connection on a smooth principal infinity-bundle

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

#Contents#
* table of contents
{:toc}

## Idea

In every [[cohesive (∞,1)-topos]] there is an intrinsic notion of ∞-Chern-Weil theory that gives rise to a notion of _connection_ on [[principal ∞-bundle]]s. We describe here details of the realization of this general abstract structure in the cohesive $(\infty,1)$-topos [[Smooth∞Grpd]] of [[smooth ∞-groupoid]]s.

For $G$ an [[∞-Lie group]], a _connection_ on a smooth $G$-[[principal ∞-bundle]] is a structure that supports the [[Chern-Weil homomorphism in Smooth∞Grpd]]: it interpolates between the [[nonabelian cohomology]] class $c \in H^1_{smooth}(X,G)$ of the bundle and the refinements to [[ordinary differential cohomology]] of its [[characteristic class]]es: the [[curvature characteristic form|curvature characteristic class]]es.

This generalizes the  notion of [[connection on a bundle]] and the ordinary [[Chern-Weil homomorphism]] in [[differential geometry]].

See the Motivation section at [[Chern-Weil theory in Smooth∞Grpd]] and the page [[∞-Chern-Weil theory introduction]] for more background.

## Definition

### For braided $\infty$-groups

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]] equippd with [[differential cohesion]] and let $\mathbb{G} \in Grp(\mathbf{H})$ be a [[braided ∞-group]]. Write

$$
  curv_{\mathbb{G}} = \theta_{\mathbf{B}\mathbb{G}}
  \;\colon\;
  \mathbf{B}\mathbb{G}
  \to
  \flat_{dR}\mathbf{B}^2 \mathbb{G}
$$

for the [[Maurer-Cartan form]] on the [[delooping]] [[∞-group]] $\mathbf{B}\mathbb{G} \in Grp(\mathbf{H})$.

Let

$$
  \Omega(-,\mathbb{G})
  \to
  \flat_{dR}\mathbf{B}^2 \mathbb{G}
$$

be the morphism out of a [[0-truncated]] object which is universal with the property that for $\Sigma \in \mathbf{H}$ any [[manifold]], the induced [[internal hom]] map

$$
  [\Sigma, \Omega(-,\mathbb{G})]
  \to
  [\Sigma, \flat_{dR}\mathbf{B}^2 \mathbb{G}]
$$

is a [[1-epimorphism]].

Then write $\mathbf{B}\mathbb{G}_{conn}$ for the [[(∞,1)-pullback]] in

$$
  \array{
    \mathbf{B}\mathbb{G}_{conn}
    &\to&
    \Omega(-,\mathbb{G})
    \\
    \downarrow
    &&
    \downarrow
    \\
    \mathbf{B}\mathbb{G}
    &\stackrel{curv_{\mathbb{G}}}{\to}&
    \flat_{dR}\mathbf{B}^2 \mathbb{G}
  }
  \,.
$$

We say that $\mathbf{B}\mathbb{G}_{conn}$ is the [[moduli ∞-stack]] of **$\mathbb{G}$-principal $\infty$-connections.

For instance for $\mathbb{G} = \mathbf{B}^{n-1}U(1)$ the [[circle n-group]] the moduli $n$-stack $\mathbf{B}^n U(1)_{conn}$ is presented by the [[Deligne complex]] for [[ordinary differential cohomology]] in degree $(n+1)$, hence is the moduli $n$-stack for [[circle n-bundles with connection]].

### For $\infty$-groups obtained by Lie integration
 {#ByLieIntegration}

We assume that the reader is familiar with the notation and constructions discussed at [[Smooth∞Grpd]]. The following definition may be understood as a direct generalization of the description of ordinary $G$-connections as cocycles in the stack $\mathbf{B}G_{conn}$ as discussed at [[connection on a bundle]], in view of the characterization of [Weil algebra in the smooth infinity-topos](Weil+algebra#CharacterizationInSmoothTopos)

[[!include Weil algebra abstractly -- table]]

We discuss now connections on those $G$-[[principal ∞-bundle]]s for which $G \in $ [[Smooth∞Grpd]] is an [[∞-Lie group|smooth ∞-group]] that arises from [[Lie integration]] of an [[L-∞ algebra]] $\mathfrak{g}$.

Let $\mathfrak{g} \in L_\infty \stackrel{CE}{\hookrightarrow} $ [[dgAlg]]${}^{op}$ be an [[L-∞ algebra]] over the [[real number]]s and of [[finite type]] with [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ and [[Weil algebra]] $W(\mathfrak{g})$.

For $X$ a [[smooth manifold]], write $\Omega^\bullet(X) \in dgAlg$ for the [[de Rham complex]] of smooth [[differential form]]s. For $k \in \mathbb{N}$ let $\Delta^k$ be the standard $k$-[[simplex]] regarded as a smooth [[manifold with corners]] in the standard way. Write $\Omega^\bullet_{si}(X \times \Delta^k)$ for the sub-[[dg-algebra]] of differential forms with sitting instants perpendicular to the boundary of the simplex, and $\Omega^\bullet_{si,vert}(X\times \Delta^k)$ for the further sub-dg-algebra of [[vertical differential form]]s with respect to the canonical projection $X \times \Delta^k \to X$.

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

A morphism

$$
  \Omega^\bullet(X)
  \leftarrow
  W(\mathfrak{g})
  :
  A
$$

in [[dgAlg]] we call an [[∞-Lie algebra valued differential forms|L-∞ algebra valued differential form]] with values in $\mathfrak{g}$, dually a morphism of [[∞-Lie algebroid]]s

$$
  A : T X \to inn(\mathfrak{g})
$$

from the [[tangent Lie algebroid]] to the [[Weil algebra|inner automorphism ∞-Lie algebra]].

Its [[curvature]] is the composite of morphisms of [[graded vector space]]s

$$
  \Omega^\bullet(X) \stackrel{A}{\leftarrow}
  W(\mathfrak{g})
  \stackrel{F_{(-)}}{\leftarrow}
  \mathfrak{g}^*[2]
  :
  F_{A}
$$

that injects the shifted generators into the [[Weil algebra]].

Precisely if the curvatures vanish does the morphism factor through the [[Chevalley-Eilenberg algebra]]

$$
  (F_A = 0)
  \;\;\Leftrightarrow
  \;\;
  \left(
  \array{
     && CE(\mathfrak{g})
     \\
     & {}^{\mathllap{\exists A_{flat}}}\swarrow & \uparrow
     \\
     \Omega^\bullet(X) &\stackrel{A}{\leftarrow}& W(\mathfrak{g})
  }
  \right)
$$

in which case we call $A$ **flat**.

The [[curvature characteristic form]]s of $A$ are the composite

$$
  \Omega^\bullet(X)
  \stackrel{A}{\leftarrow}
  W(\mathfrak{g})
  \stackrel{\langle F_{(-)} \rangle}{\leftarrow}
  inv(\mathfrak{g})
  :
  \langle F_A\rangle
  \,,
$$

where $inv(\mathfrak{g}) \to W(\mathfrak{g})$ is the inclusion of the [[invariant polynomial]]s.

=--

We define now [[simplicial presheaves]] over the [[site]] [[CartSp]]${}_{smooth} \hookrightarrow $ [[SmoothMfd]] of [[Cartesian space]]s and [[smooth function]]s between them.

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

Write $\exp(\mathfrak{g}) \in [CartSp_{smooth}^{op}, sSet]$ for the simplicial presheaf given by

$$
  \exp(\mathfrak{g}) :
  (U,[k])
  \mapsto
  \left\{
    \Omega^\bullet_{si,vert}(U \times\Delta^k)
     \stackrel{A_{vert}}{\leftarrow}
    CE(\mathfrak{g})
  \right\}
$$

(the untruncated [[Lie integration]] of $\mathfrak{g}$).

Write $\exp(\mathfrak{g})_{diff} \in [CartSp_{smooth}^{op}, sSet]$ for the simplicial presheaf given by

$$
  \exp(\mathfrak{g})_{diff} :
  (U,[k])
  \mapsto
  \left\{
     \array{
      \Omega^\bullet_{si,vert}(U \times\Delta^k)
       &\stackrel{A_{vert}}{\leftarrow}&
      CE(\mathfrak{g})
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet_{si}(U \times \Delta^k)
      &\stackrel{A}{\leftarrow}&
      W(\mathfrak{g})
    }
  \right\}
  \,.
$$

Write $\exp(\mathfrak{g})_{ChW} \in [CartSp_{smooth}^{op}, sSet]$ for the simplicial presheaf given by

$$
  \exp(\mathfrak{g})_{ChW}
  :
  (U,[k])
  \mapsto
  \left\{
     \array{
      \Omega^\bullet_{si,vert}(U \times\Delta^k)
       &\stackrel{A_{vert}}{\leftarrow}&
      CE(\mathfrak{g})
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet_{si}(U \times \Delta^k)
      &\stackrel{A}{\leftarrow}&
      W(\mathfrak{g})
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U)
      &\stackrel{\langle F_A\rangle}{\leftarrow}&
      inv(\mathfrak{g})
    }
  \right\}
  \,.
$$

Define the [[simplicial presheaf]] $\exp(\mathfrak{g})_{conn}$ by

$$
  \exp(\mathfrak{g})_{conn}(U)
  :
  [k]
  \mapsto
  \left\{
     \Omega^\bullet_{si}(U \times \Delta^k)
      \stackrel{A}{\leftarrow}
     W(\mathfrak{g})
     \;\;
     |
     \;\;
       \forall v \in \Gamma(T \Delta^k) : \iota_v F_A = 0
  \right\}
$$

=--

Here on the right we have in each case the [[set]]s of horizontal morphisms in [[dgAlg]] that make [[commuting diagram]]s in [[dgAlg]] as indicated, with the vertical morphisms being the canonical projections and inclusions. The functoriality in $f : K \to U$ and $\rho : [k] \to [l]$ is by the evident precomposition with the pullback of differential forms $\Omega^\bullet(U \times \Delta^k) \stackrel{(f,id)^*}{\to} \Omega^\bullet(K \times \Delta^k)$ and $\Omega^\bullet(U \times \Delta^l) \stackrel{(id,\rho)^*}{\leftarrow} \Omega^\bullet(U, \times \Delta^k)$.

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

There are canonical morphisms in $[CartSp_{smooth}^{op},sSet]$ between these objects

$$
  \array{
    \exp(\mathfrak{g})_{conn}
    &\hookrightarrow&
    \exp(\mathfrak{g})_{ChW}
    &\hookrightarrow&
    \exp(\mathfrak{g})_{diff}
    \\
    && && \downarrow
    \\
    && && \exp(\mathfrak{g})
  }
  \,,
$$

where the horizontal morphisms are [[monomorphism]]s of [[simplicial presheaves]] and the vertical morphism is over each $U \in CartSp$ an equivalence of [[Kan complexes]] (it is a weak equivalence between fibrant objects in the projective [[model structure on simplicial presheaves]]).

=--

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

The inclusion $\exp(\mathfrak{g})_{ChW} \hookrightarrow \exp(\mathfrak{g})_{dff}$ is clear. The weak equivalence $\exp(\mathfrak{g})_{diff} \to \exp(\mathfrak{g})$ is discussed at [[Smooth∞Grpd]] (but is also directly verified).

To see the inclusion $\exp(\mathfrak{g})_{conn} \hookrightarrow \exp(\mathfrak{g})_{ChW}$ we need to check that the horizonality condition $\iota_v F_A = 0$ on the [[curvature]] of a $\mathfrak{g}$-valued form $A$ for all [[vector field]]s $v$ tangent to the simplex implies that all the [[curvature characteristic form]]s $\langle F_A\rangle$ are _basic forms_ that "descend to $U$", hence that are in the image of the inclusion $\Omega^\bullet(U) \to \Omega^\bullet_{si}(U \times \Delta^k)$.

For this it is sufficient to show that for all $v \in \Gamma(T \Delta^k)$ we have

1. $\iota_v \langle F_A \rangle = 0$;

1. $\mathcal{L}_v \langle F_A \rangle = 0$

where in the second line we have the [[Lie derivative]] $\mathcal{L}_v$ along $v$.

The first condition is evidently satisfied if already $\iota_v F_A = 0$. The second condition follows with [[Cartan calculus]] and using that $d_{dR} \langle F_A\rangle = 0$ (which holds as a consequence of the definition of [[invariant polynomial]]):

$$
  \mathcal{L}_v \langle F_A \rangle =
  d \iota_v \langle F_A \rangle
  +
  \iota_v d \langle F_A \rangle
  = 0
  \,.
$$

=--

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

For a general [[L-∞ algebra]] $\mathfrak{g}$ the [[curvature]] forms $F_A$ themselves are not necessarily closed (rather they satisfy the [[Bianchi identity]]), hence requiring them to have no component along the simplex does not imply that they descend. This is different for abelian [[L-∞ algebra]]s: for them the curvature forms themselves are already closed, and hence are themselves already curvature characteristics that do descent.

=--

For $n \in \mathbb{N}$
let $\mathbf{cosk}_{n+1} : sSet \to sSet$ be the [[simplicial coskeleton]] functor. Its prolongation to simplicial presheaves we denote here $\tau_n$ and write

$$
 \tau_n \exp(\mathfrak{g}) \in [CartSp_{smooth}^{op}, sSet]
$$

etc. This is the [[delooping]]

$$
  \tau_n \exp(\mathfrak{g}) = \mathbf{B}G
$$

of the universal [[Lie integration]] of $\mathfrak{g}$ to an [[∞-Lie group|smooth n-group]] $G$.

