanics is best formalized in terms of [[symplectic geometry]] as described for instance in the monograoph

* [[Vladimir Arnold]], _Mathemtical Methods of Classical Mechanics_ Springer.

A classical Hamiltonian [[mechanical system]] is a pair $((X,\omega), H)$ consisting of a

* [[symplectic manifold]] $(X,\omega)$

* and a [[Hamiltonian]] function $H \in C^\infty(X)$.

Here

* $X$ is the **[[phase space]]** of the physical system;

* a curve $\gamma : \mathbb{R} \to X$ is a **[[trajectory]]** of the physical system in time;

* $(X,\omega)$ defines the **[[kinematics]]** of the system;

* $H$ is the **[[Hamiltonian]]** that defined the **[[dynamics]]** of the system.

The **dynamics** is encoded by declaring that those trajectories $\gamma : \mathbb{R} \to X$ are the physically realized trajectories that satisfy the equation

$$
  d H = \omega(\gamma', -)
$$

The components of this are **[[Hamilton's equations]]**.

In more detail this equation means that for each $t \in \mathbb{R}$ the [[differential form|1-form]]

$$
  (d H)_{\gamma(t)}
$$

and the 1-form

$$
  \omega_{\gamma(t)}((\frac{d}{d t}\gamma(t), -)
$$

coincide.

### Comments on this definition

At first, this formulation of Hamilton's mechanics is just that, an equivalent reformulation.
But as any reformulation in more abstract terms, it serves to

1. clarify a structure

1. allow more powerful thinking about that structure

1. and eventually it bears in it the seed for further developments pointing beyond this structure

Regarding the first point: this formulation of Hamiltonian mechanics makes clear what the meaning of Hamilton's equations is for systems topologically more interesting than the example $X = \mathbb{R}^{2 n}$ that many introductory physics texts concentrate on-

Regarding the second point: the differential calculus formulation lends itself much more to high-powered arguments than the traditional component-ridden presentation. Of course the latter may still be the preferred method for some concrete computations.

Regarding the second point: after Hamilton's times people started thinking about what [[quantization]] of a classical system should mean. One successful formalization is that of [[geometric quantization]] which takes a symplectic manifold with Hamiltonian function on it as input datum.

The impact that this idea of quantization from symplectic geometry eventually had is hard to underestimate. In the hands of [[Alan Weinstein]] and his school it led to [[symplectic groupoid]]s, [[Courant algebroid]]s and other higher [[Lie theory|Lie theoretic structures]].  In the hands of [[Maxim Kontsevich]] it led to the theorem on formal [[deformation quantization]] and the vast machinery nowadays associated with that.

### Examples

The symplectic-geometry description of Hamiltonian mechanics is especially well-suited to describe topologically nontrivial phase spaces that are not [[cotangent bundle]]s.

#### Vortices on the sphere

$n$ vortices on the sphere as finite dimensional limit of 2D [[equation of motion|Euler equations]]: the phase space of the system of $n$ vortices is not a [[cotangent bundle]] but is $(S^2)^n$ .

## Related entries

* [[Hamilton's equations]]

* [[Hamilton-Jacobi equation]]

* traditional [[Lagrangian mechanics]] and [[Hamiltonian mechanics]] are naturally embedding into [[local prequantum field theory]] by the notion of [[prequantized Lagrangian correspondences]]

## References

Named after [[William Rowan Hamilton]].

Formulation in [[symplectic geometry]]:

* [[Vladimir Arnol'd]], _[[Mathematical methods of classical mechanics]]_, Graduate Texts in Mathematics **60**, Springer (1978) [[doi:10.1007/978-1-4757-1693-1](https://doi.org/10.1007/978-1-4757-1693-1)]

A motivation for formulating Hamiltonian mechanics in terms of [[symplectic manifolds]]:

 * Henry Cohn, _[Why symplectic geometry is the natural setting for classical mechanics ](http://math.mit.edu/~cohn/Thoughts/symplectic.html)_


Hamiltonian n-vector field

> under construction

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## Idea

A _Hamiltonian $n$-vector field_ is the $n$-dimensional analog of a [[Hamiltonian vector field]] as one passes from [[symplectic geometry]] to [[multisymplectic geometry]]/[[n-plectic geometry]]. Roughly, the [[transgression]] of a Hamiltonian $n$-vector field to [[mapping spaces]] out of an $(n-1)$-manifold yields an ordinary [[Hamiltonian vector field]].

## Definition

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

Let $X$ be a [[smooth manifold]] equipped with a degree $(n+1)$ [[differential form]] $\omega \in \Omega^{n+1}(X)$ for $n \in \mathbb{N}$, with the pair $(X,\omega)$ regarded as a [[multisymplectic manifold]]/[[n-plectic manifold]].

Let moreover $H \;\colon\; X \longrightarrow \mathbb{R}$ be a [[smooth function]], to be regarded as an extended [[Hamiltonian function]], hence a [[de Donder-Weyl Hamiltonian]].

Then a _Hamiltonian $n$-vector field_ on $X$ is an $n$-[[multivector field]] $\mathbf{v} \in \Gamma(\wedge^{n} T X)$ satisfying the analog of [[Hamilton's equations]], namely the [[differential equation]] of [[differential 1-forms]] on $X$

$$
  \iota_{\mathbf{v}} \omega = \mathbf{d} H
  \,.
$$

=--

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

Let $\Sigma$ be a [[manifold]] to be regarded as a [[spacetime]]/[[worldvolume]] and let $E \to \Sigma$ be a [[vector bundle]], to be regarded as a [[field bundle]]. Assume for simplicity of notation that both $\Sigma$ and $E$ are in fact [[Cartesian spaces]] (which is always true locally). Write $j^1 E$ for the corresponding jet bundle]], equipped with the canonical "kinetic" [[n-plectic form]] which in local coordinates reads

$$
  \omega
    \coloneqq
  \mathbf{d}_v\phi^a_{,i} \wedge \mathbf{d}_v q^a \wedge (\iota_{\partial_i} vol_\Sigma)
$$

as discussed at [[multisymplectic geometry]].

Then for $\phi^a$ and $\phi^a_{,i}$ smooth functions on $\Sigma$ and for vector fields

$$
  v_i
  \coloneqq
  \frac{\partial}{\partial \sigma^i}
  +
  \frac{\partial \phi^a}{\partial \sigma^i}
  +
  \frac{\partial \phi^a_{,j}}{\partial \sigma^j}
  \frac{\partial}{\partial \phi^a_{,i}}
$$

the equation

$$
  \iota_{v_1 \cdots v_n} \omega = \pm \mathbf{d}H
$$

is equivalent to the [[de Donder-Weyl equations]]

$$
  \frac{\partial \phi^a}{\partial \sigma^i}
  =
  \frac{\partial H}{\partial p^i_a}
  \;\;\;\;
  \frac{\partial p^i_a}{\partial \sigma^i}
  =
  \frac{\partial H}{\partial \phi^a}
  \,.
$$

=--

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

We have

$$
  \mathbf{d}_v \phi^a = \mathbf{d}\phi^a - \phi^a_{,i}\mathbf{d}\sigma^i
$$

and

$$
  \mathbf{d}_v \phi^a_{,i}
    =
  \mathbf{d}\phi^a_{,i}
  -
  \phi^a_{,i j} \mathbf{d}\sigma^j
  \,.
$$

The claimed equation comes from contracting in $\mathbf{d}\phi^a_{,i} \wedge \mathbf{d}\phi^a \wedge (\iota_{\partial_i}vol_\Sigma)$ all but the $i$th vector with the contracted volume form. The remaining contractions are then

$$
  \frac{\partial \phi^a}{\partial \sigma^i \partial \sigma^{[j}}
  \frac{\partial \phi_a}{\partial \sigma^{i]}}
  \mathbf{d}\sigma^j
$$

and these cancel against the horizontal derivative contributions form above.

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## Interpretation in higher geometry
 {#InterpretationInHigherGeometry}

We discuss here how Hamiltonian $n$-vector fields are equivalently homorphisms from an $n$-dimensional surface into the [[Poisson bracket Lie n-algebra]] associated with the given [[n-plectic geometry]].

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

By the discussion at [[n-plectic geometry]], a (pre-)[[n-plectic manifold]] $(X,\omega)$ induces a [[Poisson bracket Lie n-algebra]] $\mathfrak{pois}(X,\omega)$, given as follows:

* in positive degree $k$ it has the degree $(n-k)$-[[differential forms]] on $X$

* in degree 0 it has pairs $(v,A)$ of [[Hamiltonian n-forms]] $A$ with their [[Hamiltonian vector fields]] $v$ (such that $\mathbf{d}A = \iota_v \omega$)

and

* whose [[L-infinity algebra|n-ary Lie bracket]] for $n \geq 2$ is non-trivial only on $n$-tuples of degree-0 elements, where it is given by

  $$
    [(v_1, A_1), \cdots, (v_n,A_n)]
    =
    \pm
    \iota_{v_n}\cdots \iota_{v_1} \omega
    \;\;\;
    \in
    \Omega^1(X)
    \,,
  $$

* and whose unary Lie bracket is given by the [[de Rham differential]], $\{-\} = \mathbf{d}$.

This means that the $n$-extended Hamilton equation of def. \ref{ExtendedHamiltonEquations} reads in terms of the [[Poisson bracket Lie n-algebra]] equivalently thus ([hgp 13, remark 2.5.10](#hgp13)):

$$
  \{v_1, \cdots, v_n\} = \pm\{H\}
  \,.
$$

=--

Let now $\Sigma$ be a [[manifold]] to be regarded as a [[spacetime]]/[[worldvolume]] and let $E \to \Sigma$ be a [[vector bundle]], to be regarded as a [[field bundle]]. Assume for simplicity of notation that both $\Sigma$ and $E$ are in fact [[Cartesian spaces]] (which is always true locally). Write $(j^1 E)^\ast$ for the coreresponding dual [[jet bundle]], equipped with the canonical "kinetic" [[n-plectic form]] which in local coordinates reads

$$
  \omega
    \coloneqq
  \mathbf{d}p^i_a \wedge \mathbf{d}q^a \wedge (\iota_{\partial_i} vol_\Sigma)
$$

as discussed at [[multisymplectic geometry]].
Write $\mathfrak{pois}(X,\omega)$ for the corresponding [[Poisson bracket Lie n-algebra]].

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

The [[L-∞ algebra]] [[homomorphisms]]

$$
  \mathbb{R}^n \longrightarrow \mathfrak{pois}((j^1 E)^\ast, \omega)
$$

which induce the canonical map under the projection to vector fields on $\Sigma$ are [[equivalence|equivalently]] [[tuples]] consisting of

* a [[smooth function]] $H$;

* $n$ [[vector fields]] $v_i$

on $(j^1 E)^\ast$, such that the $(v_i)$ satisfy the [[de Donder-Weyl equation]] for [[Hamiltonian]] $H$

$$
  \iota_{v_1 \cdots v_n} \omega = \pm \mathbf{d}H
  \,.
$$

=--

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

By the discussion at _[[Maurer-Cartan element]]_, a homomorphism as indicated is equivalently an element

$$
  \mathcal{J}
  \in
  \wedge^\bullet(\mathbb{R}^n) \otimes \mathfrak{pois}(X,\omega)
$$

of total degree 1, which satifies the [[Maurer-Cartan equation]]

$$
  \sum_k \frac{1}{k!}\{\underbrace{\mathcal{J}\wedge \cdots \wedge \mathcal{J}}_{k\; factors}\}
  = 0
  \,.
$$

If we suggestively write $\mathbf{d}\sigma^i$ for the generators of the [[Grassmann algebra]] $\wedge^\bullet(\mathbb{R}^n)$, then, by remark \ref{nExtendedHamiltonEquationAsLInfinityRelation}, $\mathcal{J}$ is of the form

$$
  \mathcal{J}
  =
  \mathbf{d}\sigma^i \otimes (v_i, J_i)
  +
  \mathbf{d}\sigma^{i_1} \wedge \mathbf{d}\sigma^{i_2}
  \otimes J_{i_1 i_2}
  +
  \cdots
  +
  \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^n
  \otimes
  H
  \,,
$$

where the $v_i$ are vector fields on the extended phase space,

$$
  J_{i_1 \cdots i_k} \in \Omega^{n-k}((j^1 E)^\ast)
$$

and $H$ is a smooth function. Again by remark \ref{nExtendedHamiltonEquationAsLInfinityRelation}, the [[Maurer-Cartan equation]] on this element is equivalent to

$$
  \iota_{v_{i_1}} \cdots \iota_{v_{i_k}} \omega
  =
  \pm
  \mathbf{d} J_{i_1 \cdots i_k}
$$

for all $1 \leq k \leq n-1$ and all $\{i_j\}_{j = 1}^k$, and

$$
  \iota_{v_{1}} \cdots \iota_{v_{n}} \omega
  =
  \pm
  \mathbf{d} H
  \,.
$$

The condition on the projection means that

$$
  v_i = \frac{\partial}{\partial\sigma^i}
  +
  \frac{\partial \phi^a}{\partial \sigma^i}
  \frac{\partial }{\partial \phi^a}
  +
  \frac{\partial \phi^a_{,j}}{\partial \sigma^i}
  \frac{\partial }{\partial \phi^a_{,j}}
  \,.
$$

This way the last equation is the [[de Donder-Weyl equation]]. But then

$$
  J_{i_1 \cdots i_k}
  \coloneqq
  \pm H \iota_{\partial_{i_1} \cdots \partial_{i_k}} vol_\Sigma
$$

=--

We can further interpret this in [[local prequantum field theory]] as follows:

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

By ([hgp 13](#hgp13)) the [[Poisson bracket Lie n-algebra]] is the [[Lie differentiation]] of the smooth [[automorphism n-group]]
of any [[prequantization]] of $(X,\omega)$ by a  [[prequantum n-bundle]] $\nabla$:

$$
  \array{
    X &\stackrel{\nabla}{\longrightarrow}& \mathbf{B}^n U(1)_{conn}
    \\
    & {}_{\mathllap{\omega}}\searrow & \downarrow^{\mathrlap{F_{(-)}}}
    \\
    && \mathbf{\Omega}^n_{cl}
  }
$$

regarded as an object in the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}$ of that [[cohesive (∞,1)-topos]] $\mathbf{H}$ of [[smooth ∞-groupoids]], hence of the [[quantomorphism n-group]])

$$
  QuantMorph(X,\nabla) = \mathbf{Aut}_{\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}}(\nabla)
  \,.
$$

In terms of this [[Lie integration|Lie integrated]] perspective, remark \ref{nExtendedHamiltonEquationAsLInfinityRelation} says that $H$-Hamitlonian $n$-vector fields are the infinitesimal [[n-morphisms]] in $\left(\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}\right)^{\Box_n}$ whose faces carry trivial labels.

Here for instance for $n = 2$ a [[2-morphism]] in $\left(\mathbf{H}_{/\mathbf{B}^2 U(1)_{conn}}\right)^{\Box_2}$ looks, "viewed from the top", like a [[diagram]] in $\mathbf{H}$ of the form

$$
  \array{
    X && \stackrel{}{\longrightarrow} && X
    \\
    & {}_{\mathllap{\nabla}}\searrow && \swarrow_{\mathrlap{\nabla}}
    \\
    \downarrow && \mathbf{B}^2 U(1)_{conn} && \downarrow
    \\
    &  {}^{\mathllap{\nabla}}\nearrow && \nwarrow^{\mathrlap{\nabla}}
    \\
    X && \underset{}{\longrightarrow} && X
  }
$$

with a big 3-[[homotopy]] filling this pyramid.

The equation of remark \ref{nExtendedHamiltonEquationAsLInfinityRelation} is the [[Maurer-Cartan equation]] exhibiting an infinitesimal such situation.

More in detai: the [[Lie integration]] of $\mathfrak{pois}(X,\omega)$ is a [[simplicial object]] which in simplicial degree $n$ has as elements the [[Maurer-Cartan elements]] of $\Omega^\bullet(\Sigma_n)\otimes \mathfrak{pois}(X,\omgea)$, where $\Sigma_n = \Delta^n$ is the $n$-dimensional [[simplex]], regarded as a [[smooth manifold]] (with boundaries and corners).

Write then $\mathbf{d}\sigma^i$ for the canonical [[basis]] of 1-forms on $\Sigma_n$, then consider an element in there is of the form

$$
  A
  =
  \mathbf{d}\sigma^i \otimes v_i
  +
  vol \otimes H
 \,.
$$

This satisfying the [[Maurer-Cartan equation]]

$$
  \mathbf{d}A + [A] +  [A \wedge A] + [A\wedge A \wedge A] + \cdots = 0
$$

then is equivalent to

$$
  [\mathbf{d}\sigma^1 \otimes v_1 \wedge \cdots \wedge \mathbf{d}\sigma^1 \otimes v_1]
  +
  \underbrace{
    \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^n
  }_{= vol}
  \otimes [H]
  = 0
$$

which is again the de Donder-Weyl-Hamilton equation of motion.

=--

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

For $(X,\omega)$ a pre-[[n-plectic manifold]] and $\mathfrak{poiss}(X,\omega)$ the corresponding [[Poisson bracket Lie n-algebra]] then
[[L-∞ algebra]] [[homomorphisms]] of the form

$$
  \mathbb{R}^k \longrightarrow \mathfrak{pois}(X,\omega)
$$

are in bijection to tuples consisting of $k$ [[vector fields]] $(v_1, \cdots, v_k)$ on $X$ and differential forms $\{J_{i_1 \cdots i_l}\}$ and a smooth function $H$ on $X$ such that

$$
  \iota_{v_{1_1} \cdots v_{i_l}} \omega = \pm \mathbf{d} J_{i_1 \cdots i_l}
$$

$$
  \iota_{v_{1_1} \cdots v_{i_n}} \omega = \pm \mathbf{d} H
$$

holds.

=--

The following is the higher/local analog of the [[symplectic Noether theorem]].