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

For any $X \in [CartSp_{smooth}^{op}, sSet]$ and $\hat X \to X$ any cofibrant [[resolution]] in the local projective [[model structure on simplicial presheaves]] (see [[Smooth∞Grpd]] for details), we say that the [[sSet]]-[[hom object]]

* $[CartSp_{smooth}^{op}, sSet](\hat X, \tau_n \exp(\mathfrak{g}))$
  is the [[∞-groupoid]] of smooth $G$-[[principal ∞-bundle]]s on $X$;

* $[CartSp_{smooth}^{op}, sSet](\hat X, \tau_n \exp(\mathfrak{g})_{diff})$
  is the [[∞-groupoid]] of smooth $G$-[[principal ∞-bundle]]s on $X$ equipped with **pseudo $\infty$-connection**;

* $[CartSp_{smooth}^{op}, sSet](\hat X, \tau_n \exp(\mathfrak{g})_{conn})$
  is the [[∞-groupoid]] of smooth $G$-[[principal ∞-bundle]]s on $X$ equipped with **$\infty$-connection**.

=--

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

In view of this definition we may read the [above](#SequenceOfInclusionsOfCoefficients) sequence of morpisms of coefficient objects as follows:

$$
  \array{
    \exp(\mathfrak{g})_{conn} &&& genuine\;connections
    \\
    \downarrow
    \\
    \exp(\mathfrak{g})_{ChW} &&& pseudo-connection\;with\;global\;curvature\;characteristics
    \\
    \downarrow
    \\
    \exp(\mathfrak{g})_{diff} &&& pseudo-connections
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \exp(\mathfrak{g}) &&& bare bundles
  }
  \,,
$$

As we shall see in more detail below, the components of an $\infty$-connection in terms of the above diagrams we may think of as follows:

$$
    \array{
      \Omega^\bullet(U \times \Delta^k)_{vert}
      &\stackrel{A_{vert}}{\leftarrow}&
      CE(\mathfrak{g})
      &&&
      gauge\;transformation
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U \times \Delta^k)
      &\stackrel{A}{\leftarrow}&
      W(\mathfrak{g})
      &&&
      \mathfrak{g}-valued\;form
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U)
      &\stackrel{\langle F_A\rangle}{\leftarrow}&
      inv(\mathfrak{g})
      &&&
      curvature\;characteristic\;forms
    }
$$

=--

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

In full [[Chern-Weil theory in Smooth∞Grpd]] the fundamental object of interest is really $\exp(\mathfrak{g})_{diff}$ -- the object of [[pseudo-connection]]s, which serves as the correspondence object for an [[∞-anafunctor]] out of $\exp(\mathfrak{g})$ that presents the differential characteristic classes on $\exp(\mathfrak{g})$. From an abstract point of view the other objects only serve the purpose of picking particularly nice representatives.

This distinction is important: over objects $X \in $ [[Smooth∞Grpd]] that are not [[smooth manifold]]s but for instance [[orbifold]]s, the genuine $\mathfrak{g}$-connections for general higher $\mathfrak{g}$ do _not_ exhaust all nonabelian differential cocycles. This just means that not every differential class in this case does have a nice representative.

=--

## Examples

### 1-Morphisms: integration of infinitesimal gauge transformations {#InfGaugeTrafo}

The 1-[[morphism]]s in $\exp(\mathfrak{g})_{conn}(U)$ may be thought of as [[gauge transformation]]s between $\mathfrak{g}$-valued forms. We unwind what these look like concretely.

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

Given a 1-morphism in $\exp(\mathfrak{g})(X)$, represented by $\mathfrak{g}$-valued forms

$$
  \Omega^\bullet(U \times \Delta^1)
  \leftarrow
  W(\mathfrak{g})
  :
  A
$$

consider the unique decomposition

$$
  A = A_U + ( A_{vert} := \lambda \wedge d s)  \; \;
  \,,
$$

with $A_U$ the horizonal differential form component and $s : \Delta^1 = [0,1] \to \mathbb{R}$ the canonical [[coordinate]].

We call $\lambda$ the **gauge parameter** . This is a function on $\Delta^1$ with values in 0-forms on $U$ for $\mathfrak{g}$ an ordinary [[Lie algebra]], plus 1-forms on $U$ for $\mathfrak{g}$ a [[Lie 2-algebra]], plus 2-forms for a Lie 3-algebra, and so forth.

=--

We describe now how this enccodes a gauge transformation

$$
  \lambda : A_0(s=0) \stackrel{}{\to} A_U(s = 1)
  \,.
$$

+-- {: .un_lemma}
###### Observation

We have

$$
  \frac{d}{d s} A_U
  =
  (d_U \lambda + [\lambda \wedge A] + [\lambda \wedge A \wedge A] + \cdots)
  +
  \iota_{\partial_s} F_A
  \,,
$$

where the sum is over all higher brackets of the [[L-∞ algebra]] $\mathfrak{g}$.

=--

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

This is the result of applying the contraction $\iota_{\partial s}$ to the defining equation for the [[curvature]] $F_A$ of $A$ using the nature of the [[Weil algebra]]:

$$
  F_A = d_{dR} A + [A \wedge A] + [A \wedge A \wedge A] + \cdots
$$

and inserting the above decomposition for $A$.

=--

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

Define the **[[covariant derivative]] of the gauge parameter** to be

$$
  \nabla_A \lambda := d \lambda + [A \wedge \lambda] + [A \wedge A \wedge \lambda] + \cdots
  \,.
$$

=--

In this notation we have

* the general identity

  \[
    \frac{d}{d s} A_U = \nabla \lambda + (F_A)_s
    \label{ShiftedGaugeTrafo}
  \]

* the **horizontality** or **[[rheonomy]]** constraint or **[[Ehresmann connection|second Ehresmann condition]]** $\iota_{\partial_s} F_A = 0$, the [[differential equation]]

  \[
    \frac{d}{d s} A_U = \nabla \lambda
    \label{GaugeTrafo}
    \,.
  \]

This is known as the equation for **infinitesimal [[gauge transformation]]s** of an $L_\infty$-algebra valued form.

+-- {: .un_lemma}
###### Observation

By [[Lie integration]] we have that $A_{vert}$ -- and hence $\lambda$ -- defines an element $\exp(\lambda)$ in the [[∞-Lie group]] that integrates $\mathfrak{g}$.

The unique solution $A_U(s = 1)$ of the above [[differential equation]] at $s = 1$ for the initial values $A_U(s = 0)$ we may think of as the result of acting on $A_U(0)$ with the gauge transformation $\exp(\lambda)$.

=--

### Ordinary connections on principal 1-bundles

+-- {: .un_prop}
###### Proposition
**(connections on ordinary bundles)**

For $\mathfrak{g}$ an ordinary [[Lie algebra]] with simply connected [[Lie group]] $G$ and for $\mathbf{B}G_{conn}$ the [[groupoid of Lie algebra-valued forms]] we have an equivalence

$$
  \tau_1 \exp(\mathfrak{g})_{conn} \simeq \mathbf{B}G_{conn}
$$

betweenn the 1-truncated coefficient object for $\mathfrak{g}$-valued $\infty$-connections and the coefficient objects for ordinary [[connections on a bundle]] (see there).

=--

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

Notice that the sheaves of [[object]]s on both sides are manifestly isomorphic, both are the sheaf of $\Omega^1(-,\mathfrak{g})$.

On [[morphism]]s, we have by the [above](#InfinitesimalGaugeTransformations) for a form $\Omega^\bullet(U \times \Delta^1) \leftarrow W(\mathfrak{g}) : A$ decomposed into a horizontal and a verical pice as $A = A_U + \lambda \wedge d t$ that the condition $\iota_{\partial_t} F_A = 0$ is equivalent to the [[differential equation]]

$$
  \frac{\partial}{\partial s} A
  =
  d_U \lambda + [\lambda, A]
  \,.
$$

For any initial value $A(0)$ this has the unique solution

$$
  \begin{aligned}
     A(t) & = g(t)^{-1} (A + d_{U}) g(t)
      \\
     & = Ad_{g(t)}(A) + g(t)^* \theta
  \end{aligned}
$$

(with $\theta$ the [[Maurer-Cartan form]] on $G$), where $g \in C^\infty([0,1], G)$ is the [[parallel transport]] of $\lambda$:

$$
  \begin{aligned}
    &
    \frac{\partial}{\partial s}
    \left(
       g_(t)^{-1} (A + d_{U}) g(t)
     \right)
     \\
     & =
     g(t)^{-1} (A + d_{U}) \lambda g(t)
     -
     g(t)^{-1} \lambda (A + d_{U}) g(t)
   \end{aligned}
$$

(where for ease of notaton we write actions as if $G$ were a [[matrix Lie group]]).

This implies that the endpoints of the path of $\mathfrak{g}$-valued 1-forms are related by the usual cocycle condition in $\mathbf{B}G_{conn}$

$$
  A(1) = g(1)^{-1} (A + d_U) g(1)
  \,.
$$

In the same fashion one sees that given 2-cell in $\exp(\mathfrak{g})(U)$ and any 1-form on $U$ at one vertex, there is a unique lift to a 2-cell in $\exp(\mathfrak{g})_{conn}$, obtained by parallel transporting the form around. The claim then follows from the previous statement of [[Lie integration]] that $\tau_1 \exp(\mathfrak{g}) = \mathbf{B}G$.

=--

### Further examples

* For $\mathfrak{g}$ [[Lie 2-algebra]], a $\mathfrak{g}$-valued differential form in the sense described here is precisely an [[Lie 2-algebra valued form]].

* For $n \in \mathbb{N}$, a $b^{n-1}\mathbb{R}$-valued differential form is the same as an ordinary differential $n$-form.

* What is called an  "extended soft group manifold" in the literature on the [[D'Auria-Fre formulation of supergravity]] is precisely a collection of $\infty$-Lie algebroid valued forms with values in a super $\infty$-Lie algebra such as the
[[supergravity Lie 3-algebra]]/[[supergravity Lie 6-algebra]] (for 11-dimensional [[supergravity]]). The way [[curvature]] and [[Bianchi identity]] are read off from "extded soft group manifolds" in this literature is -- apart from this difference in terminology -- precisely what is described above.

## Related concepts

* [[connection on a bundle]]

* [[connection on a 2-bundle]] / [[connection on a gerbe]] / [[connection on a bundle gerbe]]

* [[connection on a 3-bundle]] / [[connection on a bundle 2-gerbe]]

* **connection on an ∞-bundle**

  * [[flat ∞-connection]]

  * [[adjusted Weil algebra]]

* [[parallel transport]], [[higher parallel transport]]

* [[holonomy]]

[[!include higher Atiyah groupoid - table]]

## References

The local differential form data of $\infty$-connections was introduced in:

* [[Urs Schreiber]], *[Obstructions to $n$-Bundle Lifts Part II](https://golem.ph.utexas.edu/category/2007/10/obstructions_to_nbundle_lifts.html)* (Oct 2007) [diagram: [[Schreiber-PrincipalInfinityConnections-2007.pdf:file]]]

* {#SSSI} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:L-infinity algebra connections|L-∞ algebra connections and applications to String- and Chern-Simons n-transport]]_, in: *Quantum Field Theory*, Birkhäuser (2009) 303-424 ([arXiv:0801.3480](http://arxiv.org/abs/0801.3480), [doi:10.1007/978-3-7643-8736-5_17](https://doi.org/10.1007/978-3-7643-8736-5_17))

* {#SSSIII} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Twisted Differential String and Fivebrane Structures]]_, Communications in Mathematical Physics, October 2012, Volume 315, Issue 1, pp 169-213 ([arXiv:0910.4001](http://arxiv.org/abs/0910.4001), [doi:10.1007/s00220-012-1510-3](https://link.springer.com/article/10.1007/s00220-012-1510-3))

The global description was then introduced in

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech cocycles for differential characteristic classes]] -- An $\infty$-Lie theoretic construction_, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 ([arXiv:1011.4735](http://arxiv.org/abs/1011.4735), [euclid:atmp/1358950853](http://projecteuclid.org/euclid.atmp/1358950853))

A more comprehensive account is in sections 3.9.6, 3.9.7 of

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

For further developments see the references at *[[adjusted Weil algebra]]*.

[[!redirects ∞-connection on a principal ∞-bundle]]
[[!redirects ∞-connection on a principal ∞-bundle]]

[[!redirects connection on a principal ∞-bundle]]

[[!redirects connection on an ∞-bundle]]
[[!redirects connections on an ∞-bundle]]

[[!redirects connection on an infinity-bundle]]

[[!redirects connections on ∞-bundles]]
[[!redirects connections on infinity-bundles]]

[[!redirects connection on a smooth principal ∞-bundle]]
[[!redirects connections on smooth principal ∞-bundles]]
[[!redirects connection on a principal infinity-bundle]]

[[!redirects connections on smooth principal ∞-bundles]]

[[!redirects ∞-connection]]
[[!redirects ∞-connections]]

[[!redirects principal ∞-connection]]
[[!redirects principal ∞-connections]]

[[!redirects principal infinity-connection]]
[[!redirects principal ∞-connections]]
[[!redirects principal infinity-connections]]

[[!redirects connections on principal ∞-bundles]]

[[!redirects connection on principal L-infinity bundle]]


connection on a vector bundle

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

#Contents#
* table of contents
{:toc}

## Idea

One way to define a _connection_ on a [[vector bundle]] $E \to X$ is as a [[principal connection]] of the $GL(n)$-[[principal bundle]] that the vector bundle is an [[associated bundle]] to. Another is as a [[covariant derivative]] on the space of [[sections]] of the vector bundle.