For $(X,\omega)$ a pre-[[n-plectic manifold]], (...)

let
$H \in C^\infty(X)$ be a [[smooth function]] to be regarded
as a [[de Donder-Weyl Hamiltonian]] and let $(v_1, \cdots, v_n)$
be a Hamiltonian $n$-vector field, hence a solution to the
Hamilton-de Donder-Weyl equation of motion. By
prop. \ref{kTuplesOfCommutingDWFlows} this corresponds to an
[[L-∞ algebra]] map of the form

$$
  ((v_1, \cdots, v_n), H)
     \;\colon\;
  \mathbb{R}[n-1] \longrightarrow \mathfrak{pois}(X,\omega)
  \,.
$$

+-- {: .num_prop }
###### Proposition
**(higher symplectic Noether theorem)**

The extension of $((v_1, \cdots, v_n), H)$
to a homomorphism of the form

$$
  ((v_1, \cdots, v_n, v_0), H, J, \{K_i\})
    \;\colon\;
  \mathbb{R}[n-1]\oplus \mathbb{R}
    \longrightarrow
  \mathfrak{pois}(X,\omega)
$$

is equivalently the choice of a vector field $v_0$ on $X$ such that

$$
  \iota_{v_0} \mathbf{d}H
  =

$$

$$
  \iota_{v_0} \omega = \pm \mathbf{d}J
$$

$$
  \iota_{v_1 \cdots v_n} \omega = \pm \mathbf{d}H
$$

$$
  \iota_{v_1 \cdots v_{i-1} v_{i+1} \cdots v_n v_0} \omega
  =
  \mathbf{d} K^i
$$

=--

## Related concepts

* [[de Donder-Weyl formalism]]

## References

The specific condition of def. \ref{ExtendedHamiltonEquations} appears as equation (4) in

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

Remark \ref{nExtendedHamiltonEquationAsLInfinityRelation} appears in remark 2.5.10 of

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher geometric prequantum theory]]_ ([arXiv:1304.0236](http://arxiv.org/abs/1304.0236))
  {#hgp13}

[[!redirects Hamiltonian n-vector field]]
[[!redirects Hamiltonian n-vector fields]]
[[!redirects Hamiltonian n-vecotr field]]

[[!redirects Hamiltonian multivector field]]
[[!redirects Hamiltonian multivector fields]]

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


Hamiltonian symplectomorphism infinity-group

[[!redirects Hamiltonian symplectomorphism ∞-group]]

#Contents#
* table of contents
{:toc}

## Idea

The generalization of the traditional notion to [[symplectomorphism group]] of a [[symplectic manifold]] to [[n-plectic geometry|n-plectic]] spaces in [[higher geometry]]

## Definition

For $\mathbf{H}$ a [[cohesive (∞,1)-topos]] equipped with [[differential cohesion]], let $\mathbb{G} \in Grp(\mathbf{H})$ by an [[∞-group]] equipped with [[braided ∞-group]] structure. Write $\mathbf{B}\mathbb{G}_{conn}$ for the corresponding [[moduli ∞-stack]] of $\mathbb{G}$-[[principal ∞-connections]] (see there). For $\nabla \;\colon\; X \to \mathbf{B}\mathbb{G}_{conn}$ a $\mathbb{G}$-[[principal ∞-connection]] on some object $X \in \mathbf{H}$, which here we call a [[prequantum ∞-bundle]]. the corresponding [[quantomorphism ∞-group]] by definition comes with a canonical [[homomorphism]]

$$
  \mathbf{QuantMorph}(\nabla)
  \to
  \mathbf{Aut}(X)
$$

to the [[automorphism ∞-group]] of $X$. The **Hamiltonian symplectomorphism $\infty$-group** $\mathbf{HamSym}(\nabla)$ is the [[1-image]] of this map

$$
  \mathbf{QuantMorph}(\nabla)
  \to
  \mathbf{HamSymp}(\nabla)
  \hookrightarrow
  \mathbf{Aut}(X)
  \,.
$$

## Related concepts

[[!include geometric quantization extensions - table]]

## References

* _[[schreiber:differential cohomology in a cohesive topos]]_

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher prequantum geometry]]_


Hamiltonian vector field

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

#Contents#
* table of contents
{:toc}

## Definition

### On symplectic manifolds
 {#OnSymplecticManifolds}

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

For $(X,\omega)$ a [[symplectic manifold]], a [[vector field]] $v \in \Gamma(T X)$ is called a **Hamiltonian vector field** if its contraction with the [[differential 2-form]] $\omega$ is exact: if there exists $\alpha \in C^\infty(X)$ such that

$$
  \iota_v \omega = d \alpha
  \,.
$$

In this case $\alpha$ is called a [[Hamiltonian]] for $v$.

=--

### On $n$-plectic manifolds

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

For $(X,\omega)$ an [[n-plectic manifold]], a [[vector field]] $v \in \Gamma(T X)$ is called a **Hamiltonian vector field** if is contraction with the $(n+1)$-form $\omega$ is exact: there is $\alpha \in \Omega^{n-1}(X)$ such that

$$
  \iota_v \omega = d \alpha
  \,.
$$

In this case $\alpha$ is called a [[Hamiltonian form|Hamiltonian (n-1)-form]] for $v$.

=--

### On $n$-plectic smooth $\infty$-groupoids
 {#OnNPlecticInfinityGroupoids}

We discuss now the notion of Hamiltonian vector fields in the full generality internal to a [[cohesive (∞,1)-topos]] $\mathbf{H}$. We write out the discussion for the case $\mathbf{H} = $ [[Smooth∞Grpd]] for convenience, but any other choice of cohesive $(\infty,1)$-topos works as well.

Consider the [[circle n-group]] $\mathbf{B}^{n-1}U(1)$ and the corresponding coefficient object $\mathbf{B}^n U(1)_{conn} \in \mathbf{H}$ for $U(1)$-[[differential cohomology]] in degree $(n+1)$, the smooth [[moduli stack]] of [[circle n-bundles with connection]].

For any $X \in \mathbf{H}$,  a morphism $\omega \colon X \to \Omega^{n+1}_{cl}$ is a [[n-plectic geometry|pre-n-plectic structure]] on $X$. For instance $(X,\omega)$ might be a [[symplectic ∞-groupoid]].

A [[higher geometric prequantization]] of $(X,\omega)$ is a lift $\nabla$ in

$$
  \array{
    && \mathbf{B}^n U(1)_{conn}
    \\
    & {}^{\mathllap{\nabla}}\nearrow & \downarrow
    \\
    X &\stackrel{\omega}{\to}& \Omega^{n+1}_{cl}
  }
  \,.
$$

The [[quantomorphism n-group]] of this prequantization is

$$
  \mathbf{QuantMorph}(X,\omega) \coloneqq \prod_{\mathbf{B}^n U(1)_{conn}} \mathbf{Aut}(\nabla)
  \,,
$$

where

1. $\mathbf{Aut}(\nabla)$ is the [[automorphism ∞-group]] of $\nabla$ formed in the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}$

1. $\prod_{\mathbf{B}^n U(1)_{conn}} \colon \mathbf{H}_{/\mathbf{B}^n U(1)_{conn}} \to \mathbf{H}$ is the [[dependent product]] [[(∞,1)-functor]].

There is a canonical homomorphism of [[∞-groups]]

$$
  p \colon \mathbf{QuantMorph}(X,\omega) \to \mathbf{Aut}(X)
$$

to the [[automorphism ∞-group]] of $X$ (the [[diffeomorphism group]] of $X$), given as the restriction to invertible endomorphisms of the canonical morphism

$$
  \prod_{\mathbf{B}^n U(1)_{conn}}
  \left[
    \nabla,\nabla
  \right]
  \to
  \left[
    \sum_{\mathbf{B}^n U(1)_{conn}} \nabla,
    \sum_{\mathbf{B}^n U(1)_{conn}} \nabla
  \right]
  \simeq
  [X,X]
$$

which is discussed at _[internal hom -- Examples -- In slice categories](internal%20hom#ExampleInSliceCategories)_.

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

The [[Hamiltonian symplectomorphism n-group]] $\mathbf{HamSymp}(X,\omega)$ of $(X,\omega)$ is the [[∞-image]] of this morphism $p$, hence the factorization

$$
  p \colon \mathbf{QuantMorph}(X,\omega) \to \mathbf{HamSymp}(X,\omega)
  \hookrightarrow \mathbf{Aut}(X)
$$

of $p$ by an [[effective epimorphism in an (∞,1)-category|effective epimorphism]] followed by a [[monomorphism in an (∞,1)-category|monomorphism]].

The corresponding [[∞-Lie algebra]]

$$
  HamVect(X,\omega) \coloneqq Lie(\mathbf{HamSymp}(X,\omega))
$$

we call the $\infty$-Lie algebra of **Hamiltonian vector fields** on $(X,\omega)$.

=--

More explicitly:

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

A **Hamiltonian diffeomorphism** $\phi$ on
on $(X, \omega)$ is an element $\phi \colon X \stackrel{\simeq}{\to} X$ in the [[automorphism ∞-group]] $\phi \in \mathbf{Aut}(X)$ such that it fits into a [[diagram]] of the form

$$
  \array{
    X &&\underoverset{\simeq}{\phi}{\to}&& X
    \\
    & {}_{\mathllap{\nabla}}\searrow
    &
      \swArrow_{\alpha}
    &
    \swarrow_{\mathrlap{\nabla}}
    \\
    && \mathbf{B}^n U(1)_{conn}
  }
$$

in $\mathbf{H}$.

=--

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

For $n = 1$ and $(X, \omega)$ an ordinary
prequantizable [[symplectic manifold]] regarded as a smooth $\infty$-groupoid,
this definition reproduces the ordinary definition
of Hamiltonian vector fields [above](#OnSymplecticManifolds).

In particular it is independent of the choice of [[prequantum line bundle]].

=--

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

To compute the Lie algebra of this,
we need to consider smooth 1-parameter families of
Hamiltonian diffeomorphisms and differentiate them.

Assume first that the [[prequantum line bundle]] is
trivial as a bundle, with the connection 1-form of $\nabla$
given by a globally defined
$A \in \Omega^1(X)$ with $d A = \omega$. Then the
existence of the diagram in def. \ref{HamiltonianDiffeomorphismsOnInfinityGroupoid}
is equivalent to the condition

$$
  (\phi(t)^* A - A) = d \alpha(t)
  \,,
$$

where $\alpha(t) \in C^\infty(X)$. Differentiating this at 0 yields the
[[Lie derivative]]

$$
  \mathcal{L}_v A = d \alpha'
  \,,
$$

where $v$ is the [[vector field]] of which $t \mapsto \phi(t)$ is the
[[flow]] and where $\alpha' := \frac{d}{dt} \alpha$.

By [[Cartan calculus]] this is equivalently

$$
  d_{\mathrm{dR}} \iota_v A + \iota_v d_{dR} A  = d \alpha'
$$

and using that $A$ is the connection on a prequantum circle bundle
for $\omega$

$$
  \iota_v \omega = d (\alpha' - \iota_v A)
  \,.
$$

This says that for $v$ to be Hamiltonian,
its contraction with $\omega$ must be exact. This is precisely the definition of Hamiltonian vector fields.
The corresponding Hamiltonian function here is

$$
  h := \alpha'-\iota_v A
  \,.
$$

In the general case that the prequantum bundle is not trivial, we can present it by a [[Cech cohomology|Cech cocycle]] on the [[Cech nerve]]
$C(P_* X \to X)$ of the based [[path space]]
[[surjective submersion]] (regarding $P_* X$ as a
[[diffeological space]] and choosing one base point per connected component,
or else assuming without restriction that $X$ is connected).

Any [[diffeomorphism]] $\phi = \exp(v) : X \to X$ lifts to a
diffeomorphism $ P_*\phi : P_* X \to P_* X$ by setting $P_* \phi(\gamma) : (t \in [0,1]) \mapsto \exp(t v)(\gamma(t))$. This way the Hamiltonian diffeomorphism
is presented in the [[model structure on simplicial presheaves]] by a diagram

$$
  \array{
    C(P_*X \to X) &&\underoverset{\simeq}{\phi}{\to}&& C(P_*X \to X)
    \\
    & {}_{\mathllap{\nabla}}\searrow
    &
      \swArrow_{\alpha}
    &
    \swarrow_{\mathrlap{\nabla}}
    \\
    && \mathbf{B}^n U(1)_{conn}
  }
$$

of [[simplicial presheaves]].

Now the same argument as above applies for $P_* X$.

=--

## Properties

### Hamiltonian actions and moment maps

An [[action]] of a [[Lie algebra]] by (flows of)
Hamiltonian vector fields that can be lifted to a [[Hamiltonian action]] is equivalently given by a _[[moment map]]_. See there for details.

### Relation to symplectic vector fields
 {#RelationToSymplectic}

Every Hamiltonian vector field is in particular a [[symplectic vector field]]. Where a symplectic vector field only preserves the [[symplectic form]], a Hamiltonian vector field also preserves the connection on its [[prequantum line bundle]].

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

For $(X, \omega)$ a finite dimensional [[symplectic manifold]],
there is an [[exact sequence]]

$$
  0 \to HamVect(X, \omega) \to SympVect(X, \omega) \to H^1(X, \mathbb{R})
  \to 0
  \,.
$$

=--

This appears as ([Brylinski, 2.3.3](#Brylinski)).

### Relation to functions and Poisson brackets
 {#RelationToFunctions}

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

Let $(X, \omega)$ be a connected [[symplectic manifold]]. Then there is a [[central extension]] of [[Lie algebras]]

$$
  0 \to \mathbb{R} \to (C^\infty(X),\{-,-\}) \to HamVect(X,\omega) \to 0
  \,.
$$

=--

This is a special case of what is called the **[[Kostant-Souriau central extension]]**. See around ([Brylinski, prop. 2.3.9](#Brylinski)).

[[!include geometric quantization extensions - table]]

### Group of Hamiltonian symplectomorphisms

The auto-[[symplectomorphisms]] on a [[symplectic manifold]] form a [[group]], of which the [[symplectic vector fields]] generate the connected component. The Hamiltonian vector fields among the symplectic ones generate the group of [[Hamiltonian symplectomorphisms]].

(...)

## Related concepts

* [[Hamiltonian]], [[Hamiltonian action]], [[Hamiltonian symplectomorphism]]

* [[Reeb vector field]]

* The Hamiltonian vector field of a given function may also be called its _[[symplectic gradient]]_.

* The generalization to [[multisymplectic geometry]]/[[n-plectic geometry]]: [[Hamiltonian n-vector fields]]

## References

Named after [[William Rowan Hamilton]].

A textbook reference is section II.3 in

* [[Jean-Luc Brylinski]], _Loop spaces, characteristic classes and geometric quantization_, Birkhäuser.
 {#Brylinski}

For more references on the ordinary notion of Hamiltonian vector fields see the references at _[[symplectic geometry]]_ and _[[geometric quantization]]_.

The notion of Hamiltonian vector field in [[n-plectic geometry]] is discussed in

* [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis (2011) ([arXiv:1106.4068](http://arxiv.org/abs/1106.4068))

The notion of Hamiltonian vector field for $n$-plectic cohesive $\infty$-groupoids is discussed in section 4.8.1 of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_

See also at _[[higher geometric quantization]]_.

[[!redirects Hamiltonian vector fields]]

[[!redirects hamiltonian vector field]]
[[!redirects hamiltonian vector fields]]


Hammad Rana

##Contact
Contact me at <iltukhara@gmail.com>

## Interests
Real and Complex Algebraic geometry, essentially an analytic style of algebraic geometry type of person

##My Pages
* [[Surreal space]]
* [[Surreal geometry]]

category: people


Hamming code

#Contents#
* table of contents
{:toc}

## Idea

A certain class of [[linear codes]] used for [[error correction]] in communication protocols.

## References

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

[[!redirects Hamming codes]]


Han Yan

* [personal page](https://sites.google.com/view/hanyanphysics)

## Selected writings

On [[holography]] and [[holographic entanglement entropy]] in relation to [[tensor networks]] and [[quantum error correcting codes]] such as the [[Majorana dimer code]]:

* [[Han Yan]], *Geodesic string condensation from symmetric tensor gauge theory: a unifying framework of holographic toy models*, Phys. Rev. B 102, 161119 (2020) ([arXiv:1911.01007](https://arxiv.org/abs/1911.01007))

category: people


Hanany-Witten effect

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

#Contents#
* table of contents
{:toc}

## Idea

In [[string theory]], what has come to be known as the *Hanany-Witten-effect* (or *-process*, *-tansition*, or *-move*) is an expected phenomenon exhibited by branes in [[type II string theory]] which is roughly described as follows:

"*Moving an [[NS5-brane]] across a [[D-brane|$D (2p+\sigma)$-brane]] creates a [[D-brane|$D (2p-2 + \sigma)$-brane]] [[intersecting branes|stretching between]] the two*".

> (Here $\sigma = 0$ in [[type IIA string theory]] and $\sigma=1$ in [[type IIB string theory]].)

Something like this was informally argued in [Hanany & Witten (1997)](#HananyWitten97). Later authors have
tried to offer explanation and illustration of what is meant to be happening here, as follows:

[Nakatsu et al (1998)](#NakatsuEtAl98):

[Marolf (2001)](#Marolf01):

[Eto et al (2005)](#EtoEtAl05):

[Pelc (2008)](#Pelc08):

[Gang et al (2011)](#GangEtAl11):

\linebreak

[Marolf (2001)](#Marolf01) suggested that the Hanany-Witten effect may be understood entirely as a consequence of the famous ([[pregeometric RR-field|pregeometric]]) [[Bianchi identity]] for the [[RR-field]]-[[fluxes]] $F_{8-2p-\sigma}$ sourced by the [[D-branes]], which is [[twisted de Rham cohomology|twisted]] by the [[B-field]]-[[flux]] $H_3$ sourced by [[NS5-branes]]:

\[
  \label{GeneralBianchiIdentity}
  \mathrm{d}
  F_{8-2p-\sigma}
  \;=\;
  H_3 \wedge F_{8-(2p+2)-\sigma}
\]

and offered an argument in prose [[Marolf (2001) pp. 3](#Marolf01)].

[Below](#Examples) we spell out this approach in more detail.

## Examples
 {#Examples}

### D6 stretching between D8 and NS5
 {#D6StretchingBetweenNS5AndD8}

(cf. *[[D6-D8 brane bound state]]* and [Bergshoeff, Lozano & Ortin (1998)](#BergshoeffLozanoOrtin98))

We explain the "Hanany-Witten effect" in detail for the case of "flat" [[NS5-brane|NS5]]--[[D6-brane|D6]]--[[D8-brane|D8]]-configurations, as a consequence of the [[pregeometric RR-field|pregeometric]] [[Bianchi identities]] (eq:GeneralBianchiIdentity) satisfied by the [[flux densities]] sourced by these branes.

In this case  we are dealing with

* [[Minkowski spacetimes]]$\;$ $\mathbb{R}^{1,p}$ which we may identify with their canonical [[coordinate charts]] $(x^0, \vec x) \,\coloneqq\, (x^0, x^1, \cdots, x^p)$.