## Examples

* [[flat vector bundle]]

* [[Bott connection]]

* [[Hermitean Yang-Mills connection]]

(...)

## References

An exposition with an eye towards applications in [[gauge theory]] is in section 16.3 of

* [[Theodore Frankel]], _[[The Geometry of Physics - An Introduction]]_

[[!redirects connections on a vector bundle]]


Connections, Curvature, and Cohomology

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[[!include infinity-Chern-Weil theory - contents]]
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#### Differential cohomology
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=--
#### $\infty$-Lie theory
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[[!include infinity-Lie theory - contents]]
=--
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This entry is about the book

* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]]:

  \linebreak

  **Connections, Curvature, and Cohomology**

  \linebreak

  Academic Press (1973)

  * Volume 1:

    _De Rham Cohomology of Manifolds and Vector Bundles_

    [ISBN:978-0-12-302701-6](https://www.elsevier.com/books/connections-curvature-and-cohomology-v1/greub/978-0-12-302701-6)

  * Volume 2:

    _Lie groups, principal bundles and characteristic classes_

    ISBN:9780123027023

  * Volume 3:

    _Lie Algebras, Algebraic Connections and Characteristic Classes_

    ISBN:9780123027016

on [[Chern-Weil theory]]: [[principal bundle]]s with [[connection on a bundle|connections]] and their [[characteristic class]]es.

Related books are

* [[Theodore Frankel]], _[[The Geometry of Physics - An Introduction]]_

#Contents#
* table of contents
{:toc}

## Volume I

### 0 Algebraic and analytic preliminaries

* [[linear algebra]]

* [[homological algebra]]

* [[analysis]], [[topology]]

### 1 Basic concepts

* [[topological manifold]]

* [[smooth manifold]]

* [[fiber bundle]]

### II Vector bundles

* [[vector bundle]]

* [[section]]

### III Tangent bundle and differential forms

* [[tangent bundle]]

* [[vector field]]

* [[differential form]]

* [[orientation]]

### IV Calculus of differential forms

* [[Cartan calculus]]

* [[integration]]

* [[Stokes theorem]]

### V De Rham cohomology

* [[de Rham cohomology]]

* [[Poincare duality]]

* [[Künneth theorem]]

* [[de Rham theorem]]

### VI Mapping degree

### VII Integration over the fiber

* [[fiber integration]]

### VIII Cohomology of sphere bundles

* [[Euler class]]

### IX Cohomology of vector bundles

* [[Thom isomorphism]]

### X The Lefschetz class of a manifold

* [[Lefschetz isomorphism]]

### Appendix A The exponential map

## Volume II

### 0 Algebraic and analytic preliminaries

### I Lie groups

* [[Lie group]]

### II Subgroups and homogeneous spaces

* [[homogeneous space]]

### III Transformation groups

### IV Invariant cohomology

* [[equivariant cohomology]]

### V Bundles with structrue group

* [[principal bundle]]

### VI Principal connections and the Weil homomorphism

* [[connection on a bundle]]

* [[Chern-Weil homomorphism]]

### VII Linear connections

### VIII Characteristic homomorphism for $\Sigma$-bundles

### IX Pontrjagin, Pfaffian, Chern classes

* [[Pontryagin class]]

* [[Pfaffian]]

* [[Chern class]]

### X The Gauss-Bonnet-Chern theorem

* [[Gauss-Bonnet theorem]]

### Appendix A Characteristic coefficients and the Pfaffian

## Volume III

### 0 Algebraic preliminaries

### I Spectral sequences

* [[fiber sequence]]

* [[spectral sequence]]

### II Koszul complexes of $P$-spaces and $P$-algebras

### III Koszul complexes of $P$-differential algebras

### IV Lie algebras and differential spaces

* [[Lie algebra]]

### V Cohomology of Lie algebras and Lie groups

* [[Lie algebra cohomology]]

* [[Lie group cohomology]]

### VI The Weil algebra

* [[Weil algebra]]

### VII Operation of a Lie algebra in a graded differential algebra

### VIII Algebraic connections and principal bundles

### IX Cohomology of operations and principal bundles

### X Subalgebras

### XI Homogeneous spaces

### XII Operation of a Lie algebra on a pair

### Appendix A Characteristic coefficients and the Pfaffian

category: reference


connective

As a noun:

* [[logical connective]].

As an adjective:

* [[connective object]]

  * [[connective chain complex]]

  * [[connective spectrum]]

  * [[connective spectrum object]]

See also:

* [[connectology]]

category: disambiguation


connective chain complex

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# Contents
* table of contents
{: toc}

## Definition

An unbounded [[chain complex]] $X_\bullet \in Ch_\bullet$

$$
  \cdots \to X_2 \stackrel{\partial}{\to} X_1 \stackrel{\partial}{\to} X_0 \stackrel{\partial}{\to} X_{-1} \stackrel{\partial}{\to} X_{-2} \stackrel{\partial}{\to} \cdots
$$

is called _connective_ if it has no nontrivial [[homology groups]] in negative degree. Often one means more strictly that it is connective if it is concentrated in non-negative degree, hence if $X_{-n} \simeq 0$ for all $n \geq 1$, hence if it is of the form

$$
  \cdots \to X_2 \stackrel{\partial}{\to} X_1 \stackrel{\partial}{\to} X_0 \stackrel{\partial}{\to} 0 \stackrel{\partial}{\to} 0 \stackrel{\partial}{\to} \cdots
  \,.
$$

Such connective chain complexes are the [[connective objects]] in the [[(infinity,1)-category of chain complexes|$\infty$-category of chain complexes]]. Rearding under the [[stable Dold-Kan correspondence]] as $H R$-[[module spectra]] they are the *[[connective spectra]]*.

Hence a connective chain complex is in particular a [[bounded chain complex]], bounded from below.

## Examples

* The [[singular chain complex]] of a [[topological space]] is connective. This is directly related to the fact that the [[homotopy groups]] of topological spaces are in non-negative degree, as opposed to their [[stabilization]] by [[spectra]], which may have homotopy groups also in negative degrees.

## Properties

### Relation to simplicial abelian groups

The [[Dold-Kan correspondence]] asserts that connective chain complexes of [[abelian groups]] are equivalent to [[abelian group|abelian]] [[simplicial groups]].

## Related concepts

* [[connective object]], [[connective spectrum]]

* [[category of chain complexes]]

  * [[chain complex]]

  * [[chain map]], [[quasi-isomorphism]]

  * [[chain homotopy]]

  * [[model structure on chain complexes]]

  * [[elliptic chain complex]]

* [[cochain complex]]

* [[bounded chain complex]]

* [[filtered chain complex]]

* [[perfect chain complex]]

[[!redirects connective chain complex]]
[[!redirects connective chain complexes]]


connective cover

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#Contents#
* table of contents
{:toc}

## Idea

### General
 {#General}

Generally, given a [[stable (infinity,1)-category|stable $\infty$-category]] $\mathcal{A}$ equipped with a [[t-structure]] $\mathcal{A}_{\geq 0}, \mathcal{A}_{\leq 0} \hookrightarrow \mathcal{A}$, the [[coreflective subcategory|coreflection]] $\mathcal{A} \to \mathcal{A}_{\geq 0}$ onto the [[connective objects]] is called the *connective cover*-construction.

### For chain complexes
 {#ForChainComplexes}

Given an unbounded [[chain complex]] $V_\bullet \,\in\, Ch_\bullet(\mathcal{A})$ in some [[abelian category]] $\mathcal{A}$,

$$
  V_\bullet
  \;=\;
  \big(
    \cdots
      \to
    V_2
      \overset{\partial_2}{\to}
    V_1
      \overset{\partial_1}{\to}
    V_0
      \overset{\partial_0}{\to}
    V_{-1}
      \overset{\partial_{-1}}{\to}
    V_{-2}
      \to
    \cdots
  \big)
$$

its connective cover is obtained by

* retaining its entries in positive degrees,

* replacing its entries in negative degree by [[zero objects]],

* replacing its entry in degree 0 by the [[kernel]] of the [[differential]] in this degree:

$$
  cn_{\geq 0}
  V_\bullet
  \;=\;
  \big(
    \cdots
      \to
    V_2
      \overset{\partial_2}{\to}
    V_1
      \overset{\partial_1}{\to}
    \ker(\partial_0)
      \to
    0
      \to
    0
      \to
    \cdots
  \big)
  \,.
$$

(e.g. [Lurie, Rem. 1.2.3.4](#Lurie))

### For plain spectra
 {#ForPlainSpectra}

[[connective spectrum|Connective spectra]] form a [[coreflective sub-(∞,1)-category]] of the [[(∞,1)-category of spectra]] (as part of the canonical [[t-structure]] in spectra, see [there](t-structure#CanonicalTStructureOnSpectra)).

The [[right adjoint|right]] [[adjoint (∞,1)-functor]] from spectra to connective spectra $E \mapsto E \langle 0 \rangle$ is called the _connective cover_ construction. This comes with the [[coreflective subcategory|coreflection]] morphism of spectra

\[
  \label{CoreflectionOfSpectrumFromItsConnectiveCover}
  E\langle 0 \rangle
  \overset{
    \epsilon_E
  }{\longrightarrow}
  E
\]

which induces an [[isomorphism]] on [[homotopy groups of a spectrum|homotopy groups of spectra]] in non-[[negative number|negative]] degrees:

$$
  \pi_{\bullet \geq 0}\big( E\langle 0\rangle\big)
  \underoverset
  {\simeq}
  {
    \pi_{\bullet \geq 0}
    (\epsilon_E)
  }{\longrightarrow}
  \pi_{\bullet \geq 0}(E)
  \,.
$$

### For ring spectra
 {#ForRingSpectra}

The connective cover  [[(∞,1)-functor|functor]] extends from plain spectra to [[E-∞ ring spectra]] ([May 77, Prop. VII 4.3](#May77), [Lurie, Prop. 7.1.3.13](#Lurie)), such that the coreflection $E \langle0\rangle \longrightarrow E$ (eq:CoreflectionOfSpectrumFromItsConnectiveCover) is a [[homomorphism]] of [[E-∞ rings]].

Besides a canonically inherited ring structure, the connective cover may sometimes carry further ring structures, but in many examples of interest it is unique ([Baker-Richter 05](#BakerRichter05)).

## Examples

* [[ku]] is the connective cover of [[KU]]

## Related concepts

* [[n-connective object]]

* [[n-connected object of an (infinity,1)-topos]]

## References

For plain spectra:

* {#Schwede12} [[Stefan Schwede]], chapter II, section 8, and chapter III, section 7 of _[[Symmetric spectra]]_ (2012)

For [[ring spectra]]:

* {#May77} [[Peter May]], Prop. VII 4.3 in: _$E_\infty$-Ring spaces and $E_\infty$ ring spectra_, Lecture Notes in Mathematics 577, Springer 1977 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf), [cds:1690879](https://cds.cern.ch/record/1690879))

* {#BakerRichter05} [[Andrew Baker]], [[Birgit Richter]], _Uniqueness of $E_\infty$-structures for connective covers_,  Proc. Amer. Math. Soc. 136 (2008), 707-714  ([arXiv:math/0506422](http://arxiv.org/abs/math/0506422), [doi:10.1090/S0002-9939-07-08984-8](https://doi.org/10.1090/S0002-9939-07-08984-8))

* {#Lurie} [[Jacob Lurie]], 7.1.3.11-13 in: _[[Higher Algebra]]_

[[!redirects connective covers]]


connective spectrum

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#Contents#
* table of contents
{:toc}

## Idea

### General

A **connective spectrum** is a [[connective object]] in the [[stable (infinity,1)-category of spectra|stable $\infty$-category of spectra]],  hence a [[spectrum]] $S$ whose [[homotopy group of a spectrum|homotopy groups]] in all [[negative number|negative]] degrees are [[trivial groups|trivial]]: $\pi_{\bullet \lt 0}(S) \,=\, 0$.

These are equivalently:

* [[infinite loop spaces]];

* [[group object in an (infinity,1)-category|grouplike]]$\;$ [[symmetric monoidal (∞,1)-category|symmetric monoidal]] [[∞-groupoids]];

* [[abelian ∞-groups]].

Connective spectra form a [[sub-(∞,1)-category]] of [[spectra]]

$$
  Top \stackrel{\supset}{\leftarrow} ConnectSp(Top) \hookrightarrow Sp(Top)
  \,.
$$

There are objects in [[Spectra]], though, that do not come from "naively" [[delooping]] a [[topological space]] infinitely many times.  These are the **non-connective spectra**.  For general spectra there is a notion of [[homotopy groups]] of negative degree. The connective ones are precisely those for which all negative degree [[homotopy groups]] vanish.

### Strict

There is a subclass of connective spectra that are equivalent to possibly-more-familiar objects, namely nonnegatively graded [[chain complexes]], via the [[Dold-Kan correspondence]]. This identifies [[∞-groupoids]] that are not only connective spectra but even have a _strict_ symmetric monoidal [[group]] structure with non-negatively graded [[chain complex]]es of abelian groups.

$$
 \array{
  Ch_+ &\stackrel{Dold-Kan \; nerve}{\to}&
  ConnectSp(\infty Grp) \subset \infty Grpd
  \\
  (\cdots A_2 \stackrel{\partial}{\to} A_1 \stackrel{\partial}{\to} A_0 \to 0 \to 0 \to \cdots)
  &\stackrel{}{\mapsto}&
  N(A_\bullet)
 }
$$

The [[homology]] groups of the chain complex correspond precisely to the [[homotopy groups]] of the corresponding [[topological space]] or [[∞-groupoid]].