* A closed [[differential 3-form]]

  $$
    H_3
      \,\coloneqq\,
    k vol_{S^3}
    \,\in\,
    \Omega^3_{dR}\big(
      S^3
    \big)_{cl}
    \xrightarrow{(pr_{S^3})^\ast}
    \Omega^3_{dR}
    \big(
      \mathbb{R}^{1,5}
      \times
      \mathbb{R}_{+}
      \times
      S^3
    \big)_{cl}
    \,\simeq\,
    \Omega^3_{dR}
    \big(
      \mathbb{R}^{1,9}
      \setminus
      \mathbb{R}^{1,5}
    \big)_{cl}
  $$

  being the [[B-field]] [[flux density]] sourced by an [[NS5-brane]] with worldvolume

  $$
    \mathbb{R}^{5,1}
      \xhookrightarrow{
        \vec x \,\mapsto\, (\vec x, \vec 0)
      }
    \mathbb{R}^{9,1}
    \,.
  $$

* A closed [[differential 0-form]]  (the "[[massive type IIA supergravity|Romans mass]]")

  $$
    F_0
      \,\in\,
    \Omega^0_{dR}
    \Big(
      \mathbb{R}^{9,1}
      \setminus
      \mathbb{R}^{8,1}
    \Big)
  $$

  being the [[RR-field]] [[flux density]] sourced by a [[D8-brane]] whose worldvolume

  $$
    \mathbb{R}^{1,8}
      \overset{
       \vec x \,\mapsto\, (\vec x, d)
      }{
        \hookrightarrow
      }
    \mathbb{R}^{1,9}
  $$

  we take to be located at $x^9=d$ for some $d \in \mathbb{R}_+$.

* A [[differential 2-form]]

  $$
    F_2
      \,\in\,
    \Omega^3_{dR}
    \Big(
      \mathbb{R}^{9,1}
      \setminus
      \big(
        \mathbb{R}^{5,1}
        \cup
        \mathbb{R}^{6,1}
      \big)
    \Big)
  $$

  being the [[RR-field]] [[flux density]] sourced by a [[D6-brane]] with *potential worldvolume* ("potential" because the D6 may have $\sim$ vanishing charge carried by parts of its worldvolume, hence may "disappear" from its would-be worldvolume locus in parts, as we will see):

  \[
    \label{D6BraneWorldvolume}
    \mathbb{R}^{6,1}
      \xhookrightarrow{
        (x^0, \vec x)
        \,\mapsto\,
        \big(x^0, \vec 0, \vec x\big)
      }
    \mathbb{R}^{9,1}
  \]

  transverse to that of both the NS5-brane and the D8-brane and satisying the equation

  \[
    \label{BianchiIdentityForD6BraneFlux}
    \mathrm{d}
    F_{2}
    \;=\;
    H_3 \wedge F_{0}
    \;\;\;\;\;
    \in
    \;
    \Omega^3_{dR}
    \Big(
      \mathbb{R}^{9,1}
      \setminus
      \big(
        \mathbb{R}^{5,1}
        \cup
        \mathbb{R}^{8,1}
        \cup
        \mathbb{R}^{6,1}
      \big)
    \Big)
    \,,
  \]

  where now $H_3$ and $F_0$ are understood of the [[pullback of differential forms|pullback]] (restriction) of the above [[differential forms]] of these names to the [[complement]] of the [[D6-brane]] [[worldvolume]] (eq:D6BraneWorldvolume).

  Notice that even if/though the original forms $H_3$ is *not [[exact differential form|exact]]*, after this pullback/restriction to this it does become exact, so that the equation (eq:BianchiIdentityForD6BraneFlux) is meaningful.

In order to model the Hanany-Witten effect proper as considered by previous authors, we declare the following boundary conditions:

* The [[massive type IIA supergravity|Romans vacuum]] is to the "right" of the [[D8-brane]]

  \[
    \label{RomansVacuum}
    x^9 \gt d
    \;\;\;\;\;\;\;
    \Rightarrow
    \;\;\;\;\;\;\;
    F_0(x^9) = 0
    \,.
  \]

  Notice that the [[Bianchi identity]] for $F_0$ is $\mathrm{d} F_0 = 0$, which means that it is a [[locally constant function]] whose jump across the [[D8-brane]] worldvolume measures the charge/number $N$ of these D8-branes (cf. eg. [Fazzi (2017) p. 40](NS5-brane#Fazzi17)). Hence

  \[
    \label{RomansVacuum}
    x^9 \lt d
    \;\;\;\;\;\;\;
    \Rightarrow
    \;\;\;\;\;\;\;
    F_0(x^9) = N
    \,.
  \]

* There is no D6-brane to the far "left" of the NS5/D8,

  \[
    \label{ND6BesidesThoseCreated}
    \underset{\underset{x^9 \to -\infty}{\longrightarrow}}{\lim}
    F_2(x^9)
    \;\;
    =
    \;\;
    0
    \,,
  \]

  which just means to disregard D6-brane whose presence is not due to the Hanany-Witten effect.

With these boundary conditions set, the differential equation (eq:BianchiIdentityForD6BraneFlux) may be solved for $F_2$, and hence for the density of D6-brane charge/number which is "created" by the Hanany-Witten effect: At given $x^9$ the number/charge of D6-branes seen is the integral of $F^2$ around the 2-sphere in

$$
  \mathbb{R}^{9,1}
  \setminus
  \mathbb{R}^{6,1}
  \;\simeq\;
  \mathbb{R}^{6,1}
    \times
  \mathbb{R}_+
    \times
  S^2
$$

over $(\vec 0, x^9) \,\in\, \mathbb{R}^{6,1}$.  Now there are two cases:

\linebreak

\linebreak

\linebreak

\linebreak

\linebreak

\[
  \label{IllustrationFluxDensitityofD6OnD8}
\]

1. **$d \lt 0$ -- indicated in the top part of figure (eq:IllustrationFluxDensitityofD6OnD8)**

   In this case the NS5-brane is to the "right" of the D8-brane and hence in the Romans vacuum where $F_0 = 0$. Here the differential equation (eq:BianchiIdentityForD6BraneFlux) becomes

   $$
     \begin{array}{c}
      \mathrm{d} F_2 &=& F_0 \wedge H_3
      \\
      &=& 0
     \end{array}
   $$

   This means that $F_2$ does not change in the Romans vacuum, and hence if it vanishes to the left, as the boundary condition assumes, then it vanishes throughout.

   Since the integral of $F_2$ counts the number/charge of D6-brane, the conclsusion is that *no D6-branes are created* in this situation.

1. **$d \gt 0$ -- indicated in the bottom part of figure (eq:IllustrationFluxDensitityofD6OnD8)**

   In this case the NS5-brane is to the "left" of the D8-brane and hence in the region where $F_0 = N$ equals the number of D8-branes. Here the differential equation (eq:BianchiIdentityForD6BraneFlux) becomes

   $$
     \begin{array}{c}
      \mathrm{d} F_2 &=& F_0 \wedge H_3
      \\
      &=& N \, H_3
     \end{array}
   $$

   This means now that as approach the NS5/D8-branes from the "left", $F_2$ is continuously increasing by a rate measured by the density $H_3$ of NS5-brane flux. While this is everywhere non-vanishing, it is concentrated around the NS5-brane worldvolume locus (falling off by a power law from there). Therefore once we have "crossed the NS5-brane" and thus picked up most of its flux, $\int_{S^2} F_2$ will have increased from its asympotically vanishing value to $N \int_{S^2} \int_{-\infty}^d H_3 \,\sim\, N \int_{S^3} H_3 \,\sim\, N k$, the product of the number of NS5-branes and D8-branes. This is hence the number of D6-branes "created" by the Hanany-Witten effect.

### D3 stretching between D5 and NS5
 {#D2StretchingBetweenNS5AndD5}

(cf. *[[D3-D5 brane bound state]]*)

We discuss the "creation" of [[D3-brane]] [[flux]] "in between" a [[background field|background]] of [[D5-brane]]- and [[NS5-brane]] flux as seen from (just) the [[pregeometric RR-field|pregeometric flux]]-equations.

If we assume that there are no [[D7-branes]] present in that the [[RR-field]] [[flux density]] which they source vanishes, $F_1 = 0$, then the [[Bianchi identity]] for the [[D5-brane]]-flux is just

$$
  \begin{array}{rcl}
  \mathrm{d} \, F_3
  &=&
  H_3 \wedge F_1
  \\
  &=& 0
  \end{array}
$$

just as for the [[B-field]] [[flux density]]

$$
  \mathrm{d}\, H_3 \;=\; 0
  \,.
$$

Hence assume that there is a pair of flat parallel [[D5-branes]] and [[NS5-branes]] at a [[positive number|positive]] distance $2 d \in \mathbb{R}_+$ from each other

$$
  \array{
    \mathbb{R}^{5,1}_{NS5}
    &\xhookrightarrow{\phantom{--}}&
    \mathbb{R}^{10,1}
    \\
    (t,\vec x) &\mapsto& (t,\vec x, -d , \vec 0)
  }
  \;\;\;\;\;\;\;\;\;\;\;
  \;\;\;\;\;\;\;\;\;\;\;
  \;\;\;\;\;\;\;\;\;\;\;
  \array{
    \mathbb{R}^{5,1}_{D5}
    &\xhookrightarrow{\phantom{--}}&
    \mathbb{R}^{10,1}
    \\
    (t,\vec x) &\mapsto& (t,\vec x, +d , \vec 0)
  }
$$

in that the respective [[flux densities]] are
$$
  H_3
    \;\coloneqq\;
  vol_{S^3}
  \;\in\;
  \Omega^3_{dR}(S^3)
  \xrightarrow{ pr_{S^3}^\ast }
  \Omega^3_{dR}\big(
    \mathbb{R}^{5,1}_{NS5}
    \times
    \mathbb{R}_+
    \times
    S^3
  \big)
  \;\simeq\;
  \Omega^3_{dR}\big(
    \mathbb{R}^{9,1}
    \setminus
    \mathbb{R}^{5,1}_{D5}
  \big)
$$

and

$$
  F_3
    \;\coloneqq\;
  vol_{S^3}
  \;\in\;
  \Omega^3_{dR}(S^3)
  \xrightarrow{ pr_{S^3}^\ast }
  \Omega^3_{dR}\big(
    \mathbb{R}^{5,1}_{D5}
    \times
    \mathbb{R}_+
    \times
    S^3
  \big)
  \;\simeq\;
  \Omega^3_{dR}\big(
    \mathbb{R}^{9,1}
    \setminus
    \mathbb{R}^{5,1}_{D5}
  \big)
$$

In the following we regard both of these as further [[pullback of differential forms|pulled back]]/restricted to the joint [[complement]] $\mathbb{R}^{9,1} \setminus \big( \mathbb{R}^{5,1}_{NS5} \cup \mathbb{R}^{5,1}_{D5} \cup \mathbb{R}^{3,1}_{D3}\big)$.
Notice that as such they differ, one being the translate of the other by $2d$ along one of the axes.