The free stabilization of the [[(∞,1)-category]] of non-negatively graded chain complexes is simpy the [[stable (∞,1)-category]] of arbitrary chain complexes. There is a [[stable Dold-Kan correspondence]] that identifies these with special objects in $Sp(Top)$.

$$
 \array{
  Ch &\stackrel{Dold-Kan \; nerve}{\to}&
   Sp(\infty Grp)
  \\
  (\cdots A_2 \stackrel{\partial}{\to} A_1 \stackrel{\partial}{\to} A_0 \stackrel{\partial}{\to} A_{-1}
  \stackrel{\partial}{\to} A_{-2} \to \cdots)
  &\stackrel{}{\mapsto}&
  N(A_\bullet)
 }
$$

So it is the homologically nontrivial parts of the chain complexes in negative degree that corresponds to the non-connectiveness of a spectrum.

## Properties

### Inclusion into all spectra
 {#InclusionIntoAllSpectra}

The inclusion

$$
  Spectra_{\geq 0} \hookrightarrow Spectra
$$

of the [[full sub-(∞,1)-category]] of connective spectra into the [[(∞,1)-category of spectra]] preserves small [[(∞,1)-colimits]]. Moreover, $Spectra_{\geq 0}$ is generated under small [[(∞,1)-colimits]] by the [[sphere spectrum]].

These statements prolong to [[sheaves of spectra]].

e.g. ([[Spectral Schemes|Lurie, "Spectral Schemes", example 1.23]])

### Connective cover
 {#ConnectiveCover}

By the [above](#InclusionIntoAllSpectra), connective spectra form a [[coreflective sub-(∞,1)-category]] of the [[(∞,1)-category of spectra]]. The [[right adjoint|right]] [[adjoint (∞,1)-functor]] $\tau_{\geq 0}$ from spectra to connective spectra is called the _[[connective cover]]_ construction, part of the canonical [[t-structure]] on spectra (see [there](t-structure#CanonicalTStructureOnSpectra)).

## Related concepts

* [[connective object]], [[connective cover]]

* [[t-structure]]

* [[model structure for connective spectra]]

## Literature

* [[Jacob Lurie]], pp. 150 in: *[[Higher Algebra]]*

[[!include k-monoidal table]]

[[!redirects connective spectra]]


connectology

#Contents#
* table of contents
{:toc}

## Idea

A **connectology** on a [[set]] $X$ is a [[structure]] which abstracts the information about which [[subsets]] of $X$ are [[connected]].  Every [[topological space]] and every [[graph]] has an underlying connectology, but not every connectology is of these forms.

## Definition

A **connectology** on a set $X$ consists of a set of subsets of $X$, called *connected* sets, such that the following axioms hold.

1. If $C\subseteq \mathcal{P}(X)$ is a family of connected sets such that $\bigcap C \neq \emptyset$, then $\bigcup C$ is connected.

2. Every [[singleton]] $\{x\}$ is connected.

3. If $A$ and $B$ are nonempty, and $A$, $B$, and $A\cup B$ are connected, then there exists an $x\in A\cup B$ such that $A\cup\{x\}$ and $B\cup \{x\}$ are connected.

4. If $A,B,\{C_i\}_{i\in I}$ are disjoint connected sets such that $A\cup B\cup \bigcup_i C_i$ is connected, then there is a partition $I=J+K$ such that $A\cup \bigcup_{j\in J} C_j$ and $B\cup \bigcup_{k\in K} C_k$ are connected.

A set equipped with a connectology is sometimes called a **connective space**, although this may be confusing due to other meanings of the word [[connective]].

## Related concepts

* [[diffeology]]

## References

* Joseph Muscat and David Buhagiar, *Connective Spaces*, [PDF](http://www.math.shimane-u.ac.jp/memoir/39/D.Buhagiar.pdf).

[[!redirects connectologies]]
[[!redirects connective space]]
[[!redirects connective spaces]]


Conner-Floyd Chern class

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#Contents#
* table of contents
{:toc}

## Idea

The concept of [[Chern classes]] of [[complex vector bundles]], as [[universal characteristic classes]] in [[ordinary cohomology]], generalizes to any [[complex oriented cohomology|complex oriented]] [[generalized cohomology theory]]: the _Conner-Floyd Chern classes_ ([Conner-Floyd 66, Section 7](#ConnerFloyd66), [Adams 74, Section I.4](#Adams74), review in [Kochman 96, Section 4.3](#Kochman96), [Lurie 10, Lectures 4 and 5](#Lurie10)):

For $E$ a [[generalized cohomology theory]], the analog of the [[first Chern class]] in $E$-cohomology is what appears in the very definition of [[complex oriented cohomology]]. The higher generalized Chern classes are induced from this by the [[splitting principle]]. See also at _[complex oriented cohomology -- the cohomology ring of BU(n)](complex+oriented+cohomology+theory#TheCohomologyRingOfBUn)_.

The Conner-Floyd Chern classes in top degree serve as the generalized [[Thom classes]] that make every [[complex vector bundle]] have [[orientation in generalized cohomology]] with respect to any [[complex oriented cohomology theory]] ([Lurie 10, lecture 5, prop. 6](#Lurie10)).

## Definition
 {#Definition}

The definition of the $E$-Chern classes according to [Conner-Floyd 66, Theorem 7.6](#ConnerFloyd66) proceeds as follows.

Let $n \in \mathbb{N}$. Let $c_1^E$ in

$$
  \array{
    \mathbb{C}P^1
    &
      \overset
        { \Sigma^2 (1^E) }
        {\longrightarrow}
    &
    E_2
    \\
    \big\downarrow & \nearrow_{ \mathrlap{ c_1^E } }
    \\
    \mathbb{C}P^\infty
  }
$$

be a [[complex oriented cohomology|complex orientation]] in $E$-cohomology.

For $\mathcal{V} \longrightarrow X$ a [[complex vector bundle]] of [[rank of a vector bundle|rank]] $n + 1$ consider the diagram

\[
  \label{ProjectiveBundleInConstructionOfConnerFloydClasses}
  \array{
    &&
    \mathcal{V}'
      \oplus
    \mathcal{L}_{{}_{P(\mathcal{V})}}
    &\longrightarrow&
    \mathcal{V}
    \\
    &&
    \big\downarrow
    &
      {}^{_{(pb)}}
    &
    \big\downarrow
    \\
    \mathbb{C}P^n
    &
      \underset{
        fib_x
      }{
        \longrightarrow
      }
    &
    P(\mathcal{V})
    &
      \underset
        {\pi}
        {\longrightarrow}
    &
    X
  }
\]

where

$$
  P(\mathcal{V})
  \;\coloneqq\;
  \big(
    \mathcal{V} \setminus \{ X \times \{0\} \}
  \big) / \mathbb{C}^\times
  \;\simeq\;
  S(\mathcal{V})/\mathrm{U}(1)
$$

is the [[projective bundle]] of $\mathcal{V}$, with [[typical fiber]] the [[complex projective spaces]] $\mathbb{C}P^n$, and where in the middle we are displaying the [[splitting principle|splitting]] ([here](projective+bundle#Splitting)) of the [[pullback bundle]] of $E$ into a [[direct sum of vector bundles|direct sum]] of a [[line bundle]] $\mathcal{L}$ with a remaining vector bundle $\mathcal{V}'$ of rank just $n$.

Using the defining conditions on the total Conner-Floyd Chern class

$$
  c^E(\mathcal{V})
  \;\coloneqq\;
  1
    +
  c_1^E(\mathcal{V})
    +
  c_2^E(\mathcal{V})
    +
  c_3^E(\mathcal{V})
    +
  \cdots
$$

that

1. on a [[complex line bundle]] we have $c^E(\mathcal{L}) =  1 + c_1^E(\mathcal{L})$ for the given complex orientation $c_1^E$;

1. on a [[direct sum of vector bundles]] we have the [Whitney sum formula](Chern+class#WhitneySumFormula) $c^E(\mathcal{V} \oplus \mathcal{W}) \;=\; c^E(\mathcal{V}) \cdot c^E(\mathcal{W})$

we see from (eq:ProjectiveBundleInConstructionOfConnerFloydClasses) that

$$
  \pi^\ast
  \big(
    {\color{blue}
      c^E(\mathcal{V})
    }
  \big)
  \;=\;
  c^E(\mathcal{V}')
  \cdot
  \big(
    1
      +
    {\color{orange}
      c^E_1(\mathcal{L})
    }
  \big)
  \;\;\;
  \in
  \;
  E^\bullet\big( P(\mathcal{V}) \big)
$$

and hence

$$
  \begin{aligned}
    c^E(\mathcal{V}')
    & =\;
    \pi^\ast
    \big(
      {\color{blue}
        c^E(\mathcal{V})
      }
    \big)
    \cdot
    \big(
      1 +
      {
        \color{orange}
        c_1^E(\mathcal{L})
      }
    \big)^{-1}
    \\
    & \coloneqq\;
    \pi^\ast
    \big(
      {\color{blue}
      c^E(\mathcal{V})
      }
    \big)
    \cdot
    \underset{k}{\sum}
    (-1)^k
    \big(
      {\color{orange}
        c_1^E(\mathcal{L})
      }
    \big)^k
    \,.
  \end{aligned}
$$

But since $\mathcal{V}'$ is just of rank $n$, we must have $c^E_{n+1}(\mathcal{V}') = 0$, and hence in degree $2(n+1)$ this condition reads as follows:

\[
  \label{DefiningConditionOnConnerFloydChernClasses}
  \begin{aligned}
  0
  & =\;
  \pi^\ast
  \big(
    {\color{blue}
      c^E_{n+1}(\mathcal{V})
    }
  \big)
  -
  \pi^\ast
  \big(
    {\color{blue}
      c^E_{n}(\mathcal{V})
    }
  \big)
  \cdot
  {\color{orange}
    c_1^E(\mathcal{L})
  }
  +
  \pi^\ast
  \big(
    {\color{blue}
      c^E_{n-1}(\mathcal{V})
    }
  \big)
  \cdot
  \big(
    {\color{orange}
      c_1^E(\mathcal{L})
    }
  \big)^2
  -
  \cdots
  +
  (-1)^n
  \pi^\ast
  \big(
    {\color{blue}
      c^E_{1}(\mathcal{V})
    }
  \big)
  \cdot
  \big(
    {\color{orange}
      c_1^E(\mathcal{L})
    }
  \big)^n
  \\
  &
  \phantom{=}
  +
  (-1)^{n+1}
  \big(
    {\color{orange}
      c_1^E(\mathcal{L})
    }
  \big)^{n+1}
  \,.
  \end{aligned}
\]

It is now sufficient to observe that

1. the $E$-cohomology of $P(\mathcal{V})$ is a [[free module|free]] $E^\bullet(X)$-[[module]] [[linear span|spanned]] by the first [[cup product|cup powers]] of $c_1^E(\mathcal{L})$:

   \[
     \label{ECohomologyOfProjectiveBundle}
     E^\bullet\big( P(\mathcal{V})  \big)
       \;\simeq\;
     E^\bullet(X)
    \Big\langle
      1,
      {\color{orange}
        c_1^E(\mathcal{L})
      }
      ,
      \big(
        {\color{orange}
          c_1^E(\mathcal{L})
        }
      \big)^2
      ,
      \cdots
      ,
      \big(
        {\color{orange}
          c_1^E(\mathcal{L})
        }
      \big)^n
    \Big\rangle
    \,;
   \]

1. and in particular

   $$
     \pi^\ast
       \;\colon\;
     E^\bullet( X )
       \overset{\;\;\;\;\;}{\hookrightarrow}
     E^\bullet\big( P(\mathcal{V}) \big)
   $$

   is an [[injective function]],

because this means that (eq:DefiningConditionOnConnerFloydChernClasses) has a unique solution for the classes ${\color{blue} c_k^E(\mathcal{V})} \,\in\, E^{2k}(X)$ -- these are the Conner-Floyd Chern classes of $\mathcal{V}$.

But (eq:ECohomologyOfProjectiveBundle) holds on $\mathbb{C}P^n$ after [[pullback in cohomology|pullback]] along $fib_x$ (by standard arguments in [[complex oriented cohomology theory]], e.g. [Lurie 10, Lecture 4, Example 8](#LurieLecture)) and hence holds on $P(\mathcal{V})$ by the generalized-cohomology version of the [[Leray-Hirsch theorem]] ([Conner-Floyd 66, Thm. 7.4](#ConnerFloyd66)).

Since this construction is natural, one finds the following [[universal characteristic classes]] in $E$-cohomology:

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

Given a [[complex oriented cohomology theory]] $E$ with complex orientation $c_1^E$, then the $E$-[[generalized cohomology]] of the [[classifying space]] $B U(n)$ is freely generated over the [[graded commutative ring]] $\pi_\bullet(E)$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum)) by classes $c_k^E$ for $0 \leq k \leq n$ of degree $2k$, these are called the **[[Conner-Floyd-Chern classes]]**

$$
  E^\bullet(B U(n))
    \;\simeq\;
  \pi_\bullet(E)[ [ c_1^E, c_2^E, \cdots, c_n^E ] ]
  \,.
$$

Moreover, pullback along the canonical inclusion $B U(n) \to B U(n+1)$ is the [[identity]] on $c_k^E$ for $k \leq n$ and sends $c_{n+1}^E$ to zero.

For $E = $ [[HA|HZ]] this reduces to the standard [[Chern classes]].