Now the [[Bianchi identity]] for the [[RR-field]] [[flux density]] $F_5$ sourced by any [[D3-branes]] is

\[
  \label{BianchiIdentityForD3InBetweenNS5AndD5}
  \mathrm{d} \, F_{5}
  \;=\;
  H_3 \wedge F_3
  \,,
\]

Imposing the [[boundary condition]] that $F_5$ [[vanishing at infinity|vanishes at infinity]],  the differential equation will imply the presence of non-vanishing $F_5$-flux and hence of $D3$-branes "in between" the D5/NS5 iff the [[wedge product]] $H_3 \wedge F_3$ has the form of a "multipole" roughly supported in between these branes.

The following graphics, which shows a 2-dimensional shadow of the situation, qualitatively indicates that and why this is the case:

Here

* the strength of the circular lines indicates the [[absolute value]] of the [[flux densities]] $H_3$, $F_3$ (vanishing at infinity and spiked at the locus of the sourcing 5-brane),

* the little arrows indicate the [[orientation of a vector|orientation]] of the two flux densities, projected to the plane that is being shown

* the little parallelograms indicate the [[orientation]] of the [[wedge product]] $H_3 \wedge F_3$ in the projection to the plane being shown.

(Hence what the graphics is really showing, approximately, is the wedge product of a pair of translates of $dvol_{S^1}$ pulled back to around either puncture of the 2-punctured plane.)

The point is to observe that --- besides the abolute value of the wedge product $F_3 \wedge H_3$ evidently being concentrated near the branes, hence particularly between them when they are close --- the sign of the orientation of the wedge product changes as the transverse coordinate passes through the axis between the branes.

Hence as we solve the differential equation $\mathrm{d} F_5 \,=\, H_3 \wedge F_3$ incrementally by running, say, from the top to the bottom in the above graphics, we see that the absolute value of $F_5$ --- indicated in $\color{red}\text{red shading}$ above ---  fist incrementally picks up positive contributions as we approach the area between the branes, and then dimishies again as we leave that area. In total, the absolute value of the solution $F_5$ will be supported roughly in between the two 5-branes.

Since $F_5$ is the flux sourced by [[D3-branes]], this means that D3-branes are seen to be "created" in between the 5-branes.

However, from this analysis of the fluxes alone, it does not follow that the D3-branes would disappear as the NS5- and D5-brane "swap positions", as Hanany-Witten would suggest. In fact there is no intrinsic such directedness in the problem in the first place with respect to which one could even speak about the branes "swapping positions".

Hence to see the full Hanany-Witten effect in this case, if it exists, one may have to look beyond the bare fluxes and include also the field of gravity. A suggestion for how this might work is given, for the analogous situation of [[M2-M5 brane bound state|M2-M5 brane]] [[brane intersection|intersections]], in [Hosomichi (2000), §5](#Hosomichi00).

\linebreak

## Related concepts

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

## References

The original article:

* {#HananyWitten97} [[Amihay Hanany]], [[Edward Witten]], *Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics*, Nucl. Phys.B **492** (1997) 152-190 [[arXiv:hep-th/9611230](https://arxiv.org/abs/hep-th/9611230), doi:10.1016/S0550-3213(97)80030-2]

Further discussion:

* {#NakatsuEtAl98} Toshio Nakatsu, Kazutoshi Ohta, Takashi Yokono, Yuhsuke Yoshida, *A Proof of Brane Creation via M theory*,  	Mod. Phys. Lett. A **13** (1998) 293-302 [[arXiv:hep-th/9711117](https://arxiv.org/abs/hep-th/9711117), [doi:10.1142/S0217732398000358](https://doi.org/10.1142/S0217732398000358)]

* {#KubotaZhou00} Takahiro Kubota, Jian-Ge Zhou, *RR charges of D2-branes in group manifold and Hanany-Witten effect*, JHEP 0012 (2000) 030 [[arXiv:hep-th/0010170](https://arxiv.org/abs/hep-th/0010170), [doi:10.1088/1126-6708/2000/12/030](https://doi.org/10.1088/1126-6708/2000/12/030)]

* {#Marolf01} [[Donald Marolf]], §2 of: *T-duality and the case of the disappearing brane*, JHEP 0106:036 (2001) [[arXiv:hep-th/0103098](https://arxiv.org/abs/hep-th/0103098), [doi:10.1088/1126-6708/2001/06/036](https://doi.org/10.1088/1126-6708/2001/06/036)]

* {#EtoEtAl05} Minoru Eto, Youichi Isozumi, Muneto Nitta, Keisuke Ohashi, Kazutoshi Ohta, Norisuke Sakai, §3.3 in: *D-brane Construction for Non-Abelian Walls*, Phys.Rev. D **71** (2005) 125006 [[arXiv:hep-th/0412024](https://arxiv.org/abs/hep-th/0412024), [doi:10.1103/PhysRevD.71.125006](https://doi.org/10.1103/PhysRevD.71.125006)]

* {#Pelc08} [[Oskar Pelc]], , *On the Quantization Constraints for a D3 Brane in the Geometry of NS5 Branes*,  	JHEP 0008 (2000) 030 [[arXiv:hep-th/0007100](https://arxiv.org/abs/hep-th/0007100), [doi:10.1088/1126-6708/2000/08/030](https://doi.org/10.1088/1126-6708/2000/08/030)]

* {#GangEtAl11} Dongmin Gang, Eunkyung Koh, Kimyeong Lee, Jaemo Park, *ABCD of 3d $\mathcal{N}=8$ and 4 Superconformal Field Theories* [[arXiv:1108.3647](https://arxiv.org/abs/1108.3647)]

See also:

* Wikipedia, *[Hanany-Witten transition](https://en.wikipedia.org/wiki/Hanany%E2%80%93Witten_transition)*

The case of [[D6-D8 brane bound states]]:

* {#HananyZaffaroni98} [[Amihay Hanany]], [[Alberto Zaffaroni]], §2.4 in: *Chiral Symmetry from Type IIA Branes*, Nucl. Phys. B **509** (1998) 145-168 [[arXiv:hep-th/9706047](https://arxiv.org/abs/hep-th/9706047), doi:10.1016/S0550-3213(97)00595-6]

* {#BergshoeffLozanoOrtin98} [[Eric Bergshoeff]], [[Yolanda Lozano]], [[Tomas Ortin]], p. 60 of: *Massive Branes*, Nucl. Phys. B **518** (1998) 363-423 [[arXiv:hep-th/9712115](https://arxiv.org/abs/hep-th/9712115), doi:10.1016/S0550-3213(98)00045-5]

Discussion for [[M2-M5 brane bound state|M2-M5 brane]] [[brane intersection|intersections]]:

* {#Hosomichi00} [[Kazuo Hosomichi]], *On Branes Ending on Branes in Supergravity*, JHEP 0006 (2000) 004 [[arXiv:hep-th/0002069](https://arxiv.org/abs/hep-th/0002069), [doi:10.1088/1126-6708/2000/06/004](https://doi.org/10.1088/1126-6708/2000/06/004)]

[[!redirects Hanany-Witten effects]]

[[!redirects Hanany-Witten process]]
[[!redirects Hanany-Witten processes]]

[[!redirects Hanany-Witten move]]
[[!redirects Hanany-Witten moves]]

[[!redirects Hanany-Witten transition]]
[[!redirects Hanany-Witten transitions]]


Handbook of Algebraic Topology

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

This page collects material and links related to

* [[Ioan Mackenzie James]],

  _The Handbook of Algebraic Topology_

  Oxford 1995

  [ISBN:9780080532981](https://www.elsevier.com/books/handbook-of-algebraic-topology/james/978-0-444-81779-2)

  [doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7)

collecting survey articles on subjects in [[algebraic topology]]/[[homotopy theory]].

#Contents#
* table of contents
{:toc}

## Chapters

* Foreword.

* List of Contributors.

1. [[Homotopy Types]] (H.-J. Baues).

1. [[Homotopy theories and model categories]] (W.G. Dwyer, J. Spalinski).

1. [[Proper Homotopy Theory]] (T. Porter).