=--

## References

The concept was introduced for:

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Theorems 7.5, 7.6 of: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

  > (with the example of [[MSp]]- and of [[MU]]-cohomology theory in Section 8)

An early account in a broader context of [[complex oriented cohomology theory]]:

* {#Adams74} [[Frank Adams]], part I.4, part II.2, part III.10 of _[[Stable homotopy and generalised homology]]_, 1974

More recent accounts:

* {#Kochman96} [[Stanley Kochman]], Section 4.3 of: _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#TamakiKono06} [[Dai Tamaki]], [[Akira Kono]], Section 3.4 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

* {#Lurie10} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, 2010, lecture 4 ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture4.pdf)) and lecture 5 ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture5.pdf))

[[!redirects Conner-Floyd Chern classes]]

[[!redirects Conner-Floyd class]]
[[!redirects Conner-Floyd classes]]

[[!redirects Conner-Floyd-Chern class]]
[[!redirects Conner-Floyd-Chern classes]]

[[!redirects generalized Chern class]]
[[!redirects generalized Chern classes]]

[[!redirects generalised Chern class]]
[[!redirects generalised Chern classes]]


Conner-Floyd isomorphism

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#### Algebraic topology
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[[!include algebraic topology - contents]]
=--
#### Cohomology
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[[!include cohomology - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

The __Conner-Floyd isomorphism__ ([Conner-Floyd 66, Thm. 10.1](#ConnerFloyd66), [Conner-Smith 69, Thm. 9.1](#ConnerSmith69)) is a [[natural isomorphism]]

$$
  KU^\bullet(X)
    \cong
  MU^\bullet(X)
    \otimes_{\Omega^U}
  \mathbb{Z}
$$

between the [[complex topological K-theory]] [[cohomology group|group]] of a finite [[CW-complex]] $X$ and the [[extension of scalars]] of the [[MU]]-[[cobordism cohomology]] of $X$ along the [[Todd genus]] $\Omega^U \overset{Td}{\longrightarrow} \mathbb{Z}$ (where $\Omega^U_{-\bullet} = MU^\bullet(\ast)$ is the [[MU]]-[[cobordism ring]] of stably almost complex manifolds $\Sigma$, and $Td(\Sigma) \in \mathbb{Z}$ is their [[Todd class]]).

A slightly more abstract way of saying the same is

$$
  KU^\bullet(X)
    \cong
  MU^\bullet(X)
    \otimes_{MU^\bullet}
  KU^\bullet\,
$$

which  -- thinking now of the [[Todd genus]] as coming from the canonical [[complex oriented cohomology|complex orientation]] $MU \longrightarrow KU$ (see at _[[universal complex orientation of MU]]_) -- shows that the Conner-Floyd isomorphism is a special case of the [[Landweber exact functor theorem]].

The analogous statement holds

* for [[MSp]] and [[KO]] ([Conner-Floyd 66](#ConnerFloyd66))

and

* for [[MSpin]] and [[KO]] as well as [[MSpin^c]] and [[KU]] under the [[Atiyah-Bott-Shapiro orientation]] ([Hopkins-Hovey 92, Thm. 1](#HopkinsHovey92))

However, the analogous statement for

* for [[MSU]] and [[KO]]

(via the [[Conner-Floyd orientation]]) _fails_, or rather does hold with a small modification ([Ochanine 87](#Ochanine87)).

## Related concepts

* [[Landweber exact functor theorem]]

* [[Conner-Floyd Chern class]]

## References

The original articles on the cases [[MU]]$\to$[[KU]] and [[MSp]]$\to$[[KO]]:

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Theorem 10.1 in: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

  \linebreak

  Russian transl.: Коннер П., Флойд Э. 0 соотношении теории кобордизмов и К-теории. - Дополнение к кн.: Гладкие периодические отображения. - М.: Мир, 1969 ([djvu](http://gen.lib.rus.ec/get?nametype=orig&md5=11459C40D0E52BA45BDAD92BC8AC0ACA))

with an alternative proof for [[MU]]$\to$[[KU]] in:

* {#ConnerSmith69} [[Pierre Conner]], [[Larry Smith]], Theorem 9.1 in: _On the complex bordism of finite complexes_, Publications Mathématiques de l'IHÉS, Tome 37 (1969) , pp. 117-221 ([numdam:PMIHES_1969__37__117_0](http://www.numdam.org/item/?id=PMIHES_1969__37__117_0))

Review:

* {#TamakiKono06} [[Dai Tamaki]], [[Akira Kono]], Theorem 6.35 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

See also:

* Gerhard Wolff, _Der Einfluss von $K^{\ast} (-)$ auf $U^{\ast} (-)$_, Manuscripta Math. __10__ (1973), 141–-161 ([doi:10.1007/BF01475039](https://doi.org/10.1007/BF01475039))

* Gerhard Wolff,  _Vom Conner-Floyd Theorem zum Hattori-Stong Theorem_,  Manuscripta Math. __17__ (1975), no. 4, 327–-332 ([doi:10.1007/BF01170729](http://dx.doi.org/10.1007/BF01170729), [MR388420](http://www.ams.org/mathscinet-getitem?mr=388420))

The (failure of the) version for [[MSU]]$\to$[[KO]] is due to:

* {#Ochanine87} [[Serge Ochanine]], _Modules de SU-bordisme. Applications_, Bulletin de la Société Mathématique de France, Tome 115 (1987) , pp. 257-289 ([doi:10.24033/bsmf.2078]( https://doi.org/10.24033/bsmf.2078))

The version for [[MSpin^c]]$\to$[[MU]] and [[MSpin]]$\to$[[KO]] is due to:

* {#HopkinsHovey92} [[Michael Hopkins]], [[Mark Hovey]], _Spin cobordism determines real K-theory_, Mathematische Zeitschrift volume 210, pages 181–196 (1992) ([doi:10.1007/BF02571790](https://doi.org/10.1007/BF02571790), [[HopkinsHoveyCobordismK.pdf:file]])

Generalization to [[equivariant cohomology theory]], now relating [[equivariant cobordism cohomology]] to [[equivariant K-theory]]:

* [[Steven Costenoble]], *Equivariant Conner-Floyd isomorphism*, Trans. Amer. Math. Soc. **304** 2 (1987) 801--818, [[jstor:2000742](http://www.jstor.org/stable/2000742)]

Discussion in [[motivic cohomology]]:

*  [[David Gepner]], [[Victor Snaith]], _On the motivic spectra representing algebraic cobordism and algebraic K-theory_, Doc. Math., 14:359–396 (electronic), 2009, [pdf](http://www.math.uni-bielefeld.de/documenta/vol-14/14.pdf)

*  [[Ivan Panin]], Konstantin Pimenov, Oliver Röndings, _On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory_, Invent. Math., __175__ (2009), no. 2, 435--451., [MR2470112](http://www.ams.org/mathscinet-getitem?mr=2470112)

On a construction of [[algebraic cobordism]] as a non-[[A1-homotopy theory|$\mathbb{A}^1$-invariant]] [[cohomology theory]] on [[derived schemes]], and the resulting [[Conner-Floyd isomorphism]]:

* Toni Annala, [[Marc Hoyois]], Ryomei Iwasa, _Algebraic cobordism and a Conner-Floyd isomorphism for algebraic K-theory_ [[arXiv:2303.02051](https://arxiv.org/abs/2303.02051)]


Connes distribution

Connes distribution space is a ccertain analogue of the theory of generalized functions ([[distribution]]s) for functional spaces.

Let $\alpha_(n)$, $n\in \mathbf{N}$ be a sequence of real strictly positive numbers and $H_k$, $k\gt 0$ the Sobolev space
in the corresponding geometric setip (typically of certain sections of a Hermitean bundle $E$ on a compact Riemannian manifold $M$). Let
$$
\sigma = \sum_n \sigma_n
$$
where $\sigma_k \in H^{\otimes n}_k$. For every $C\gt 0$ set
$$
\|\sigma\|_{1,C,k} := \sum C^n \alpha(n) \| \sigma_n \|_{H^{\otimes n}_k}
$$
Banach space $Co_{C,k}$ is the space of $\sigma$ for which $\|\sigma\|_{1,C,k}\lt \infty$. The space of Connes functionals $Co_{\infty -}:= \cap_{C\gt 0, k\gt 0} Co_{C,k}$. The Connes distribution space $Co_{-\infty}$ will be its topological dual.

The Potthoff–Streit theorem allows to define
flat Feynman [[path integral]]s as distributions.

* [[Alain Connes]], _On the Chern character of θ summable Fredholm modules_, Comm. Math. Phys. __139__ (1991), no. 1, 171–181, [MR92i:19003](http://www.ams.org/mathscinet-getitem?mr=1116414), [euclid](http://projecteuclid.org/getRecord?id=euclid.cmp/1104203140)
* [[Rémi Léandre]], _Path integrals in non-commutative geometry_, in Encyclopedia of Mathematical Physics (Elsevier, 2006); _Stochastic analysis without probability: study of some basic tools_, J. Pseudo-Differ. Oper. Appl. 1 (2010), no. 4, 389–400, preprint version [pdf](http://math.u-bourgogne.fr/IMB/prepub/prepub2010/leandre601.pdf); _Theory of distributions in the sense of Connes-Hida and Feynman path integral on a manifold_, [ps](http://www.iecn.u-nancy.fr/Preprint/publis/Textes/2002-44.ps); _Connes-Hida calclus in index theory_, [ps](http://www.iecn.u-nancy.fr/Preprint/publis/Textes/2003-30.ps)
* [[Ezra Getzler]], _Cyclic homology and the path integral of the Dirac operator_, 1988, Unpublished Preprint.
* J D S Jones, Rémi Léandre, _$L_p$ Chen forms on loop spaces_,
In: Barlow M, Bingham N (eds.) Stochastic Analysis, 104–162, Cambridge Univ. Press 1991

A closely related approach is that of a Hida in the theory of [[white noise]].

* T. Hida, H.-H. Kuo, J. Potthoff, L. Streit, _White Noise. An infinite dimensional calculus_, Kluwer 1993.
* M. de Faria1, J. Potthoff, L. Streit, _The Feynman integrand as a Hida distribution_, J. Math. Phys. __32__, 2123 (1991); [doi](http://dx.doi/org/10.1063/1.529184)

[[Gel'fand triple]]s are often used in spectral analysis and distribution theory on infinite-dimensional spaces:

* Yu.M. Berezansky, Yu.G. Kondratiev, _Spectral Methods in Infinite-Dimensional Analysis_, Kluwer 1995. Originally in Russian, Naukova Dumka, Kiev, 1988.
* Sebastian Jung, _d-dimensional Feynman Integrands as Hida Distributions_, slides, [pdf](http://ccm.uma.pt/mme38/files/Sebastian-jung.pdf)

[[!redirects Hida distribution]]


Connes fusion

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

## Idea

The operation called _Connes fusion_ (after [[Alain Connes]]) is a tensor product operation of [[Hilbert space]] [[bimodule]]s over [[von Neumann algebra]]s that adjusts the naive algebraic tensor product.

## Applications

Connes fusion is used to define fusion of positive energy representations of the [[loop group]] $\mathcal{L}S U(N)$ in

* Antony Wassermann, _Operator algebras and conformal field theory III_ ([arXiv](http://arxiv.org/abs/math.OA/9806031)) .

It is used in a proposal for a construction of [[string group]]-[[principal bundle]]s in the context of finding geometric models for [[elliptic cohomology]] in

* [[Stephan Stolz]] and [[Peter Teichner]], _What is an elliptic object?_ ([link](http://math.berkeley.edu/~teichner/Papers/Oxford.pdf))

## References

The fusion operation on [[Hilbert space|Hilbert]] [[module]]s of [[von Neumann algebra]]s has been described in

* [[Alain Connes]], _On the spatial theory of von Neumann algebras_ , J. of functional analysis 35 (1980) 153--164;

and in 1983 studied further by V. Jones in context of subfactors.

A useful review of Connes fusion observing some simplifications is

* Andreas Thom, _A remark about the Connes fusion tensor product_ ([arXiv:math.OA/0601045](http://arxiv.org/abs/math.OA/0601045))

An attempt to summarize that in a blog entry is [here](http://golem.ph.utexas.edu/string/archives/000718.html)

R. M Brower and N. P. Landsman described a [[bicategory]] of von Neumann algebras, Connes' correspondences as morphisms and (bounded) intertwiners as 2-cells. They proved that the equivalence in that bicategory is the same as usual [[Morita equivalence]] of von Neumann algebras as studied by [[Mark Rieffel]] and others.


Connes-Lott-Chamseddine-Barrett model

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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Connes-Lott-Chamseddine-Barrett model_ ([Connes-Lott 91](#ConnesLott91), [Barrett 07](#Barrett07), [Chamseddine-Connes-Marcolli 07](#ChamseddineConnesMarcolli07)) is a [[spectral triple]] that spectrally encodes a [[spacetime]] which is the [[product]] of 4d [[Minkowski spacetime]] $X$ with a _[[non-geometric background|non-geometric]]_ ([[Isbell duality|formal dually]]: [[non-commutative geometry]])  space $F$ whose classical dimension is that of a point, but which has [[KO-dimension]] 6 (mod 8). Hence this is a non-commutative version of a [[Kaluza-Klein compactification]] of a spacetime of [[KO-dimension]] $4 + 6$. (See also the discussion at _[[2-spectral triple]]_).

As in all ([[supersymmetry|super]]-)[[KK-compactification|Kaluza-Klein theory]], what is pure [[pseudo-Riemannian geometry]] (albeit "[[spectral geometry|spectral]]") in 4+6 (KO-)dimension, hence pure [[gravity]], effectively looks like a configuration of [[Einstein-Yang-Mills-Dirac-Higgs theory]] down in 4 dimensions, hence of [[gravity]] coupled to [[gauge fields]] and [[fermions]] and a [[Higgs boson]].