1. [[Introduction to fibrewise homotopy theory]] (I.M. James).

1. Coherent homotopy over a fixed space (K.A. Hardie, K.H. Kamps).

1. [[Modern foundations for stable homotopy theory]] (A.D. Elmendorf et al.)

1. Completions in algebra and topology (J.P.C. Greenlees, J.P. May).

1. [[Equivariant stable homotopy theory]] (J.P.C. Greenlees, J.P. May).

1. The stable homotopy theory of finite complexes (D.C. Ravenel).

1. The EHP sequence and periodic homotopy (M. Mahowald, R.D. Thompson).

1. Introduction to nonconnective $Im(J)$-theory (M.C. Crabb, K. Knapp).

1. Applications of nonconnective $Im(J)$-theory (K. Knapp).

1. Stable homotopy and iterated loop spaces (G. Carlsson, R.J. Milgram).

1. Stable operations in generalized cohomology ([[J. M. Boardman]]).

   [pdf](https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf), [[Boardman-StableOperations.pdf:file]]

1. Unstable operations in generalized cohomology ([[J. M. Boardman]], [[David Copeland Johnson|D. C. Johnson]], [[W. S. Wilson]])

   ([pdf](https://hopf.math.purdue.edu/Boardman-Johnson-Wilson/bjw.pdf))

1. Differential graded algebras in topology (Y. Felix et al.).

1. {#BrownSzczarba} Real and rational homotopy theory ([[Edgar H. Brown]], [[Robert H. Szczarba]])

   on [[real homotopy theory]]

   and [[Borel-equivariant rational homotopy theory]] applied to [[covering spaces]] of non-[[nilpotent spaces]]

   [doi:10.1016/B978-044481779-2/50018-3](https://doi.org/10.1016/B978-044481779-2/50018-3)

1. Cohomology of groups (D.J. Benson, P.H. Kropholler).

1. Homotopy theory of Lie groups (M. Mimura).

1. Computing $v_1$-periodic homotopy groups of spheres and some compact Lie groups (D.M. Davis).

1. Classifying spaces of compact Lie groups and finite loop spaces (D. Notbohm).

1. H-spaces with finiteness conditions (J.P. Lin).

1. [[co-H-space|Co-H-spaces]] ([[Martin Arkowitz]]).

1. Fibration and product decompositions in nonstable homotopy theory (F.R. Cohen).

1. Phantom maps (C.A. McGibbon).

1. Wall's finiteness obstruction (G. Mislin).

1. Lusternik-Schnirelmann category (I.M. James).

## Related entries

* [[Handbook of Homotopy Theory]]

category:reference


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

This page compiles pointers related to:

* [[Eric Schechter]]:

  \linebreak

  **Handbook of Analysis and its Foundations**

  \linebreak

  Academic Press (1996)

  \linebreak

  [doi:10.1016/B978-0-12-622760-4.X5000-6](https://doi.org/10.1016/B978-0-12-622760-4.X5000-6)

  [webpage](https://math.vanderbilt.edu/schectex/ccc/)

  [pdf](http://staffwww.dcs.shef.ac.uk/people/R.Chisholm/docs/Eric%20Schechter%20Handbook%20of%20Analysis%20and%20Its%20Foundations%20%201996.pdf)

  [errata](http://www.math.vanderbilt.edu/~schectex/ccc/addenda/)

on [[analysis]].

***

Schechter\'s _[Handbook of Analysis and its Foundations](http://www.math.vanderbilt.edu/~schectex/ccc/)_ is a large book, intended for self study by beginning graduate students or senior-level undergraduates, on all of the basic topics of abstract [[analysis]] and then some.  Its logical flow is very much like that of [[Bourbaki]], but focussed on that which applies to analysis and written in a modern style.  Except for the slightly broken index (check the [errata](http://www.math.vanderbilt.edu/~schectex/ccc/addenda/)!), it is very user-friendly, with sketched proofs phrased as exercises with hints, many examples (eventually), and abundant cross references.  It is also extremely self contained; the only prerequisite is mathematical maturity, and the first section even helps with that!

It begins, as the name implies, with [[foundations]]: not only the usual naïve [[set theory]], but also a discussion of [[ZFC]], [[constructive mathematics]], and enough [[model theory]] to do [[nonstandard analysis]].  There is special emphasis on the [[axiom of choice]]; throughout the book, it is explicitly pointed out whenever anything beyond [[dependent choice]] and [[excluded middle]] is required.  (The axioms of [[axiom of replacement|replacement]] and [[axiom of foundation|foundation]], on the few occasions when they appear, are also pointed out, so logically the book takes place in ${Z^-} + {DC}$.)

The book then moves on to [[algebra]], moving from [[monoid]]s to [[field]]s, on the grounds that such algebra also serves as a foundation for analysis.  This culminates in a treatment of [[category theory]]; this is somewhat unsatisfactory (although very good for an analysis book!) and is not much more than Bourbaki\'s theory of structures reinterpreted as a theory of [[concrete category|concrete categories]].  After algebra comes [[topology]], and then analysis proper.

Those aspects of analysis that do not depend on the theory of the [[real number]]s are also covered when appropriate in the set-theory and algebra sections of the book; thus [[topological space]]s and convex sets (for example) are defined early, although they are not thoroughly studied until later (after the foundational parts are finished).  The analysis in the book is soft and abstract, on the grounds that this material serves as the proper foundation for hard results in concrete cases.  However, many concrete examples are given to illustrate the abstract ideas.  The breadth of topics covered, even within analysis itself, is quite wide; from [[convergence space]]s to [[ultrabarrel]]s, from the [[Henstock integral]] to the [[Brouwer fixed-point theorem]], it has it all.

But everything must stop somewhere; it does not cover [[complex analysis]].

## Contents

*  Part A:  Sets and Orderings (Chapters 1--7)
*  Part B:  Algebra (Chapters 8--14)
*  Part C:  Topology and Uniformity (Chapters 15--21)
*  Part D:  Topological Vector Spaces (Chapters 22--30)

1.  Sets
2.  Functions
3.  Relations and Orderings
4.  More About Sups and Infs
5.  Filters, Topologies, and Other Sets of Sets
6.  Constructivism and Choice
7.  Nets and Convergences
8.  Elementary Algebraic Systems
9.  Concrete Categories
10. The Real Numbers
11. Linearity
12. Convexity
13. Boolean Algebras
14. Logic and Intangibles
15. Topological Spaces
16. Separation and Regularity Axioms
17. Compactness
18. Uniform Spaces
19. Metric and Uniform Completeness
20. Baire Theory
21. Positive Measure and Integration
22. Norms
23. Normed Operators
24. Generalized Riemann Integrals
25. Fréchet Derivatives
26. Metrization of Groups and Vector Spaces
27. Barrels and Other Features of TVS's
28. Duality and Weak Compactness
29. Vector Measures
30. Initial Value Problems

category: reference, analysis

[[!redirects HAF]]
[[!redirects Handbook of analysis and its foundations]]


Handbook of Categorical Algebra

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

This entry is to record the monograph

* [[Francis Borceux]],

  **Handbook of Categorical Algebra**

  * Vol. 1:

    **Basic Category Theory**

    [doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)

  * Vol. 2:

    **Categories and Structures**

    [doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865)

  * Vol. 3:

    **Categories of Sheaves**

    [doi:10.1017/CBO9780511525872](https://doi.org/10.1017/CBO9780511525872)

  Encyclopedia of Mathematics and its Applications **50**

  Cambridge University Press (1994)

on [[category theory]], [[categorical algebra]] and [[topos theory]].

Book review:

* [[Peter Johnstone]], Bull. London Math. Soc. __28__:4, 440-442 ([doi:10.1112/blms/28.4.440](http://dx.doi.org/10.1112/blms/28.4.440))

## Contents

Vol. 1: Basic Category Theory

> Preface; Introduction; 1. The language of categories; 2. [[limits|Limits]]; 3. [[adjoint functor|Adjoint functors]]; 4. Generators and projectives; 5. Categories of fractions; 6. Flat functors and Cauchy completeness; 7. Bicategories and distributors; 8. [[internal category|Internal category theory]]; Bibliography; Index.

Vol. 2: Categories and Structures

> Preface; Introduction; 1. [[abelian category|Abelian categories]]; 2. [[regular category|Regular categories]]; 3. [[algebraic theory|Algebraic theories]]; 4. [[monad|Monads]]; 5. [[accessible category|Accessible categories]]; 6. [[enriched category theory|Enriched category theory]]; 7. [[topological category|Topological categories]]; 8. [[fibered category|Fibred categories]]; Bibliography; Index.

Vol. 3: Categories of Sheaves

> Preface; Introduction; 1. [[locale|Locales]]; 2. [[sheaf|Sheaves]]; 3. [[Grothendieck toposes]]; 4. [[classifying topos|The classifying topos]]; 5. [[elementary topos|Elementary toposes]]; 6. [[internal logic|Internal logic of a topos]]; 7. [[the law of excluded middle|The law of excluded middle]]; 8. The axiom of infinity; 9. [[internal sheaf|Sheaves in a topos]]; Bibliography; Index.

category: reference


Handbook of Homotopy Theory

This page compiles material related to the book collection:

* [[Haynes Miller]] (ed.)