The interest in the model lies in the fact that a comparatively simple algebraic choice in the spectral triple of the Connes-Lott-Chamseddine model this way reproduces the [[standard model of particle physics]], and does so in quite some fine detail (even if some issues remain open).

For instance the fiber space $F$ is modeled as the formal dual to the algebra

$$
  \mathbb{C} \oplus \mathbb{H}_L \oplus \mathbb{H}_R \oplus M_3(\mathbb{C})
$$

which is the [[direct sum]] of the $\mathbb{R}$-[[associative algebra|algebras]] of [[complex numbers]], [[quaternions]] and $3x3$ complex [[matrices|matrix algebra]].

Moreover, the entire field content in one [[generation of fundamental particles|generation]] of [[fermions]] (i.e. [[electrons]], [[quarks]], [[neutrinos]]) is claimed to be encoded precisely in the [[bimodule]] over this algebra which is the direct sum of all inequivalent irreducible odd bimodules ([Connes 06, prop. 2.2 - prop 2.5](#Connes06)).

The Connes-Lott models realize a non-commutative version of [[gauge-Higgs unification]].

## Related entries

* [[Gepner model]]

* [[higher category theory and physics]]: _[Spectral standard model and gravity](higher+category+theory+and+physics#SpecStandModAndGravity)_

## References

### Review
 {#ReferencesReview}

* Robert Brout, _Notes on Connes' Construction of the Standard Model_, Nucl. Phys. Proc. Suppl. 65 (1998) 3-15 ([arXiv:hep-th/9706200](https://arxiv.org/abs/hep-th/9706200))

* Agostino Devastato, Maxim Kurkov, [[Fedele Lizzi]], _Spectral Noncommutative Geometry, Standard Model and all that_ ([arXiv:1906.09583](https://arxiv.org/abs/1906.09583))

Relation to the "[[neutrino]]-paradigm":

* {#G-B19} [[José Gracia-Bondia]], _On Marshak's and Connes' views of chirality_, in: [[E. C. G. Sudarshan]] (ed.), _A Gift of Prophecy -- Essays in Celebration of the Life of [[Robert Eugene Marshak]]_, pp. 208-217, World Scientific  (1995)  ([arXiv:hep-th/9706200](https://arxiv.org/abs/hep-th/9706200), [doi:10.1142/9789812831408_0017](https://doi.org/10.1142/9789812831408_0017))

Exposition highlighting the relation to [[KK-compactification]] and [[string theory]]-[[perturbative string theory vacuum|vacua]] ([[2-spectral triples]]) includes

* {#Schreiber16} [[Urs Schreiber]], _[Spectral Standard Model and String Compactifications](https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/)_, 2016

Other commentary in view of [[D-brane]]-physics:

* [[John Iliopoulos]], _Gauge Theories and non-Commutative Geometry: A review_, EPJ Web Conf. Volume 182, 2018 6th International Conference on New Frontiers in Physics (ICNFP 2017) ([doi:10.1051/epjconf/201818202055](https://doi.org/10.1051/epjconf/201818202055))

See also

* Wikipedia, _[Noncommutative standard model](https://en.wikipedia.org/wiki/Noncommutative_standard_model)_

### Original articles

The basic mechanism was originally laid out in

* [[Alain Connes]], _Gravity coupled with matter and foundation of non-commutative geometry_, Commun.Math.Phys. 182 (1996) 155-176 ([arXiv:hep-th/9603053](https://arxiv.org/abs/hep-th/9603053))

(see also the references at _[[spectral action]]_).

The early version of the model is due to

* {#ConnesLott91} [[Alain Connes]], [[John Lott]], _Particle models and noncommutative geometry_, Nuclear Physics B - Proceedings Supplements 18(2): 29-47 (1991) ([web](https://deepblue.lib.umich.edu/handle/2027.42/29524))

The modern version of the model that produces the correct fermionic content (and finds the [[KO-dimension]] of the compactification space to be 6 mod 8) is due to

* {#Barrett07} [[John Barrett]], _A Lorentzian version of the non-commutative geometry of the standard model of particle physics_, J.Math.Phys.48:012303,2007 ([arXiv:hep-th/0608221](http://arxiv.org/abs/hep-th/0608221), [doi:10.1063/1.2408400](https://doi.org/10.1063/1.2408400))

* {#Connes06} [[Alain Connes]], _Noncommutative Geometry and the standard model with neutrino mixing_, JHEP0611:081,2006 ([arXiv:hep-th/0608226](https://arxiv.org/abs/hep-th/0608226))

* {#ChamseddineConnesMarcolli07} [[Ali Chamseddine]], [[Alain Connes]], [[Matilde Marcolli]], _Gravity and the standard model with neutrino mixing_, Adv. Theor. Math. Phys.11:991-1089,2007 ([arXiv:hep-th/0610241](http://arxiv.org/abs/hep-th/0610241))

A more succinct version of the axioms of the model is claimed in

* [[Latham Boyle]], [[Shane Farnsworth]],  _Non-Commutative Geometry, Non-Associative Geometry and the Standard Model of Particle Physics_ New J. Phys. 16, 123027 (2014) ([arXiv:1401.5083](http://arxiv.org/abs/1401.5083))

* [[Shane Farnsworth]], [[Latham Boyle]], _Rethinking Connes' approach to the standard model of particle physics via non-commutative geometry_, New J. Phys. 17, 023021 (2015) ([arXiv:1408.5367](https://arxiv.org/abs/1408.5367))

* Christian Brouder, Nadir Bizi, Fabien Besnard, _The Standard Model as an extension of the noncommutative algebra of forms_ ([arXiv:1504.03890](https://arxiv.org/abs/1504.03890))

Introduction of a [[scalar field]] to fix the prediction of the [[Higgs particle]] [[mass]]:

* {#ChamseddineConnes12} [[Ali Chamseddine]], [[Alain Connes]], _Resilience of the Spectral Standard Model_, JHEP 1209 (2012) 104 ([arXiv:1208.1030](https://arxiv.org/abs/1208.1030))

Implementation of [[Pati-Salam model]] ($SU(5)$-[[GUT]]):

* [[Ali Chamseddine]], [[Alain Connes]], Walter D. van Suijlekom, _Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification_, JHEP 1311 (2013) 132 ([arXiv:1304.8050](https://arxiv.org/abs/1304.8050))

* Hosein Karimi Khozani, _Symmetry Breaking and Proton Decay in Spectral Pati-Salam Model_ ([arXiv:1905.04533](https://arxiv.org/abs/1905.04533))

and specifically discussion of [[leptoquarks]] and possible relation to [[flavour anomalies]]:

* Ufuk Aydemir, Djordje Minic, Chen Sun, Tatsu Takeuchi, _$B$-decay anomalies and scalar leptoquarks in unified Pati-Salam models from noncommutative geometry_, JHEP 09 (2018) 117 ([arXiv:1804.05844](https://arxiv.org/abs/1804.05844))

See also:

* Fabien Besnard, _Extensions of the noncommutative Standard Model and the weak order one condition_ ([arXiv:2011.02708](https://arxiv.org/abs/2011.02708))

Relation to actual [[quantum gravity]] induced by quantized maps from [[spacetime]] to the [[4-sphere]] (see also at _[[Cohomotopy]]_) is claimed in

* {#ChamseddineConnesMukhanov14} [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, _Quanta of Geometry: Noncommutative Aspects_, Phys. Rev. Lett. 114 (2015) 9, 091302 ([arXiv:1409.2471](https://arxiv.org/abs/1409.2471))

* {#ChamseddineConnesMukhanov14} [[Ali Chamseddine]], [[Alain Connes]], Viatcheslav Mukhanov, _Geometry and the Quantum: Basics_, JHEP 12 (2014) 098 ([arXiv:1411.0977](https://arxiv.org/abs/1411.0977))

* {#Connes17} [[Alain Connes]], section 4 of _Geometry and the Quantum_, in _Foundations of Mathematics and Physics One Century After Hilbert_, Springer 2018. 159-196 ([arXiv:1703.02470](https://arxiv.org/abs/1703.02470), [doi:10.1007/978-3-319-64813-2](https://www.springer.com/gp/book/9783319648125))

With [[spin^c structure]]:

* Arkadiusz Bochniak, Andrzej Sitarz, _A spectral geometry for the Standard Model without the fermion doubling_ ([arXiv:2001.02902](https://arxiv.org/abs/2001.02902))

Discussion of [[spectral triples]] over [[Jordan algebras]] in the [[Connes-Lott model]]:

* [[Latham Boyle]], [[Shane Farnsworth]],
_The standard model, the Pati-Salam model, and "Jordan geometry"_ ([arxiv:1910.11888](https://arxiv.org/abs/1910.11888))

* [[Shane Farnsworth]], _The geometry of physical observables_ ([arXiv:2003.01708](https://arxiv.org/abs/2003.01708))

Discussion in Lorentzian signature:

* Fabien Besnard, Christian Brouder, _Noncommutative geometry, the Lorentzian Standard Model and its B-L extension_ ([arXiv:2010.04960](https://arxiv.org/abs/2010.04960))

* Manuele Filaci, Pierre Martinetti, *A critical survey of twisted spectral triples beyond the Standard Model* [[arXiv:2301.08346](https://arxiv.org/abs/2301.08346)]

[[!redirects Connes-Lott-Chamseddine-Barrett models]]

[[!redirects Connes-Lott-Chamseddine model]]
[[!redirects Connes-Lott-Chamseddine models]]

[[!redirects Connes-Lott model]]
[[!redirects Connes-Lott models]]

[[!redirects non-commutative standard model]]
[[!redirects non-commutative standard models]]


Conor J. Houghton

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=67548)

## Selected writings

On [[Skyrmions]] via [[rational maps]] from the [[complex plane]], hence [[holomorphic maps]] from the [[Riemann sphere]], to itself:

* [[Conor J. Houghton]], [[Nicholas Manton]], [[Paul Sutcliffe]], *Rational Maps, Monopoles and Skyrmions*, Nucl. Phys. B510 (1998) 507-537 ([arXiv:hep-th/9705151](https://arxiv.org/abs/hep-th/9705151), doi:10.1016/S0550-3213(97)00619-6)

category: people

[[!redirects Conor Houghton]]


Conor McBride

* [personal page](http://strictlypositive.org)

* [institute page](https://www.strath.ac.uk/staff/mcbrideconordr/)

## Selected writings

On [[functional programming]] with [[dependent types]]:

* [[Conor McBride]], *Dependently Typed Functional Programs and their Proofs*, PhD thesis (1999) [[pdf](https://www.lfcs.inf.ed.ac.uk/reports/00/ECS-LFCS-00-419/ECS-LFCS-00-419.pdf), [pdf](https://ncatlab.org/nlab/files/conor-mcbride-thesis.pdf)]

On [[fibered heterogeneous identity types]]:

* {#McBride99} [[Conor McBride]], *Dependently Typed Functional Programs and their Proofs*, (1999) [[pdf](https://www.lfcs.inf.ed.ac.uk/reports/00/ECS-LFCS-00-419/ECS-LFCS-00-419.pdf)]

On [[dependent linear type theory]]:

* [[Conor McBride]], *I Got Plenty o’ Nuttin’*, in: *A List of Successes That Can Change the World*, Lecture Notes in Computer Science **9600**, Springer (2016) [[doi:10.1007/978-3-319-30936-1_12](https://doi.org/10.1007/978-3-319-30936-1_12)]

Introducing [[observational type theory]]:

* [[Thorsten Altenkirch]], [[Conor McBride]], _Towards observational type theory_, draft (2006) [[pdf](http://strictlypositive.org/ott.pdf), [pdf](https://www.cs.nott.ac.uk/~psztxa/publ/ott-conf.pdf), [[AltenkirchMcBrided-TowardsOTT.pdf:file]]]

* {#AltenkirchMcBrideSwierstra07} [[Thorsten Altenkirch]], [[Conor McBride]], [[Wouter Swierstra]], *Observational Equality, Now!*, PLPV '07: Proceedings of the 2007 workshop on Programming languages meets program verification (2007) 57-68 [ISBN:978-1-59593-677-6, [doi:10.1145/1292597.1292608](http://doi.org/10.1145/1292597.1292608), [pdf](https://www.cs.nott.ac.uk/~psztxa/publ/obseqnow.pdf)]

category: people


conormal bundle

#Contents#
* table of contents
{:toc}

## Definition

### Of submanifolds

In [[differential geometry]] a __conormal bundle__ of an
[[embedding|embedded]] submanifold is the (fiberwise linear) dual of the [[normal bundle]].

### Of locally ringed subspaces

The phrase conormal bundle is also used for more general [[conormal sheaf]] in the study of [[locally ringed spaces]], especially of [[analytic space]]s and [[algebraic schemes]].

### Of abelian subcategories

Even more generally, [[Alexander Rosenberg]] defines a conormal bundle of a [[topologizing subcategory]] $S$ of an [[abelian category]] $A$ as follows.

He first modifies the notion of the [[defining sheaf of ideals]] of a closed subscheme to the notion of [[defining ideal of a topologizing subcategory]] as the endofunctor $\mathcal{I}=\mathcal{I}_S\in End(A)$ which is the subfunctor of identity $Id_A$ assigning to any $M\in A$ the intersection of kernels $Ker(f)$ of all morphisms $f: M\to N$ where $N\in Ob(S)$. One can show that if $T\subset S$ is an inclusion of topologizing subcategories, then $\mathcal{I}_{S}\subset \mathcal{I}_{T}$. In particular, for [[Gabriel multiplication]] of topologizing subcategories we have $\mathcal{I}_{S\circ S} \subset \mathcal{I}_S$.

Then the conormal bundle is simply $\Omega_S = \mathcal{I}_S/\mathcal{I}_{S\circ S}$, similarly to the sheaf case.

## Related concepts

* [[normal bundle]]

* **conormal bundle**

[[!redirects conormal bundles]]


conormal sheaf

Given a [[closed immersion of schemes]] $f= (f,f^\sharp): (Y,O_Y)\to (X,O_X)$ of (or of more general [[locally ringed spaces]], e.g. of analytic varieties), the kernel $\mathcal{I} = O_X$ of the comorphism $f^\sharp:O_X\to f_* O_Y$ is a sheaf of ideals, called the [[defining sheaf]] of the closed immersion $f$.

The __conormal sheaf__ is the [[quotient sheaf]] of $O$-modules given by $\Omega_{X/Y} := \mathcal{I}/\mathcal{I}^2$. In the algebraic setting, it is also called the sheaf of relative [[Kähler differentials]]. It is always [[quasicoherent sheaf|quasicoherent]].
Most important example is the case of the diagonal $\Delta:X\to X\times_S X$ of an $S$-scheme $X$. For example, when $S = Spec k$ and $X = Spec R$ where $R$ is a $k$-algebra then the global sections of $\Omega_{X\times_S X/X}$ form the kernel of the multiplication map $R\times_k R\to R$ quotiented by its square.

One sometimes says [[conormal bundle]] meaning a conormal sheaf, though the bundle and sheaf are two points of view in the cases when the conormal sheaf is locally free of constant rank.

[[!redirects conormal sheaves]]


consequent

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#### Type theory
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# Contents
* table of contents
{: toc}

## Idea

A _consequent_ or _conclusion_ [[judgement]] on the right of the "$\vdash$"-symbol of a [[sequent]]/[[hypothetical judgement]]: a [[judgement]] that is a consequence of the [[antecedent]] judgements on the left of the symbol.

If there is more than one [[disjunction|disjunctively]] connected consequent, these are called the _[[succedents]]_.

## Related concepts

[[antecedents]] $\vdash$ [[succedents]]

[[hypothesis]] $\vdash$ **consequent**

[[!include mathematical statements --- contents]]

[[!redirects consequent]]
[[!redirects consequents]]

[[!redirects consequence]]
[[!redirects consequences]]

[[!redirects conclusion]]
[[!redirects conclusions]]


conservation law

#Contents#
* table of contents
{:toc}

## Idea

Given a [[dynamical system]], a **conservation law** is the statement that some observable is time independent *in every solution*. Thus it is the same as an [[observable]] whose value is time-independent in each solution. Of course, in some particular solution there can be a conserved quantity which is not coming from a conservation law, but is an incidental feature of just that particular solution of equations of motion. Thus a conservation law is the same as a "universally" **conserved observable**.

In [[quantum mechanics]] an observable is conserved in time if it commutes with the [[Hamiltonian operator]]. In [[classical mechanics]], there is a famous theorem of [[Emmy Noether]] -- [[Noether's theorem]] -- which assigns a conservation law to any smooth symmetry of system. For example the isotropy of a space is related to the conservation of angular momentum, and the homogeneity of the space to the conservation of usual momentum.

## Related concepts

* [[conserved current]]
* [[Noether theorem]]
* [[symplectic integrator]]

## References

For 1-parameter groups of symmetries in classical mechanics, the formulation and the proof of [[Noether's theorem]] can be found in the monograph

* [[Vladimir Arnold]], _Mathematical methods of classical mechanics_

For more general case see the books by Peter Olver.

[[!redirects conservation law]]
[[!redirects conservation laws]]


conservative (∞,1)-functor

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#Contents#
* table of contents
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## Definition

An [[(∞,1)-functor]] is _conservative_ if its [[reflected limit|reflects]] [[equivalences in an (∞,1)-category]]. This means that on [[homotopy category of an (∞,1)-category|homotopy categories]] it is a [[conservative functor]].

## Examples

+-- {: .un_theorem}
###### Theorem
In an [[(∞,1)-topos]], [[(∞,1)-pullback]] along an [[effective epimorphism in an (∞,1)-category|effective epimorphism]] is a conservative (∞,1)-functor between [[slice (∞,1)-categories]].
=--
+-- {: .proof}
###### Proof

We write in the conjectural [[internal logic]] of an (∞,1)-topos expressed as [[homotopy type theory]], with [[slice (∞,1)-categories]] represented using [[dependent types]].  The reader is free to translate this manually into an [[(∞,1)-category theory|(∞,1)-category-theoretic]] proof.

Thus, suppose $f:A\to B$ is a surjection (i.e. $\prod_{b:B} \exists_{a:A} f(a)=b$, where $\exists$ denotes the [[propositional truncation]] of $\sum$).  Let $P,Q:B\to Type$ be dependent types with a fiberwise map $g:\prod_{b:B} P(b) \to Q(b)$ which becomes a fiberwise [[equivalence in homotopy type theory|equivalence]] upon restricting to $A$, i.e. such that $\prod_{a:A} IsEquiv(g(f(a)))$.  We must show that $g$ is itself a fiberwise equivalence.

Thus, let $b:B$; we must show $IsEquiv(g(b))$.  By assumption we have $\exists_{a:A} f(a)=b$.  But $IsEquiv(g(b))$ is an [[h-proposition]], so we may strip the truncation and obtain an $a:A$ such that $f(a)=b$.  Transporting along this equality, it suffices to show $IsEquiv(g(f(a)))$, but this follows from the other assumption.
=--

See also Lemma 6.2.3.16 in [[Higher Topos Theory]].

## References

In the context of [[quasi-categories]], def. 6.2.1 in

* {#RiehlVerity13} [[Emily Riehl]], [[Dominic Verity]], _Homotopy coherent adjunctions and the formal theory of monads_ ([arXiv:1310.8279](http://arxiv.org/abs/1310.8279))

[[!redirects conservative (∞,1)-functors]]

[[!redirects conservative (infinity,1)-functor]]
[[!redirects conservative (infinity,1)-functors]]


conservative cocompletion

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#### Limits and colimits
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#### Yoneda lemma
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[[!include Yoneda lemma - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

Given a category $C$, we may construct the [[free cocompletion]] of $C$, freely adding some class of [[colimits]]. Often, however, $C$ will already have some colimits, which we wish to preserve. A **conservative cocompletion** of a category $C$ is a cocompletion that preserves the colimits in $C$.

Explicitly, given a class $\Phi$ of colimits, and a $\Phi$-cocomplete [[small category]] $C$, the **$\Phi$-conservative cocompletion** of $C$ is given by taking the closure of $[C^\circ, Set]_\Phi$ under small colimits, where  $[C^\circ, Set]_\Phi$ is the [[full subcategory]] of the [[presheaf category]] $[C^\circ, Set]$ of presheaves sending $\Phi$-colimits in $C$ to $\Phi$-limits in $Set$. When $C$ is instead a [[locally small category]], we consider instead the [[small presheaves]].

## Related concepts

- [[free cocompletion]]

## References

- [[Jiřı́ Velebil]] and [[Jiřı́ Adámek]]. _A remark on conservative cocompletions of categories_. Journal of Pure and Applied Algebra 168.1 (2002): 107-124.

See also Theorem 11.5 of:

- [[Marcelo Fiore]]. _Enrichment and representation theorems for categories of domains and continuous functions_. University of Edinburgh.

[[!redirects conservative cocompletions]]
[[!redirects free conservative cocompletion]]
[[!redirects free conservative cocompletions]]


conservative functor

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## Definition

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

A [[functor]] $F \colon C\to D$ is **conservative** if it is "isomorphism-[[reflected limit|reflecting]]", i.e. if $g \colon a\to b$ is a [[morphism]] in $C$ such that $F(g)$ is an [[isomorphism]] in $D$, then $g$ is an isomorphism in $C$.

=--

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

Sometimes (e.g. in the [[Elephant]]) conservative functors are assumed to be [[faithful functor|faithful]] as well.  If $C$ has, and $F$ preserves, [[equalizer]]s, then conservativity implies faithfulness.

=--

See [[conservative morphism]] for a generalization to an arbitrary [[2-category]].

## Examples

\begin{example}\label{FullyFaithfulFunctorIsConservative}
Every [[fully faithful functor]], and more generally any [[pseudomonic functor]], is a conservative functor.

(Every [[fully faithful functor]] is [[pseudomonic]].)
\end{example}

But further would-be converses of Exp. \ref{FullyFaithfulFunctorIsConservative} fail: not every conservative functor is full or faithful:

\begin{example}
An example of a functor that is conservative but not full is the inclusion of the [[groupoid core]] $Core(\mathcal{C}) \longrightarrow \mathcal{C}$ of a [[category]] $\mathcal{C}$ that is itself not a groupoid, into that category.
\end{example}

\begin{example}
An example of a functor that is conservative but not faithful is the unique functor from any groupoid with two distinct isomorphisms $f, g : x \to y$ to the [[terminal]] groupoid.
\end{example}

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

Every [[monadic functor]] is a conservative functor (see also at *[[monadicity theorem]]*):

For a [[algebra over a monad|$T$-algebra]] [[homomorphism]] given by an [[invertible morphism]] $f \colon A \to B$, the [[inverse]] $f^{-1} \colon B \to A$ is easily seen to also be a $T$-algebra homomorphism.

=--

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

Let $C$ be a [[category]] with [[pullbacks]]. Given any [[morphism]] $f \colon X \longrightarrow Y$ in $C$ write

$$
  f^\ast \colon C_{/Y} \longrightarrow C_{/X}
$$

for the [[functor]] of [[pullback]] along $f$ between [[slice categories]] ("[[base change]]"). If [[strong epimorphisms]] in $\mathcal{C}$ are preserved by pullback, then the following are equivalent:

1. $f$ is a [[strong epimorphism]];

1. $f^\ast$ is conservative.

=--

(e.g. [Johnstone, lemma A.1.3.2](#Johnstone))

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

When $C$ and $D$ are [[pretopos|pretoposes]], a pretopos morphism $F \colon C \to D$ is conservative if and only if for every object $c \in C$, the induced map between [[poset of subobjects | subobject lattices]] $F^{(c)} : \operatorname{Sub}(c) \to \operatorname{Sub}(F(c))$ is injective.

=--

## Properties

\begin{proposition}
\label{ConservativeFunctorsReflectLimitsWhichTheyPreserve}
A conservative functor $F \colon C \to D$ [[reflected limit|reflects]] all [[limits]] and [[colimits]] that it [[preserved limit|preserves]] and which exist in the source category.
\end{proposition}

\begin{proof}
We discuss the case of limits (the argument for colimits is [[formally dual]]):

Let $K \colon J \to C$ be a [[diagram]] in $C$ whose limit $\lim K$ exists and such that $\lim (F\circ K) \,\simeq\, F (\lim K)$. Now if $const_c \to K$ is a [[cone]] in $C$ that is sent to a limiting cone $F const_c$ in $D$, then by the [[universal property]] of the limit in $D$ the morphism $F( c \to \lim K)$ is an [[isomorphism]] in $D$, hence must have been an isomorphism in $C$ (by the assumption that $F$ is conservative), hence $const_c$ must have been a limiting cone in $C$.
\end{proof}

## Related concepts

* [[monadicity theorem]]

* [[conservative (∞,1)-functor]]

## Literature

* Geun Bin Im, [[Gregory Maxwell Kelly]], _Some remarks on conservative functors with left adjoints_,  J. Korean Math. Soc. __23__ (1986),  no. 1, 19–33, [MR87i:18002b](http://www.ams.org/mathscinet-getitem?mr=843247), [pdf](http://mathnet.kaist.ac.kr/mathnet/thesis_file/JKMS-23-1-19-33.pdf); _On classes of morphisms closed under limits_, J. Korean Math. Soc. __23__ (1986), no. 1, 1–18, _Adjoint-triangle theorems for conservative functors_, Bull. Austral. Math. Soc. __36__ (1987),  no. 1, 133–136, [MR88k:18005](http://www.ams.org/mathscinet-getitem?mr=897429), [doi](http://dx.doi.org/10.1017/S000497270002637X)

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]]_

Formalization in [[cubical Agda]]:

* [[1lab]], *[Functors](https://1lab.dev/Cat.Functor.Base.html#functors)*

For an example of a conservative, but not faithful, functor $f: A\to Set$ having a left adjoint:

* [[Reinhard Börger]], [[Walter Tholen]], Example 2.4 in _Strong regular and dense generators_,  [[Cahiers de Topologie et Géométrie Différentielle Catégoriques]] **32** 3 (1991) 257-276 [[numdam:CTGDC_1991__32_3_257_0](http://www.numdam.org/item?id=CTGDC_1991__32_3_257_0), [MR1158111](http://www.ams.org/mathscinet-getitem?mr=1158111)]

[[!redirects conservative functors]]
[[!redirects conservative]]


conservative morphism

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#Contents#
* table of contents
{:toc}

## Definition

A [[morphism]] $f\colon A\to B$ in a [[2-category]] $K$ is said to be **(representably) conservative** if for all objects $X$, the induced [[functor]]
$$K(X,A) \to K(X,B)$$
is [[conservative functor|conservative]].  In [[Cat]], this is equivalent to $f$ being conservative in the usual sense.

## Remarks

* Conservative morphisms often form the right class of a [[factorization system in a 2-category|factorization system]].  In [[Cat]], the left class consists of (possibly [[transfinite composition|transfinitely]]) iterated [[localizations]].