  _Handbook of Homotopy Theory_

  Chapman and Hall/CRC Press 2019

  982 pages

  [ISBN:9780815369707](https://www.routledge.com/Handbook-of-Homotopy-Theory/Miller/p/book/9780815369707)

  [doi:10.1201/9781351251624](https://doi.org/10.1201/9781351251624)

  ([table of contents](http://math.mit.edu/~hrm/papers/handbook-toc.pdf))

on [[homotopy theory]]/[[algebraic topology]], including [[higher algebra]] and [[higher category theory]].

## Terminology

The editor, [[Haynes Miller]], comments in the introduction on the choice of title:

> This volume may be regarded as a successor to the "[[Handbook of Algebraic Topology]]", edited by [[Ioan James]] and published a quarter of a century ago. In calling it the "Handbook of Homotopy Theory", I am recognizing that the discipline has expanded and deepened, and traditional questions of topology, as classically understood, are now only one of many distinct mathematical disciplines in which it has had a profound impact and which serve as sources of motivation for research directions within homotopy theory proper.

## Contributions

1. [[Gregory Arone]], [[Michael Ching]], _Goodwillie Calculus_, ([arXiv:1902.00803](https://arxiv.org/abs/1902.00803))

   \linebreak

   on [[Goodwillie calculus]]

   \linebreak

1. [[David Ayala]], [[John Francis]], _A factorization homology primer_, ([arXiv: 1903.10961](https://arxiv.org/abs/1903.10961))

   \linebreak

   on [[factorization homology]]

   \linebreak

1. [[Anthony Bahri]], [[Martin Bendersky]], and [[Fred Cohen]], _Polyhedral products and features of their homotopy theory_, ([arXiv:1903.04114](https://arxiv.org/abs/1903.04114))

   \linebreak

1. [[Paul Balmer]], _A guide to tensor-triangular classification_, ([arXiv:1912.08963](https://arxiv.org/abs/1912.08963))

   \linebreak

   on [[tensor triangulated categories]]

   \linebreak

1. [[Tobias Barthel]] and [[Agnès Beaudry]], _Chromatic structures in stable homotopy theory_, ([arXiv:1901.09004](https://arxiv.org/abs/1901.09004))

   \linebreak

   on [[chromatic homotopy theory]]

   \linebreak

1. [[Mark Behrens]], _Topological modular and automorphic forms_ ([arXiv:1901.07990](https://arxiv.org/abs/1901.07990))

   \linebreak

   on [[topological modular forms]]

   \linebreak

1. [[Julia Bergner]], _A survey of models for $(\infty,n)$-categories_ ([arXiv:1810.10052](https://arxiv.org/abs/1810.10052))

   \linebreak

   on [[(infinity,n)-categories]]

   \linebreak

1. [[Gunnar Carlsson]], _Persistent homology and applied homotopy theory_ ([arXiv:2004.00738](https://arxiv.org/abs/2004.00738))

   \linebreak

   on [[persistent homology]] in [[topological data analysis]]

   \linebreak

1. [[Natàlia Castellana]], _Algebraic models in the homotopy theory of classifying spaces_

   \linebreak

   on [[classifying spaces]]

   \linebreak

1. [[Ralph Cohen]], _Floer homotopy theory, revisited_ ([arXiv:1901.08694](https://arxiv.org/abs/1901.08694))

   \linebreak

1. {#Fresse18} [[Benoit Fresse]], _Little discs operads, graph complexes and Grothendieck--Teichmüller groups_, ([arXiv:1811.12536](https://arxiv.org/abs/1811.12536))

   \linebreak

   on [[little disc operads]], [[graph complexes]] and the [[Grothendieck-Teichmueller group]]

   \linebreak

1. [[Søren Galatius]], [[Oscar Randal-Williams]], _Moduli spaces of manifolds: a user's guide_, ([arXiv:1811.08151](https://arxiv.org/abs/1811.08151))

   \linebreak

   on [[cobordism categories]]

   \linebreak

1. [[David Gepner]], _An Introduction to Higher Categorical Algebra_, ([arXiv:1907.02904](https://arxiv.org/abs/1907.02904))

   \linebreak

   on [[higher algebra]]

   \linebreak

1. [[Moritz Groth]], _A short course on $\infty$-categories_, ([arXiv:1007.2925](https://arxiv.org/abs/1007.2925))

   \linebreak

   on [[(infinity,1)-categories]]

   \linebreak

1. [[Lars Hesselholt]], [[Thomas Nikolaus]], _Topological cyclic homology_, ([arXiv:1905.08984](https://arxiv.org/abs/1905.08984))

   \linebreak

   on [[topological cyclic homology]]

   \linebreak

1. [[Gijs Heuts]], _Lie algebra models for unstable homotopy theory_, ([arXiv:1907.13055](https://arxiv.org/abs/1907.13055))

   \linebreak

1. [[Michael Hill]], _Equivariant stable homotopy theory_

   \linebreak

   on [[equivariant stable homotopy theory]]

   \linebreak

1. [[Daniel Isaksen]], [[Paul Arne Østvær]], _Motivic stable homotopy groups_, ([arXiv:1811.05729](https://arxiv.org/abs/1811.05729))

   \linebreak

   on [[motivic homotopy theory]]

   \linebreak

1. [[Tyler Lawson]], _$E_n$-ring spectra and Dyer-Lashof operations_, ([arXiv:2002.03889](https://arxiv.org/abs/2002.03889), differently-formatted [author pdf](http://www-users.math.umn.edu/~tlawson/papers/dyerlashof.pdf))

   \linebreak

   on [[En-algebra|E_n]]-[[ring spectra]] and [[power operations]]

   \linebreak

1. [[Wolfgang Lueck]], _Assembly Maps_, ([arXiv:1805.00226](https://arxiv.org/abs/1805.00226))

   \linebreak

   on [[assembly maps]]

   \linebreak

1. [[Nathaniel Stapleton]], _Lubin-Tate theory, character theory, and power operations_, ([arXiv:1810.12339](https://arxiv.org/abs/1810.12339))

   \linebreak

   on [[Lubin-Tate theory]] and [[power operations]]

   \linebreak

1. [[Kirsten Wickelgren]], [[Ben Williams]], _Unstable Motivic Homotopy Theory_, ([arXiv: 1902.08857](https://arxiv.org/abs/1902.08857))

   \linebreak

   on [[motivic homotopy theory]]

   \linebreak

## Related entries

* [[Handbook of Algebraic Topology]]

category:reference


Handbook of K-theory

* [[Eric Friedlander]], [[Daniel Grayson]],

  _Handbook of K-theory_,

  Springer 2005, in 2 vols.

  [webpage](http://k-theory.org/handbook)

#Contents#
* table of contents
{:toc}

(...)

## Bott periodicity

On [[Bott periodicity]]:

* {#Karoubi05} [[Max Karoubi]], _Bott Periodicity in Topological, Algebraic and Hermitian K-Theory_,  In: [[Eric Friedlander]], [[Daniel Grayson]] (eds.) _[[Handbook of K-Theory]]_, Springer 2005 ([doi:10.1007/978-3-540-27855-9_4](https://doi.org/10.1007/978-3-540-27855-9_4))

category: reference

[[!redirects Handbook of K-Theory]]


Handbook of Logic in Computer Science

This page compiles pointers related to:

\linebreak

* [[Samson Abramsky]] et al. (eds.)

  \linebreak

  **Handbook of Logic in Computer Science**

  \linebreak

  Oxford University Press (1995)

  \linebreak

on [[formal logic]] in [[computer science]].

\linebreak

\tableofcontents

## Volume 1 -- Background: Mathematical Structures

* [[ISBN:9780198537359](https://global.oup.com/academic/product/handbook-of-logic-in-computer-science-9780198537359)]

(...)

## Volume 2 -- Background: Computational structures

* [[ISBN:9780198537618](https://global.oup.com/academic/product/handbook-of-logic-in-computer-science-9780198537618)]

(...)

## Volume 3 -- Semantic structures

* [[ISBN:9780198537625](https://global.oup.com/academic/product/handbook-of-logic-in-computer-science-9780198537625)]

  [TOC pdf](http://geca.area.ge.cnr.it/files/213070.pdf)

\linebreak

On [[domain theory]]:

* [[Samson Abramsky]], [[Achim Jung]]: *Domain Theory*  [[pdf](https://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf)]

\linebreak

On [[denotational semantics]]:

* [[Robert D. Tennet]]: *Denotational semantics*

\linebreak

* Eric Wagner: *Algebraic semantics*

\linebreak

On [[data type|data-]][[type theory]] in [[programming languages]]:

* {#Gunter} [[Carl A. Gunter]]: *The semantics of types in programming languages* [[pdf](http://seclab.illinois.edu/wp-content/uploads/2011/04/Gunter94.pdf)]

\linebreak

## Volume 4 -- Semantic Modelling

(...)

\linebreak

## Volume 5 -- Logic and Algebraic Methods

* [doi:10.1093/oso/9780198537816.001.0001](https://doi.org/10.1093/oso/9780198537816.001.0001)

On [[categorical semantics]] and the [[categorical model of dependent types]]:

* [[Andrew M. Pitts]], *Categorical logic*, in *[[Handbook of Logic in Computer Science]]* **5** Oxford Univ. Press (2000) 39-128 [[doi:10.1093/oso/9780198537816.003.0002](https://doi.org/10.1093/oso/9780198537816.003.0002), [pdf](https://www.cl.cam.ac.uk/~amp12/papers/catl/catl.pdf), [MR1859470](http://www.ams.org/mathscinet-getitem?mr=1859470)]

(...)

\linebreak

category: reference