* Of course, any [[fully faithful morphism]] is also conservative, and in particular any [[inverter]] or [[equifier]] is conservative.  Moreover, any [[inserter]] is also conservative (and [[faithful morphism|faithful]]), though not generally fully-faithful.

* If $f$ is conservative in $K^{op}$ (the [[opposite 2-category|1-cell dual]]), then $f$ is said to be **liberal**.  This joke is due to Carboni, Johnson, Street, and Verity; see [[codiscrete cofibration]].

## Related pages

* [[subcategory]]
* [[fully faithful morphism]]
* [[faithful morphism]]
* [[pseudomonic morphism]]
* [[discrete morphism]]

[[!redirects conservative morphisms]]
[[!redirects liberal morphism]]
[[!redirects liberal morphisms]]


conserved current

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#### Variational calculus
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#### Physics
+--{: .hide}
[[!include physicscontents]]
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# Contents
* table of contents
{: toc}

## Via the variational bicomplex

The following discusses the formulation of conserved currents in terms of [[variational calculus]] and the [[variational bicomplex]].

### The context

Let $X$ be a [[spacetime]] of [[dimension]] $n$, $E \to X$ a [[bundle]], $j_\infty E \to X$ its [[jet bundle]] and

$$
  \Omega^{\bullet,\bullet}(j_\infty E), (D = \delta + d)
$$

the corresponding [[variational bicomplex]] with $\delta$ being the vertical and $d = d_{dR}$ the horizontal [[differential]].

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

For $L \in \Omega^{n,0}(j_\infty E)$ a [[Lagrangian]] we have that

$$
  \delta L = E(L) + d \Theta
$$

for $E$ the [[Euler-Lagrange equations|Euler-Lagrange operator]].

=--

The [[covariant phase space]] of the Lagrangian is the locus

$$
  \{\phi \in \Gamma(E) | E(L)(j_\infty \phi) = 0\}
  \,.
$$

For $\Sigma \subset X$ any $(n-1)$-dimensional submanifold,

$$
  \delta \theta := \delta \int_\Sigma \Theta
$$

is the [[presymplectic structure]] on covariant phase space

### Definition

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

A **conserved current** is an element

$$
  j \in \Omega^{n-1, 0}(j_\infty E)
$$

which is horizontally closed on [[covariant phase space]]

$$
  d j|_{E(L) = 0} = 0
  \,.
$$

=--

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

For $\Sigma \hookrightarrow X$ a submanifold of dimension $n-1$, the **[[charge]]** of the conserved current $j$ with respect to $\Sigma$ is the [[integral]]

$$
  Q_\Sigma := \int_\Sigma j
  \,.
$$

=--

### Properties

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

If $\Sigma, \Sigma' \subset X$ are homologous, the associated charge is the same

$$
  Q_{\Sigma} = Q_{\Sigma'}
  \,.
$$

=--

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

By [[Stokes' theorem]].

=--

+-- {: .num_theorem}
###### Theorem

Every [[symmetry of the Lagrangian]] induces a conserved current.

=--

This is [[Noether's theorem]]. See there for more details.

## In higher prequantum geometry
 {#InHigherPrequantumGeometry}

The following discusses conserved currents in the context of [[higher prequantum geometry]], closely related to [Azcarraga-Izquierdo 95, section 8.1](#AzcarragaIzquierdo95). This follows ([classicalinhigher, section 3.3.](#classicalinhigher), going back to [Schreiber 13](#Schreiber13)). Similar observations have been made by [[Igor Khavkine]].

> this section needs much polishing. For the moment better see [classicalinhigher, section 3.3](#classicalinhigher)

### Context

Let $\mathbf{H}$ be the ambient [[(∞,1)-topos]]. For $\mathbf{Fields} \in \mathbf{H}$ a [[moduli ∞-stack]] of [[field (physics)|fields]] a [[local Lagrangian]] for an $n$-dimensional [[prequantum field theory]] is equivalently a [[prequantum n-bundle]] given by a map

$$
  \mathbf{L} \;\colon\; \mathbf{Fields} \longrightarrow \mathbf{B}^n U(1)_{conn}
$$

to the [[moduli ∞-stack]] of smooth [[circle n-bundles with connection]]. The local connection [[differential n-form]] is the [[local Lagrangian]] itself as in traditional literature, the rest of the data in $\mathcal{L}$ is the [[higher gauge theory|higher]] [[gauge symmetry]] [[equivariance|equivariant]] structure.

The following is effectively the direct higher geometric analog of the [Hamiltonian version of Noether's theorem](Noether's+theorem#HamiltonianNoetherTheorem).

### Symmetries

A transformation of the [[field (physics)|fields]] is an [[equivalence]]

$$
  \mathbf{Fields}
    \underoverset{\simeq}{\phi}{\longrightarrow}
  \mathbf{Fields}
  \,.
$$

That the [[local Lagrangian]] $\mathcal{L}$ be preserved by this, up to ([[gauge symmetry|gauge]]) equivalence, means that there is a [[diagram]] in $\mathbf{H}$ of the form

$$
  \array{
    \mathbf{Fields} &&\underoverset{\simeq}{\phi}{\longrightarrow}&&
    \mathbf{Fields}
    \\
    & {}_{\mathllap{\mathbf{L}}}\searrow &\swArrow^\simeq_\alpha& \swarrow_{\mathrlap{\mathbf{L}}}
    \\
    && \mathbf{B}^n U(1)_{conn}
  }
  \,.
$$

(With $\mathbf{L}$ equivalently regarded as [[prequantum n-bundle]] this is equivalently a [higher quantomorphism](quantomorphism%20group#InHigherGeometry). These are the transformations studied in ([Fiorenza-Rogers-Schreiber 13](#FiorenzaRogersSchreiber13)))

For $\phi$ an [[infinitesimal object|infinitesimal]] operation an $L$ locally the Lagrangian $n$-form, this means that the [[Lie derivative]] $\mathcal{L}_{\delta \phi}$ of $L$ has a potential,

$$
  \mathcal{L}_{\delta \phi} L = \mathbf{d} \alpha
$$

hence that the Lagrangian changes under the [[Lie derivative]] by an exact term, hence by a [[divergence]] on the [[worldvolume]] (since the degree of the Lagrangian form is the [[dimension]] of the worldvolume). This defines an _[[infinitesimal symmetry of the Lagrangian]]_. See also ([Azcarraga-Izquierdo 95 (8.1.13)](#AzcarragaIzquierdo95)).

This is the situation of the [[Noether theorem]] for the general case of "weak" symmetries (see at [Noether theorem -- schematic idea  -- weak symmetries](Noether+theorem#WeakSymmetrySchematicIdea)).

By [[Cartan's magic formula]] the above means

$$
  \mathbf{d}\left(
    \alpha - \iota_{\delta\phi} \mathbf{L}
  \right)
   =
  \iota_{\delta \phi} \omega
  \,.
$$

and hence the combination $j \coloneqq \alpha - \iota_{\delta\phi} \mathbf{L}$ (a _[[Hamiltonian form]]_ for $\delta \phi$ with respect to $\omega$) is conserved on trajectories in the kernel of the [[n-plectic form]] $\omega$ (which are indeed the classical trajectories of $\mathbf{L}$, see ([Azcarraga-Izquierdo 95 (8.1.14)](#AzcarragaIzquierdo95))).

This is the first stage in the [[Poisson bracket Lie n-algebra]], the _[[current algebra]]_ (see there at _[As a homotopy Lie algebra](current%20algebra#AsAHomotopyLieAlgebra)_).

## Examples

### Energy-momentum tensor

* [[energy-momentum tensor]] -- see at _[[geometry of physics -- A first idea of quantum field theory]]_ [this example](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#ScalarFieldEnergyMomentum)

### Dirac current

* [[Dirac current]] -- see at [[geometry of physics -- A first idea of quantum field theory]] [this example](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#DiracCurrent)

### Of Green-Schwarz super $p$-brane sigma models

The WZW term of the [[Green-Schwarz super p-brane sigma models]] is invariant under [[supersymmetry]] only up to a [[divergence]], hence here the general [[Noether theorem]] for "weak" symmetries applies and yields a current algebra which is an [polyvector extension](super+Poincare+Lie+algebra#PolyvectorExtensions) of the [[supersymmetry]] algebra. See at _[Green-Schwarz action functional -- Conserved currents](Green-Schwarz+action+functional#ConservedCurrents)_ for more.

## Related concepts

* [[conservation law]]

* [[current algebra]]

* [[hadron current]]

* [[chiral anomaly]]

* [[classical observable]]

* [[Noether theorem]]

* [[charge]]

## References

### Dickey-Lie bracket on currents
 {#ReferencesDickeyBracket}

The [[Dickey Lie bracket]] on conserved currents is due to

* [[Leonid Dickey]], _Soliton equations and Hamiltonian systems_, Advanced Series in Mathematical Physics, Vol. 12 (World Scientific 1991).

and is reviewed in

* [[Glenn Barnich]], [[Marc Henneaux]],  section 3 of _Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket_, Journal of Mathematical Physics 37, 5273-5296 (1996) ([arXiv:hep-th/9601124](http://arxiv.org/abs/hep-th/9601124), [DOI 10.1063/1.531726](http://dx.doi.org/10.1063/1.531726))

The statement that the Dickey bracket Lie algebra of currents is a central [[Lie algebra extension]] of the algebra of symmetries by [[de Rham cohomology]] of the jet bundle appears as theorem 11.2 in (Part II of)

* {#Vinogradov84a} [[Alexandre Vinogradov]], _The $\mathcal{C}$-spectral sequence, Lagrangian formalism, and conservation laws. I. the linear theory_, Journal of Mathematical Analysis and Applications **100**, 1-40 (1984) ([doi](http://dx.doi.org/10.1016/0022-247x(84)90071-4))

* {#Vinogradov84b} [[Alexandre Vinogradov]], _The $\mathcal{C}$-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory_, Journal of Mathematical Analysis and Applications **100**, Issue 1, 30 April 1984, Pages 41-129 ([publisher](http://www.sciencedirect.com/science/article/pii/0022247X84900726))

and is stated as exercise 2.28 on p. 203 of

* [[Alexandre Vinogradov]], [[Joseph Krasil'shchik]] (eds.), _Symmetries and conservation laws for differential equations of mathematical physics_, vol. 182 of Translations of Mathematical Monographs, AMS (1999)

A lift of the Dickey Lie bracket on cohomologically trivial spaces to an equivalent _[[model structure for L-infinity algebras|L-infinity equivalent]]_ [[L-infinity algebra|L-infinity bracket]] is constructed, under some assumptions, in

* {#BarnichFulpLadaStasheff97} [[Glenn Barnich]], [[Ronald Fulp]], [[Tom Lada]], [[Jim Stasheff]], _The sh Lie structure of Poisson brackets in field theory_, Communications in Mathematical Physics 191, 585-601 (1998) ([arXiv:hep-th/9702176](http://arxiv.org/abs/hep-th/9702176))

* [[Martin Markl]], [[Steve Shnider]], _Differential Operator Endomorphisms of an Euler-Lagrange Complex_, Contemporary Mathematics, Volume 231, 1999 ([arXiv:9808105](http://arxiv.org/abs/math/9808105))

The cohomologically non-trivial lift is discussed in

* [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Lie n-algebras of BPS charges]]_, J. High Energ. Phys. (2017) 2017: 87 ([arXiv:1507.08692](http://arxiv.org/abs/1507.08692))

### In variational calculus

A general discussion as above is around definition 9 of

* [[Gregg Zuckerman|G. J. Zuckerman]], _Action Principles and Global Geometry_ , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259–284. ([[ZuckermanVariation.pdf:file]])

The relation of conserved currents to [[moment maps]] in [[symplectic geometry]] is highlighted for instance in

* Huijun Fan, _Lecture 8, Moment map and symplectic reduction_ ([pdf](http://www.math.pku.edu.cn/teachers/fanhj/courses/symp-8.pdf))
 {#Fan}

### Higher conserved currents

Higher conserved currents are discussed for instance in

* [[Glenn Barnich]], [[Friedemann Brandt]], _Covariant theory of asymptotic symmetries, conservation laws and central charges_,  	Nucl.Phys.B633:3-82, 2002 ([arXiv:hep-th/0111246](http://arxiv.org/abs/hep-th/0111246))

### In higher prequantum theory

Conserved currents for Lagrangians written as [[WZW terms]] are discussed in

* {#AzcarragaIzquierdo95} [[José de Azcárraga]], José M. Izquierdo, section 8.1 of _[[Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics]]_ , Cambridge monographs of mathematical physics, (1995)

Building on that, in the context of [[higher prequantum geometry]] conserved currents of the [[WZW model]] and in [[schreiber:∞-Wess-Zumino-Witten theory]] are briefly indicated on the last page of

* {#Schreiber13} [[Urs Schreiber]], _Higher geometric prequantum theory and The Brane Bouquet_, notes for a [[schreiber:The brane bouquet|talk]] at [Bayrischzell 2013](http://hep.itp.tuwien.ac.at/~miw/bzell2013/) ([pdf notes](http://ncatlab.org/schreiber/files/hpqWZWintro.pdf))

* {#classicalinhigher} [[Urs Schreiber]], _[[schreiber:Classical field theory via Cohesive homotopy types]]_

The same structure is considered in

* {#FiorenzaRogersSchreiber13} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher geometric prequantum theory]]_, 2013

as higher [[quantomorphisms]] and [[Poisson bracket Lie n-algebras]] of local currents.

category: physics
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